# Analysis on Image Set Visual Question Answering

Abhinav Khattar  Aviral Joshi  Har Simrat Singh  Pulkit Goel  Rohit Prakash
Barnwal

###### Abstract

We tackle the challenge of Visual Question Answering in multi-image setting
for the ISVQA dataset. Traditional VQA tasks have focused on a single-image
setting where the target answer is generated from a single image. Image set
VQA, however, comprises of a set of images and requires finding connection
between images, relate the objects across images based on these connections
and generate a unified answer. In this report, we work with 4 approaches in a
bid to improve the performance on the task. We analyse and compare our results
with three baseline models - LXMERT, HME-VideoQA and VisualBERT - and show
that our approaches can provide a slight improvement over the baselines. In
specific, we try to improve on the spatial awareness of the model and help the
model identify color using enhanced pre-training, reduce language dependence
using adversarial regularization, and improve counting using regression loss
and graph based deduplication. We further delve into an in-depth analysis on
the language bias in the ISVQA dataset and show how models trained on ISVQA
implicitly learn to associate language more strongly with the final answer.

Machine Learning, ICML

##  1 Introduction

Image Set Visual Question Answering (ISVQA) (Bansal et al., 2020) is the task
of answering a question given a set of Images. In this paper we explore the
task of ISVQA, identify relevant research challenges and propose well defined
strategies inspired by error analysis and thorough investigation of Visual
Question Answering (Antol et al., 2015) a closely related task. Tasks that
combine vision and natural language continue to inspire considerable research
at the boundary of computer vision and natural language processing. Since its
introduction, VQA has attracted significant attention from the research
community as answering natural language questions about images requires
understanding a wide range of detailed semantics of an image and how they are
referred to in natural language. Furthermore, solving VQA is of practical
importance given its utility in assisting visually impaired people. While the
problem of ISVQA is quite similar to that of VQA, the former can be
significantly more challenging than vanilla VQA since the answer is not
necessarily contained within a single image and thus reasoning over objects
and concepts in different images becomes paramount for achieving good
performance on ISVQA. For example, In Fig 1, a model has to find the
relationship between the bed in the left image and the mirror in the right,
via pillows which are common to both the images.

Figure 1: Image from the ISVQA dataset. The associated question with the image
is, ”What is hanging above the bed?” and with the corresponding answer is,
”mirror”.

ISVQA reflects information retrieval from multiple images of relevance but
with no obvious continuous correspondences. A model for solving this type of
problems has to understand the question, find the connections between
different images, use these connections to relate objects across images and
generate the answer. ISVQA leads to new research challenges, including: a) How
to use natural language to guide scene understanding across multiple
views/images. b) How to fuse information from relevant images to reason about
relationships among entities. These challenges associated with scene
understanding do not exist in single-image VQA settings but frequently happen
in the real world. While ISVQA task includes answering questions about images
taken at different times (e.g. like in camera trap photography, or images
under same story on Facebook/Instagram), images taken at different locations
(e.g. security cameras capturing different locations within a store, or
pictures of same house on real estate websites), or images taken from
different viewpoints (e.g. live sports coverage), we focus on images from
multiple views of an environment.

In this paper, we explore the specific challenges and issues faced while
solving the problem proposed in ISVQA. To the best of our knowledge, this is
the first work on solving the problem of Image Set Visual Question Answering,
apart from the initial ISVQA dataset paper. Initially, we study the LXMERT
baseline for ISVQA and discuss where the model lacks. Based on our findings,
we focus on the issues of counting, color awareness, spatial awareness, and
language dependence, proposing approaches to tackle each of these issues and
finally improving the performance over the LXMERT baseline. We also discuss
in-depth the problem of language bias in ISVQA, show how it significantly
affects a model being trained on the dataset, and demonstrate how our
adversarial regularization strategy stops the model from being overconfident
in its prediction in the absence of visual modality. We finally call for work
on a VQA-CP (Agrawal et al., 2018) like dataset for the problem of ISVQA that
helps evaluate how ISVQA models will perform in “real-life” setting.

In the following section 2, we present the works that closely relate to ISVQA.
We then analyze the ISVQA dataset and evaluation metrics in section 3. We also
formulate the problem statement and introduce the Multimodal baselines for
ISVQA dataset in this section. Section 4 discusses the 4 research ideas that
we propose in this paper; namely the adversarial regularization, enhanced VQA,
count-aware ISVQA and Regression loss based improvements. In section 5 we
present results obtained on the 4 proposed models and present an in-depth
anaysis of adversarial regularization. Finally, we conclude and discuss future
work in section 6.

##  2 Related Work

Being a relatively new task, the only available work on ISVQA is the dataset
paper. The paper describes 4 baseline architectures and concludes that their
Transformer-based architecture - which derives from LXMERT (Tan & Bansal,
2019) \- performs the best. The problem of ISVQA lies somewhere at the
intersection of VideoQA and VQA where instead of having a single image (as in
VQA) we have multiple images (which is the case with VideoQA), but the images
do not form a continuous scene (as is the case with VideoQA). Given the
similarity of our problem with Visual Question Answering (VQA) and Video
Question Answering (VideoQA), in this section, we discuss existing works in
both these fields and use existing methodologies from both of these as our
baselines.

###  2.1 VQA

VQA is a well-researched task with multiple datasets including VQA 1.0 (Antol
et al., 2015), VQA 2.0 (Goyal et al., 2017), DAQUAR (Malinowski & Fritz,
2014), and COCO-QA (Ren et al., 2015). VQA 2.0 is the largest, and most
commonly used dataset for VQA with over 1.1M questions and over 11M ground-
truth answers.

While most current research in VQA involves using some sort of masked pre-
training objective, for the sake of completeness, we discuss the 2 major types
of models that exist to tackle VQA:

####  2.1.1 RNN based models without masked pre-training

Anderson et al. (2018) introduced an approach which combines top-down(soft)
and bottom-up (hard) attention mechanisms to achieve state-of-the-art
performance on image captioning and obtain first place in the VQA 2017
challenge. The proposed approach utilized the Faster-RCNN model to identify
objects of interest in images and a ResNet model to extract vector
representations of those objects. The answer is generated using a 2 layer LSTM
network where the first layer is referred to as top-down attention layer and
the second deals with generating the final output for both image captioning
and VQA. The top-down LSTM attention layer involved utilizes a concatenation
of word embedding, image embedding and previous time-steps hidden-state of the
2nd layer as input. The hidden-state generated by the top-down attention layer
is then fed to the Language LSTM layer which attends to the embeddings of the
extracted regions generated by the bottom-up attention model. Finally the
output of the Language LSTM layer is used to compute a probability
distribution over the vocabulary to generate the caption in the case of Image
Captioning and the answer in the case of Visual Question Answering.

####  2.1.2 Transformer based models with masked pre-training

Recent works in the field of NLP have brought to light the positive effects of
Language Model pre-training on various downstream tasks (Devlin et al., 2018;
Peters et al., 2018; Lample & Conneau, 2019). Taking inspiration from such
architectures, recent VQA models adopt a BERT (Devlin et al., 2018) style of
pre-training for Transformer models (Vaswani et al., 2017). Some of these VQA
models include: Visual BERT (Li et al., 2019), VL-BERT (Su et al., 2019),
UNITER (Chen et al., 2019), and OSCAR (Li et al., 2020). Using a Transformer
with a pre-training objective is a common theme that runs across all of these
models.

VisualBERT, which is probably the most simplistic architecture out of the
three techniques mentioned, was one of the first VQA architectures to come out
with a BERT-like architecture. In an attempt to learn implicit alignments
between text and images, the model is trained on 2 pre-training objectives -
(1) Masked Language Objective, (2) Sentence Image prediction. While the first
objective is completely identical to BERT, the model, in this case, takes as
input both the text and the image regions. The second objective, on the other
hand, is unique and attempts to answer if the text provided with the image is
the image’s true caption. The authors of VisualBERT first pre-train the model
on MS-COCO, followed by pre-training on the VQA dataset, finally followed by
task-specific fine-tuning.

Works using Transformers that came after VisualBERT focus on how to learn
better alignments and relationships between text and images. VL-BERT throws
away the Sentence Image prediction brought in by VisualBERT and add a Masked
Region of Interest Classification task which masks out Region of Interests
(RoI) and then asks the model to predict the entity that may have been masked
out. This objective helps learn stronger alignments and as a result, helps VL-
BERT outperform VisualBERT. To further improve over VL-BERT, UNITER
incorporates Image-Text Matching, Word-Region Alignment, Masked Region Feature
Regression, Masked Region Classification, and Masked Region Classification
with KL-Divergence objectives. OSCAR - which is the current SOTA on VQA 2.0 -
outperforms UNITER by further helping image-text alignment using object-tags
detected from images while just using Masked Token and Contrastive Loss.
Another recent model, LXMERT (Tan & Bansal, 2019), diverges a bit from the
standard Transformer-Encoder architecture and uses an attention based
architecture with different initial model segments for both the modalities.
The architecture is also pre-trained on various pre-training objectives
including Masked LM, Masked Object Precition, Cross-Modality Matching, and
Image Question Answering. We use this LXMERT architecture as on of the
baselines for ISVQA task as it is easy to directly use this model for our
task.

Owing to the strong positive effects of transfer learning and pre-training,
recent Transformer-based pretrained models outperform the non pretrained RNN-
based ones. The Transformer models, though, use the same visual features as
the ones used by RNN models which are most commonly extracted using an R-CNN
based feature extractor. As a whole, the modern Transformer-based
architectures provide a more expressive and elegant solution to the problem of
VQA learning from a glut of other information sources using an array of
objectives that help the model learn better and more generalizable
representations.

###  2.2 VideoQA

Compared to image-based VQA, there has been less work done on video-based VQA.
In past few years, several video QA datasets have been proposed, e.g. MovieFIB
(Chen et al., 2017), MovieQA (Tapaswi et al., 2016), TGIF-QA (Jang et al.,
2017), MarioQA (Mun et al., 2017) and TVQA (Lei et al., 2018). TVQA is one of
the largest video QA datasets, providing a large video QA dataset built on top
of 6 famous TV series containing 152.5K multiple choice questions from 21.8K,
60-90 second long video clips. The image sets in ISVQA dataset are not akin to
video frames which are temporally continuous. Also, unlike embodied QA (Das et
al., 2018), ISVQA does not have an agent interacting with the environment.

Fan et al. (2019) propose a novel Multimodal Fusion based architecture to
solve the task of Video QA. The architecture takes as input 2 sets of visual
features as appearance features extracted frame by frame using a standard CNN
based architecture such as ResNet(He et al., 2015) or VGG(Simonyan &
Zisserman, 2015) and motion features extracted from clips of the video using
the C3D(Tran et al., 2015) model. GloVe 300-D(Pennington et al., 2014)
embeddings based features extracted from an LSTM network form the inputs of
the textual modality. The architecture incorporates an external heterogeneous
memory to integrate the two visual features and learn the joint attention. A
similar external memory is also implemented for the question features. This
helps to understand the global as well as the local context and also helps in
multimodal fusion of the text and video modalities. The multimodal fusion
consists of a core LSTM controller which takes the hidden memory of the
combined video and question based features as the input. During each
iteration, the controller attends to different parts of the video features and
question features with temporal attention mechanism, and combines the attended
features.

Our work focuses on the problem of ISVQA, and to the best of our knowledge,
ours is the first work exploring the techniques to solve this problem, apart
from the initial dataset paper. As a result, none of our related works try to
tackle the problem of ISVQA - adding to the novelty of our work. In addition,
some of the techniques that we propose including enhanced pre-training and
adversarial regularization are novel and have not been previously explored in
other related contexts.

##  3 Experimental Setup

###  3.1 Dataset

We propose to use the dataset provided by ISVQA. The dataset comprises of 2
modalities - image and language. The authors of ISVQA use existing image set
datasets of Gibson (Xia et al., 2018) and nuScenes (Caesar et al., 2020) as
sources to build an ISVQA dataset. Gibson dataset provides 3D indoor scans of
buildings, rooms and offices. The scans are of 2 types 1) Gibson Building and
2) Gibson Room. nuScenes contain outdoor scenes generated by a $360^{\circ}$
camera mounted on a car in city streets. The details for the dataset are
presented in Table 1.

Dataset | Train Set | Test Set | Unique Answers
---|---|---|---
Indoor - Gibson | 69,207 | 22,272 | 961
Outdoor - nuScenes | 33,973 | 15,644 | 650
Table 1: Dataset specifications

Currently, we are only working with the NuScenes data as the Gibson data has
not entirely been made available by the ISVQA dataset authors.

###  3.2 Dataset Analysis

Most of the questions phrases contain between 5 and 10 words. Most frequent
questions are about physical properties of objects, and spatial relationships
between different entities. The questions can be broadly said to be about the
color of objects in the images, relative spatial properties and the count of
the objects.

The dataset tries to ensure that there are a significant number of data points
which require more than one image to answer a question. This forces the model
to learn the entire scene rather than look at just one region of interest.
About 7,000 such samples exist in Gibson-Room and about 3,000 in Gibson-
Building. In nuScenes only about 1,000 such examples exist but the number of
data points which require more than two images is greater than the indoor
ones.

###  3.3 Evaluation Metric

Given that the questions are multiple-choice, we propose to use VQA-accuracy
as the evaluation metric. In the ISVQA setting, each image set has been
annotated by 3 annotators. VQA-accuracy for an ISVQA sample is thus, 1 when it
is supported by 2 or more annotators, 0.5 when supported by 1 annotator, and 0
otherwise. Human performance using standard accuracy measure was calculated in
ISVQA. VQA-accuracy of 91.88% for nuScenes and 88.80% for Gibson was obtained.
This makes the task quite challenging and also provides an interesting
observation of why the human performance was not close to 100%. In many cases,
humans gave a similar answer to the question which although conveyed the same
meaning, was not semantically similar to the provided ”true labels”. For
example, the answer could be “black and white” but “white and black”, though
same would not be considered semantically similar.

###  3.4 Problem Statement

The problem of ISVQA can thus be formulated as the following. Given an image
set $S=[I_{1},I_{2},\cdots,I_{N}]$, where $N$ is the total number of images
associated with a data point and a question $Q=[q_{1},q_{2},\cdots,q_{T}]$,
where $q_{i}$ denotes the $i^{th}$ word in the question of total $T$ words,
the model should provide an answer $a$ such that $a=f(S,Q)$. The task is a
discriminative question answering task which implies that the model picks the
best possible answer from a set of all possible answers. Thus, the function
$f$ ideally generates a set of scores or probabilities over all the possible
answers in the answer set $A=[a_{1},a_{2},\cdots,a_{M}]$. For nuScenes
dataset, there are 6 images associated with each data point i.e. $N=6$ and the
total possible answers $M=650$.

###  3.5 Multimodal Basline

Given the lack of work on ISVQA specific baseline models, we implemented and
adapted one Video-QA and two VQA models to work in an ISVQA setting. In
specific, we tried: HME-Video VQA, VisualBERT, and LXMERT models as our
baselines. The HME-Video VQA and LXMERT baselines have been proposed by the
authors of the ISVQA dataset, and our re-implementation of these models on
ISVQA yields results in the same domain as has been observed by authors of the
dataset.

To adapt HME-VideoQA model to ISVQA, we used the images in the set as the
frames of the video so that the appearance features could be calculated
directly. We used VGG19 to extract the features. For motion features, we
assumed the images in the set to be a part of a 3D video. We used C3D model to
extract the features. The visual features were of size 4096. Adam was used as
the optimizer with learning rate of $1e-3$. A memory of size $256$ was used
for the Heterogeneous memory model. LSTM of hidden memory size $512$ was used
in the multimodal fusion module.

For the VQA models, we stitched the images in the image-set together to get
object features and positions. For LXMERT, we used the same amount of layers,
embedding dimensions as have been used in LXMERT and a learning rate of $2e-5$
and trained our model for 4 epochs. Adam optimizer with linear-decayed
learning-rate schedule was used for this task. For VisualBERT, all the
hyperparameters utilized for training the model were kept the same as in the
default implementation provided by the authors.

For all the baseline models we use the Categorical Cross-Entropy loss for
training.

##  4 Proposed Approach

An analysis of the relationship between the model performance and the nature
of questions shows that certain question categories such as count-based and
colour-based questions perform worse as compared to other question categories
such as object-based and true/false questions. This is also shown in Fig. 2
where we analyze the 15 most frequent types of questions in our dataset using
LXMERT. This can be attributed to the inherent difficult nature of these
tasks, often involving a multi-step process; including object detection,
recognition and counting; to perform optimally while also limiting the impact
of common issues such as double-counting.

Figure 2: Question-based error analysis for LXMERT baseline

We propose models using pre-training (Section 4.2) and count-aware feature
generation (Section 4.3 and 4.4) for mitigating these issues. Further, the
questions in our dataset have been susceptible to dataset biases. We propose
using Adversarial regularization to remove language dependence in LXMERT in
Section 4.1.

###  4.1 Removing Language Dependence using Adversarial Regularization

Adversarial regularization is the process of marginalizing biases which may be
detrimental to performance of a model when evaluated on out of distribution
data. The procedure of performing regularization adversarially involves
introducing a competing framework which penalize overconfident predictions of
the model when data with insufficient information is provided to the model.

The idea to incorporate adversarial regularization stems from the observations
made on the Question only Analysis. A model which captures strong language
biases limits its ability to generalize to data which is not observed during
training and hence introducing a regularizing component can prove beneficial
to improve the performance of the ISVQA baselines discussed in section 3.5. A
similar observation to the one made in the Question only BERT baseline is
observed in the LXMERT model. Here the LXMERT model which achieves the best
result for ISVQA is able to accurately answer questions even when the
information from the visual modality is completely removed by zeroing out
object features extracted from the RCNN module. The Table 3 shows the result
from our experiment on adversarial regularization.

The procedure for adversarial regularization involves eliminating features of
objects in the image that contain the answer to the question. This is done by
parsing the question to identify objects (ex. car, person, truck, etc.) and
removing their image feature representation information extracted by the
Faster-RCNN module. The training example generated by the above process lacks
important information needed to answer the question and hence the model must
not be able to predict the correct answer for such an example. Correct answers
to such questions might indicate the presence of unwanted biases learnt by the
model during training and to eliminate them we reverse the direction of
gradient on adversarial examples and add an additional term to the original
loss formulation which maximizes the loss on adversarial examples, thus
eliminating biases in the process. The equation 4.1 shows the formulation of
the loss function with an additional loss term for adversarial examples.

$L_{(}q,I,I^{\prime};\theta)=L_{(}q,I;\theta)-\lambda_{R}*L_{(}q,I^{\prime};\theta)$

Here, $L$ corresponds to the Loss of the ISVQA model. $\lambda_{R}$ is the
regularization constant and controls the strength of the regularization. $q$
represents the question, $I$ represents the true image features extracted from
Faster-RCNN, $I^{\prime}$ represents the augmented Image features and $\theta$
represents the model parameters. For all our experiments on adversarial
regularization we fix $\lambda_{R}$ to be $0.1$.

We also experiment with 2 different loss functions for adversarial
regularization and analyze their effectiveness in reducing bias. The Cross-
Entropy (CE) loss obtained on an adversarial example given the ground truth
label gives us a measure of the extent to which the model is overconfident in
its prediction hence we choose to maximize this loss. Another loss function
that we try out is the per class Binary Cross-Entropy (BCE) loss which is used
to determine the cumulative loss per class given an adversarial example and we
choose to minimize this loss as its magnitude indicates how certain the model
is of the predicted answer. Hence a higher value indicates more certainty
which corresponds to overconfident predictions.

###  4.2 Enhancing Pre-Training with Color and Position Questions

Fig. 2 shows that color based questions perform sub-optimally for ISVQA. We
try to tackle this, along with the general problem of spatial-awareness (which
is pivotal to ISVQA) using enhanced pre-training. Recent works in the field of
NLP have focussed on different positional embedding strategies to improve
performance with Transformer-based models. While pre-training using novel
positional embedding strategies would have been an interesting path to pursue,
the computational complexity of the problem would have been difficult to
manage within the resources available (LXMERT pre-training took 10 days on 4
Titan Xp). In order to achieve the benefits of having a better spatial and
color-awareness, we take advantage of the pre-training strategy used by
LXMERT. In particular, we take advantage of the fact that LXMERT’s last 10
epochs of pre-training only take place on the Image Question-Answering task.
In order to provide better spatial and color information, we continue pre-
training the previously pre-trained LXMERT on artificially created color and
position based VQA questions.

In order to create artificial color VQA questions, we used the attributes
predicted by R-CNN. For all the color-based attributes, we created a template
question What is the color of X where X is the object. For generating position
questions, we looked into the left, right, top, and bottom-most objects, found
the nearest objects to these objects based on Euclidean distance, and
generated artificial question of the form what is below X. We only consider
the left-most object as the answer for a question of the format what is to the
left of X to ensure that there is no ambiguity. Further, we also ensure that
the questions are generated for unique objects - i.e. objects that appear in
the questions only appear once in the set of the images. Fig. 3 gives an
example of questions generated for an image set using this technique.

For this apporach, we use the same hyperparameter, loss, and optimizer as with
the LXMERT baseline.

Figure 3: The given image generates two questions: What is the color of the
sign? (Green) and What is below the wall? (Sidewalk)

###  4.3 Count Aware ISVQA using graph based deduplication

We observe that all the baseline models struggle to perform optimally on the
count-based questions. Traditionally, these questions have been susceptible to
dataset biases (Jabri et al., 2016) and often require a multi-step process
involving object detection, recognition and counting to perform optimally,
while also maintaining the generalizability of the model. We propose an
improvement on the baseline model by adapting and extending the work of (Zhang
et al., 2018) specifically to handle count based questions. In their work
(Zhang et al., 2018) propose that simply using soft attention for the VQA task
does not help with the objective of counting. The major reason for this is
attributed to the fact that the soft attention mechanism gives an equal weight
to an object category. Thus if $n$ objects of the same class are present in an
image the attention weights for each are $1/n$ times reduced (with all of them
still summing to 1).

The authors present a differentiable mechanism for counting from attention
weights, while also dealing with the problem of overlapping object regions of
interest to reduce double-counting of objects. The central idea is to create a
graph and subsequently an adjacency matrix from these object proposals. Edges
of the graph can then be scaled and pruned in a specific way such that a count
of the number of underlying objects is found. The model takes in input as the
region of interest features and the corresponding bounding boxes extracted
from a Faster R-CNN network and the question encoded using an LSTM network.
Adjacency graph $A$ is obtained from the attention matrix by computing the
outer product - $A=aa^{T}$. In this graph, the $i^{th}$ vertex represents the
object region associated with $a_{i}$ and the edge between any pair of
vertices $(i,j)$ has weight $a_{i}a_{j}$. In the regions associated with
bounding boxes, the matrix $A$ forms the basis for the graph between objects.
Further pruning is done in two steps using an inter edge pruning and an intra
edge pruning technique.

Bounding boxes are compared using, intersection-over-union (IoU) metric. Hence
the distance matrix $D$ is defined as $D_{ij}=1-IoU(b_{i},b_{j})$. Intra
object edges are removed by elementwise multiplying the distance matrix with
the attention matrix $A^{\prime}=f_{1}(A)f_{2}(D)$ where $f_{1},f_{2}$
represent piecewise linear functions. For removing inter-object edges a
similarity metric is computed over the rows of $A^{\prime}$. The whole process
results in the generation of the score matrix $C$ which is the count based
score for each of the objects.

We adapt this approach to the task of ISVQA as an intermediate step to improve
the features extracted by the RCNN and score each ROI according to the
computed matrix $C$. Each of the image features is multiplied by its
respective score and then fused with the question representation using soft
attention. This output is considered as the updated ‘count-aware’ feature. The
count aware features are used as an input to the LXMERT model. The count-aware
embedding model uses the RCNN extracted features of dimension 2048 with 36
regions of interest in each image along with the corresponding bounding boxes.
The questions are encoded with LSTM layers of hidden dimension 1024. The model
is trained for 100 epochs using Adam optimizer using a learning rate of
$1.5e-3$ with an Exponential Learning Rate scheduler. The model is trained on
negative log likelihood loss. The LXMERT model is trained from scratch with
AdamW optimizer using an initial learning rate of $2e-5$. We compare the
prediction results of this model with an LXMERT model trained from scratch
using base RCNN features and observe a visible improvement.

###  4.4 Improving Performance on Count-based Questions using Regression Loss

To address the poor performance on count-based questions we also experiment
with another approach which approaches such questions from the prospective of
regression. Contrary to the approach in section 4.3 which involves processing
inputs to the LXMERT model this approach introduces a new Mean Squared Error
loss along with the classification loss for count based questions. The process
for training this model involves utilizing both classification and regression
heads to predict the answer and at inference time the answer to the question
is determined by using only the classification head.

Model | Model Type | Train Accuracy (%) | Test Accuracy (%)
---|---|---|---
LXMERT | Baseline | 89.95 | 64.55
LXMERT without pretraining | Baseline | 79.37 | 57.05
Visual BERT | Baseline | 74.3 | 55.55
HME - Video QA | Baseline | 97.1 | 49.9
LXMERT with enhanced pre-training | Ours | 88.74 | 64.93
LXMERT with Count-based VQA | Ours | 79.56 | 57.50
LXMERT with Regression Loss | Ours | 92.6 | 64.38
Adversarial Regularization | Ours | 82.62 | 63.24
Table 2: Baseline model results

##  5 Results and Discussion

Table 2 tabulates the results obtained for the baseline and the advanced
models proposed by us on the ISVQA dataset. We further assess the performance
of each research enhancement in the following sections.

###  5.1 Removing Language Dependence using Adversarial Regularization (In-
Depth Analysis)

To better understand the bias present in the text dataset and its implications
on the model performance, we analyze the distribution of answer categories
across each questions and find that colour-based and count-based questions are
often dominated by a smaller number of answer categories, as shown in Figure
4. We can see that for count-based questions like how many
people/cars/pedestrian etc, same 2 answer categories occurred about 75% times
(denoted by blue and green stacks), while all the remaining classes accounted
for about 25% of the occurrences, thus denoting a high answer bias for count-
based questions. Meanwhile, object-based questions like what does the/what are
the show a high diversity of answer classes, and thus low text-induced bias.
This indicates the presence of bias in the ISVQA dataset and can lead the
models learned on this data to not generalize well and under-utilize visual
information when answering questions as the information needed to answer the
questions can be learnt from just the language modality features. This is
further highlighted from Table 4 which shows the results for a BERT-based
question-only model. We observe that this unimodal model is able to achieve
over 52% accuracy.

Figure 4: Frequency of answer categories for 15 most frequent questions. Each
column bar corresponds to a question-type and each colored-stack in the bar
refers to an answer category

High occurrence of certain answer categories can result in model being biased
towards those answers. This is evident in Figure 5 where we show the accuracy
of count-based answer categories for the baseline LXMERT model. We can see
that more frequent answer categories such as Three/Two had accuracy as high as
55%, while rare categories such as nine/fifteen were never predicted
correctly.

Figure 5: Accuracy of count-based answer categories

(a) Train set

(b) Test set

Figure 6: Most Frequent Answers | Base LXMERT | w/ CE Reg. | w/ BCE Reg.
---|---|---|---
L + V | 64.55% | 63.24% | 63.05%
L | 42.87% | 5.66% | 12.8%
Table 3: Results for Adversarial Regularization. Here L corresponds to
Language and V corresponds to visual modality

To mitigate the presence of such biases we introduce a new adversarial
regularization formulation. Here we experiment with training the LXMERT model
using 2 different regularization loss terms which penalizes overconfident
predictions on adversarial examples and to compare the effect of adversarial
regularization we utilize the baseline LXMERT model which was trained on both
Language and Visual Modalities. Our results presented in Table 3 indicate that
when visual information is scrubbed (zeroed out) during inference time (but is
trained with both Visual and Language modality present) the LXMERT model
trained without adversarial regularization is able to obtain considerably high
accuracy on the test dataset showing its over reliance on language modality
and the presence of bias in the test dataset. However, when the model is
trained with adversarial regularization, performance on test set with srubbed
out visual information is reduced drastically 37% while little impact is seen
when both the visual and text data are provided which indicates that
regularized model does not over rely on language biases. We also observe that
using the label-wise Binary Cross-Entropy loss is less effective in removing
bias when compared to using the Cross-Entropy adversarial loss formulation.

Train Accuracy (%) |  Test Accuracy (%)
---|---
72.55 |  52.69
Table 4: Question-only BERT result

###  5.2 Enhancing Pre-Training with Color and Position Questions

Table 2 shows that our LXMERT model with enhanced pre-training performs
slightly better than the baseline LXMERT model. This shows the efficacy of
pre-training on our artificially created data. Further, from Table 5, we see
that our performance on the color-based questions is significantly better than
the baseline LXMERT model. We do not report results on position-specific
questions as the aggregator for such questions is not provided in the dataset.
These results re-iterate the need to provide better color representations to
our model as they can significantly enhance the performance. More importantly,
these results provide motivation on developing and spending computational
resources on training better pre-training objectives that incorporate position
and color as they can help improve accuracy on ISVQA even further.

While using such artificially generated questions leads a decent performance
improvement for Color based questions, there is a clear disadvantage of using
this approach as the errors in object or attribute prediction from R-CNN get
cascaded to the questions generated, making our model susceptible to these
errors. An of example of this is the question what is the color of the train
with answer orange for image in Fig. 3. In this case, our R-CNN mis-detects
the bus as a train and leads to the creation of an incorrect question.

Model | Color-Only Accuracy
---|---
LXMERT | 67.72
LXMERT w/ enhanced PT | 70.09
Table 5: Model performance on Color-based Questions only

###  5.3 Count Aware ISVQA using graph based deduplication

The results for the Count based ISVQA is presented in table 2. The results are
obtained on an LXMERT model trained from scratch rather than adopting a
pretrained one. Thus the comparison will be done on an LXMERT model without
pretraining using base RCNN features. This is done because the input features
for the Count based model are changed and thus we cannot adapt a pretrained
model as is (and pre-training our model is computationally unfeasible). A
slight improvement over the baseline is observed on the overall results
whereas Table 6 compares the model performance specifically on count based
questions. This shows a clear improvement over the baseline confirming that
’count-aware’ features perform better. The claims made by (Zhang et al., 2018)
that the soft attention based models generally do not perform well on count
based question is proven. This also shows that a graph based modelling of the
regions of interest in an image is a good approach to resolve disambiguities
and solve the double counting problem especially in datasets such ISVQA where
such anomalies are present in abundance.

Model | Count-Only Accuracy
---|---
LXMERT w/o pre-training | 42.7%
LXMERT w/ Count Based VQA | 46.5%
Table 6: Model performance on Count-based Questions only

###  5.4  Improving Performance on Count-based Questions using Regression Loss

The results for this experiment in Table 2 and 7 show similar results to the
LXMERT baseline and a marginal 0.2% improvement in Count-only question
accuracy was observed. The results suggest that introducing an additional
regression loss is enough to improve accuracy on count-based questions. And
additional modifications such as changes in the features extraction process,
better attention mechanisms to improve alignment and overall enhancements to
the model architecture are required to improve accuracy on count based
questions.

Model | Count-Only Accuracy
---|---
LXMERT | 49.84%
LXMERT w/ Regression Loss | 50.02%
Table 7: Model performance on Count-based Questions only

##  6 Conclusion and Future Work

In this work we discuss 4 research ideas to improve performance on Image Set
Visual Questions Answering and present an in depth analysis for the problem of
Language Dependence along with testing 2 different adversarial regularization
formalizations. We utilize three established baselines for the task and
perform error analysis to provide a strong motivation for our research ideas.
Our presented approaches include an enhanced pertaining strategy for the task
of ISVQA that incorporates color and spatiality. We then describe ideas to
incorporate count awareness in deep learning models for ISVQA. Finally, we
discuss an adversarial regularization technique for bias removal technique as
we observe the presence of significant bias in both training and testing data.
We observe that our proposed approaches are able to improve performance over
the LXMERT baseline, with a 0.38% absolute performance increase using our
enhanced pre-training strategy. Our count-based models were able to improve
performance on count-based questions by 3.8% while enhanced pre-training
helped improved the performance on color-based questions by 2.3%.
Additionally, our adversarial regularization technique is able to reduce
dependence on the language modality.

As a future work, we suggest the development of a ISVQA dataset with changing
priors similar to VQA-CP to study the generalizability of ISVQA models.
Furthermore, work needs to be done to improve the performance on count based
questions as our experiments reveal that count based questions are
particularly harder for Visual Questions Answering models to handle. Finally,
our results from the enhanced pre-training model show that incorporating more
pre-training objectives that cater to spatial and color-awareness can help
improve the performance even further and running such experiments can be worth
the computational efforts.

## References

  * Agrawal et al. (2018) Agrawal, A., Batra, D., Parikh, D., and Kembhavi, A.  Don’t just assume; look and answer: Overcoming priors for visual question answering.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp. 4971–4980, 2018.
  * Anderson et al. (2018) Anderson, P., He, X., Buehler, C., Teney, D., Johnson, M., Gould, S., and Zhang, L.  Bottom-up and top-down attention for image captioning and visual question answering.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pp. 6077–6086, 2018.
  * Antol et al. (2015) Antol, S., Agrawal, A., Lu, J., Mitchell, M., Batra, D., Lawrence Zitnick, C., and Parikh, D.  Vqa: Visual question answering.  In _Proceedings of the IEEE international conference on computer vision_ , pp. 2425–2433, 2015.
  * Bansal et al. (2020) Bansal, A., Zhang, Y., and Chellappa, R.  Visual question answering on image sets.  _arXiv preprint arXiv:2008.11976_ , 2020.
  * Caesar et al. (2020) Caesar, H., Bankiti, V., Lang, A. H., Vora, S., Liong, V. E., Xu, Q., Krishnan, A., Pan, Y., Baldan, G., and Beijbom, O.  nuscenes: A multimodal dataset for autonomous driving.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp. 11621–11631, 2020.
  * Chen et al. (2017) Chen, J., Shao, J., Shen, F., He, C., Gao, L., and Shen, H. T.  Movie fill in the blank with adaptive temporal attention and description update.  In _Proceedings of the 2017 ACM on Conference on Information and Knowledge Management_ , pp. 1039–1048, 2017.
  * Chen et al. (2019) Chen, Y.-C., Li, L., Yu, L., Kholy, A. E., Ahmed, F., Gan, Z., Cheng, Y., and Liu, J.  Uniter: Learning universal image-text representations.  _arXiv preprint arXiv:1909.11740_ , 2019.
  * Das et al. (2018) Das, A., Datta, S., Gkioxari, G., Lee, S., Parikh, D., and Batra, D.  Embodied question answering.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops_ , pp. 2054–2063, 2018.
  * Devlin et al. (2018) Devlin, J., Chang, M.-W., Lee, K., and Toutanova, K.  Bert: Pre-training of deep bidirectional transformers for language understanding.  _arXiv preprint arXiv:1810.04805_ , 2018.
  * Fan et al. (2019) Fan, C., Zhang, X., Zhang, S., Wang, W., Zhang, C., and Huang, H.  Heterogeneous memory enhanced multimodal attention model for video question answering.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp. 1999–2007, 2019.
  * Goyal et al. (2017) Goyal, Y., Khot, T., Summers-Stay, D., Batra, D., and Parikh, D.  Making the v in vqa matter: Elevating the role of image understanding in visual question answering.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp. 6904–6913, 2017.
  * He et al. (2015) He, K., Zhang, X., Ren, S., and Sun, J.  Deep residual learning for image recognition, 2015.
  * Jabri et al. (2016) Jabri, A., Joulin, A., and Van Der Maaten, L.  Revisiting visual question answering baselines.  In _European conference on computer vision_ , pp. 727–739. Springer, 2016.
  * Jang et al. (2017) Jang, Y., Song, Y., Yu, Y., Kim, Y., and Kim, G.  Tgif-qa: Toward spatio-temporal reasoning in visual question answering.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp. 2758–2766, 2017.
  * Lample & Conneau (2019) Lample, G. and Conneau, A.  Cross-lingual language model pretraining.  _arXiv preprint arXiv:1901.07291_ , 2019.
  * Lei et al. (2018) Lei, J., Yu, L., Bansal, M., and Berg, T. L.  Tvqa: Localized, compositional video question answering.  _arXiv preprint arXiv:1809.01696_ , 2018.
  * Li et al. (2019) Li, L. H., Yatskar, M., Yin, D., Hsieh, C.-J., and Chang, K.-W.  Visualbert: A simple and performant baseline for vision and language.  _arXiv preprint arXiv:1908.03557_ , 2019.
  * Li et al. (2020) Li, X., Yin, X., Li, C., Zhang, P., Hu, X., Zhang, L., Wang, L., Hu, H., Dong, L., Wei, F., et al.  Oscar: Object-semantics aligned pre-training for vision-language tasks.  In _European Conference on Computer Vision_ , pp. 121–137. Springer, 2020.
  * Malinowski & Fritz (2014) Malinowski, M. and Fritz, M.  A multi-world approach to question answering about real-world scenes based on uncertain input.  In _Advances in neural information processing systems_ , pp. 1682–1690, 2014.
  * Mun et al. (2017) Mun, J., Hongsuck Seo, P., Jung, I., and Han, B.  Marioqa: Answering questions by watching gameplay videos.  In _Proceedings of the IEEE International Conference on Computer Vision_ , pp. 2867–2875, 2017.
  * Pennington et al. (2014) Pennington, J., Socher, R., and Manning, C. D.  Glove: Global vectors for word representation.  In _Empirical Methods in Natural Language Processing (EMNLP)_ , pp. 1532–1543, 2014.  URL http://www.aclweb.org/anthology/D14-1162.
  * Peters et al. (2018) Peters, M. E., Neumann, M., Iyyer, M., Gardner, M., Clark, C., Lee, K., and Zettlemoyer, L.  Deep contextualized word representations.  _arXiv preprint arXiv:1802.05365_ , 2018.
  * Ren et al. (2015) Ren, M., Kiros, R., and Zemel, R.  Exploring models and data for image question answering.  In _Advances in neural information processing systems_ , pp. 2953–2961, 2015.
  * Simonyan & Zisserman (2015) Simonyan, K. and Zisserman, A.  Very deep convolutional networks for large-scale image recognition, 2015\.
  * Su et al. (2019) Su, W., Zhu, X., Cao, Y., Li, B., Lu, L., Wei, F., and Dai, J.  Vl-bert: Pre-training of generic visual-linguistic representations.  _arXiv preprint arXiv:1908.08530_ , 2019.
  * Tan & Bansal (2019) Tan, H. and Bansal, M.  Lxmert: Learning cross-modality encoder representations from transformers.  _arXiv preprint arXiv:1908.07490_ , 2019.
  * Tapaswi et al. (2016) Tapaswi, M., Zhu, Y., Stiefelhagen, R., Torralba, A., Urtasun, R., and Fidler, S.  Movieqa: Understanding stories in movies through question-answering.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pp. 4631–4640, 2016.
  * Tran et al. (2015) Tran, D., Bourdev, L., Fergus, R., Torresani, L., and Paluri, M.  Learning spatiotemporal features with 3d convolutional networks, 2015\.
  * Vaswani et al. (2017) Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, Ł., and Polosukhin, I.  Attention is all you need.  In _Advances in neural information processing systems_ , pp. 5998–6008, 2017.
  * Xia et al. (2018) Xia, F., Zamir, A. R., He, Z., Sax, A., Malik, J., and Savarese, S.  Gibson env: Real-world perception for embodied agents.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pp. 9068–9079, 2018.
  * Zhang et al. (2018) Zhang, Y., Hare, J., and Prügel-Bennett, A.  Learning to count objects in natural images for visual question answering.  _arXiv preprint arXiv:1802.05766_ , 2018.


††∗ corresponding author

# Almost diagonalization of $\Psi$DO’s over various generalized function
spaces

Stevan Pilipović∗ Stevan Pilipović, Serbian Academy of Sciences and Arts,
Serbia<EMAIL_ADDRESS>,  Nenad Teofanov  Nenad Teofanov,
Department of Mathematics and Informatics, Faculty of Sciences University of
Novi Sad, Serbia<EMAIL_ADDRESS>and  Filip Tomić  Filip Tomić,
Faculty of Technical Sciences, University of Novi Sad, Serbia
<EMAIL_ADDRESS>
###### Abstract.

Inductive and projective type sequence spaces of sub- and super-exponential
growth, and the corresponding inductive and projective limits of modulation
spaces are considered as a framework for almost diagonalization of pseudo-
differential operators. Moreover, recent results of the first author and B.
Prangoski related to the almost diagonalization of pseudo-differential
operators in the context of Hörmander metrics are reviewed.

###### Key words and phrases:

modulation spaces, Gelfand-Shilov spaces, pseudo-differentail operators,
Hörmander metric

###### 2010 Mathematics Subject Classification:

47G30, 46F05, 42C15, 58J40

##  1\. Introduction

The main goal of this paper is to offer a brief review of some recent results
on the almost diagonalization of pseudo-differential operators with symbols in
various projective and inductive limits of modulation spaces, and spaces of
generalized functions. The results were presented at the conference in
Sarajevo dedicated to the 75. anniversary of academician Mirjana Vuković.

Properties of pseudo-differential operators depend on the assigned classes of
symbols. Here we consider the Weyl correspondence between operators and
symbols, see (3). Apart from the classical Hörmander classes ([19]), certain
modulation spaces are recognized to be useful symbol classes, see [13] where
the tools of time-frequency analysis are used in approximate diagonalization
of related operators. This approach is thereafter developed and successfully
used in different contexts, see [5] and the references given there. Let us
just mention sparse decompositions for Schrödinger-type propagators given in
[4], and diagonalization in the framework of tempered ultra-distributions,
[24].

General results from [14] are recently extended to Hörmander metrics by the
first author and B. Prangoski in [23]. It turns out that the class of weights
used in [23] could be extended to the class of moderate weights (see
subsection 2.2). The main aim of this paper is to provide necessary background
material for investigations in that direction. This includes the introduction
of new symbol classes as well as exposition of results for approximate
diagonalization in the context of Gelfand-Shilov spaces.

We end this section with an explanation of the idea behind the notion of
approximate diagonalization of operators.

###  1.1. Motivation

Let us start with a general and simple example of a matrix type operator on a
Hilbert space. Let $\psi_{n},n\in\mathbb{N}$ be a basis for a separable
Hilbert space $\mathcal{H}$,
$f=\sum_{n\in\mathbb{N}}a_{n}\psi_{n}\in\mathcal{H}$, and
$A:\mathcal{H}\rightarrow\mathcal{H}$ be linear and continuous. Then

$Af=\sum_{n\in\mathbb{N}}a_{n}A\psi_{n}=\sum_{n\in\mathbb{N}}a_{n}\sum_{m\in\mathbb{N}}b_{n,m}\psi_{m}=\sum_{m\in\mathbb{N}}\sum_{n\in\mathbb{N}}b_{n,m}a_{n}\psi_{m}.$

So, $A$ can be viewed as the action of an infinite matrix on a space of
sequences:

$(a_{n})_{n\in\mathbb{N}}\to(Af)_{m\in\mathbb{N}}=\big{(}\sum_{n\in\mathbb{N}}b_{n,m}a_{n}\big{)}_{m\in\mathbb{N}},$

More generally, instead of $\mathbb{N}\times\mathbb{N}$ one can observe
indices in $\Lambda$, a discrete subgroup of $\mathbb{R}^{2d}$. Such group is
often represented as $A\,\mathbb{Z}^{2d}$ ($\mathbb{Z}^{2d}$ is the set of
integer points in $\mathbb{R}^{2d}$), where $A$ is a $2d$-dimensional, regular
matrix with determinant ${\rm det}A<1$. We will also use the term lattice for
such $\Lambda$.

For the sake of simplicity, consider the lattice points of the form

$\lambda=(\alpha k,\beta i)\in\Lambda,\qquad k,i\in\mathbb{Z}^{d}.$

i.e. $\Lambda=\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d}$. Then the time-
frequency shifts of $g\in L^{2}(\mathbb{R}^{d})$ are given by

$\pi(\lambda)g=\pi_{\alpha k,\beta i}g=e^{2\pi\alpha k\cdot t}g(t-\beta
i),\;\;\lambda=(\alpha k,\beta i),\;\;k\cdot t=\langle
k,t\rangle=\sum_{j}^{d}k_{j}t_{j}.$

Recall that the set
$\mathcal{G}(g,\Lambda)=\\{\pi(\lambda)g;\lambda\in\Lambda\\}$ is a Gabor
frame in $L^{2}(\mathbb{R}^{d})$ if for every $f\in L^{2}(\mathbb{R}^{d})$
there exist $c_{1},c_{2}>0$ such that

$c_{1}\sum_{\lambda\in\Lambda}|\langle
f,\pi(\lambda)g\rangle|^{2}\leq||f||_{L^{2}}^{2}\leq\
c_{2}\sum_{\lambda\in\Lambda}|\langle f,\pi(\lambda)g\rangle|^{2}$ (1)

($\langle\cdot,\cdot\rangle$ here denotes the scalar product in
$L^{2}(\mathbb{R}^{d})$). If $c_{1}=c_{2}$ then $\mathcal{G}(g,\Lambda)$ is
called a tight frame and $\gamma=g$. If $\mathcal{G}(g,\Lambda)$ is a frame,
then there exists a dual window $\gamma\in L^{2}(\mathbb{R}^{d})$ such that

$f=\sum_{\lambda\in\Lambda}\langle f,\pi(\lambda)g\rangle\pi(\lambda)\gamma\in
L^{2}(\mathbb{R}^{d}),\qquad f\in L^{2}(\mathbb{R}^{d}).$ (2)

If $H_{0}(x)=e^{-a|x|^{2}}$, $x\in\mathbb{R}^{d},$ $a>0$, and
$\lambda\in\Lambda=\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d}$,
$\alpha\beta<1$, then $\mathcal{G}(H_{0},\lambda)$ is a frame in
${L^{2}}(\mathbb{R}^{d})$, and its dual frame $\gamma$ satisfies

$|\gamma(x)|+|\widehat{\gamma}(x)|\leq Ce^{-c|x|^{2}},\quad
x\in\mathbb{R}^{d}.$

Such frames are also referred as superframes. For the details we refer to [15,
21, 25]. Moreover, it is well known ([5]Theorem 3.2.21.) that for
$g(x)=e^{-\pi x^{2}}$, $x\in\mathbb{R}^{d}$,the set

${\mathcal{G}}(g,\Lambda)={\mathcal{G}}(g,\alpha,\beta)=\\{\pi(\lambda)g;\lambda\in\Lambda\\}$

is a Gabor frame for $L^{2}(\mathbb{R}^{d})$ if and only if $\alpha\beta<1$.
We also mention that if $\alpha>0$ and $g\in W(\mathbb{R}^{d})$ are such that

$a\leq\sum_{k\in\mathbf{Z}^{d}}|g(x-\alpha k)|^{2}\leq b,\quad
x\in\mathbb{R}^{d},$

then there exists $\beta_{0}=\beta_{0}(\alpha)$ such that
${\mathcal{G}}(g,\alpha,\beta)$ is the frame for all $\beta\leq\beta_{0}$.
Here, $W(\mathbb{R}^{d})$ is the Wiener space which consists of function that
are locally bounded, and globally in $L^{1}(\mathbb{R}^{d})$. For details we
refer to [5] (see Section 3).

Next we introduce pseudo-differential operators. Let $\pi(\lambda)g$,
$\lambda\in\Lambda\subset\mathbb{R}^{2d}$ be a tight frame, and for $f,\phi\in
L^{2}(\mathbb{R}^{d})$ we consider expansions

$f(t)=\sum_{\lambda\in\Lambda}a_{\lambda}\pi(\lambda)g(t),\qquad\phi(t)=\sum_{\lambda\in\Lambda}b_{\lambda}\pi(\lambda)g(t),\quad
t\in\mathbb{R}^{d}.$

The Weyl-Hörmander pseudodifferential operator (or the Weyl transform) $a^{w}$
with the symbol $a\in\mathcal{S}^{\prime}\left(\mathbb{R}^{2d}\right)$ (see
subsection 2.1 for the notation) is defined by

$a^{w}f(x)=\int_{\mathbb{R}^{2d}}a\left(\frac{x+y}{2},\xi\right)e^{2\pi
i(x-y)\cdot\xi}f(y)dyd\xi,\quad{f\in\mathcal{S}(\mathbb{R}^{d})}.$ (3)

Then we have

$\langle a^{w}(t,D)f(t),\phi(t)\rangle\\\
=\langle\sum_{(k,i)\in\mathbb{Z}^{2d}}a_{k,j}a^{w}(t,D)\pi_{\alpha k,\beta
i}g(t),\sum_{(p,q)\in\mathbb{Z}^{2d}}b_{p,q}\pi_{\alpha p,\beta
q}g(t)\rangle\\\ =\sum_{k,i}\sum_{(p,q)\in\mathbb{Z}}a_{k,i}b_{p,q}\langle
a^{w}(t,D)\pi_{\alpha k,\beta i}g(t),\pi_{\alpha p,\beta q}g(t)\rangle.$

If we denote by $A=A_{\mathbb{Z}^{2d}\times\mathbb{Z}^{2d}}$ the infinite
dimensional matrix of the dimension $\mathbb{Z}^{2d}\times\mathbb{Z}^{2d}$
with elements

$\langle a^{w}(t,D)\pi_{\alpha k,\beta i}g(t),\pi_{\alpha p,\beta
q}g(t)\rangle,\qquad(k,i),(p,q)\in\mathbb{Z}^{2d},$

then

$\langle
a^{w}(t,D)f(t),\phi(t)\rangle=(a_{k,i})_{1\times\mathbb{Z}^{2d}}\,A\,(b_{p,q})_{\mathbb{Z}^{2d}\times
1},\qquad(k,i),(p,q)\in\mathbb{Z}^{2d},$

where the expansion of $f$ and $\phi$ is clear.

In such a way we may represent $a^{w}$ as a matrix type operator whose
properties are determined by the matrix elements. Asymptotic decay estimates
of these elements away from diagonal are related to mapping properties of the
corresponding operator. For that reason the term approximate diagonalization
is used to describe techniques based on these observations.

Thus, the goal is to characterize the decrease of the matrix $A$ far from its
diagonal. In the continuous case, instead of a matrix, we use an integral
representation, namely the short-time Fourier transform (STFT) as it will be
explained in Section 2.

##  2\. Preliminaries

In this section we fix general notation, and then proceed with basic facts on
weight functions, short-time Fourier transform, modulation spaces, and pseudo-
differential operators.

###  2.1. Notation

The sets of all all integers, positive integers, nonnegative integers, real
and complex numbers are denoted by
$\mathbb{Z},\mathbb{N},\mathbb{N}_{0},\mathbb{R}$ and $\mathbb{C}$,
respectively. For $x=\left(x_{1},\ldots,x_{d}\right)\in\mathbb{R}^{d}$, and a
multi-index
$\alpha=\left(\alpha_{1},\ldots,\alpha_{d}\right)\in\mathbb{N}_{0}^{d}$, we
use the notation: $|x|:=\left(x_{1}^{2}+\ldots+x_{d}^{2}\right)^{1/2}$,
$x^{\alpha}:=\prod_{j=1}^{d}x_{j}^{\alpha_{j}}$,
$|\alpha|:=\alpha_{1}+\ldots+\alpha_{d}$,
$\alpha!:=\alpha_{1}!\cdots\alpha_{d}!$,
$D^{\alpha}=D_{x}^{\alpha}:=D_{1}^{\alpha_{1}}\cdots D_{d}^{\alpha_{d}}$,
where $D_{j}^{\alpha_{j}}:=\left(-i\partial/\partial
x_{j}\right)^{\alpha_{j}}(j=1,\ldots,d)$. We write capital letters $X,Y,Z...$
for elements in $\mathbb{R}^{2d}$, and $A\lesssim B$ means $A\leq cB$ for a
suitable constant $c>0$.

The symbol $K\subset\subset V$ for an open $V\subset\mathbb{R}^{d}$ means that
$K$ is a compact subset of $V$. By $\hookrightarrow$ we denote continuous
embeddings between two Banach spaces. The norm in
$L^{p}\left(\mathbb{R}^{d}\right)$ is denoted by $\|\cdot\|_{p},\,1\leq
p\leq\infty$, and the corresponding sequence spaces will be denoted by
$l^{p}$. Fourier transform $\mathcal{F}$ of a function $f\in L^{1}$ will be
denoted by

$\displaystyle{\hat{f}}(\xi)=\int_{\mathbb{R}^{d}}f(x)e^{-2\pi
ix\xi}dx\qquad(\mathcal{F}^{-1}f(\xi)=\mathcal{F}f(-\xi)).$

$\mathcal{S}(\mathbb{R}^{d})$ denotes the Schwartz space of infinitely smooth
($C^{\infty}(\mathbb{R}^{d})$) functions which, together with their
derivatives, decay at infinity faster than any inverse polynomial. Its dual
space of tempered distributions is denoted by
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$.

Function $f$ belongs to weighted space $L^{p}_{v}(\mathbb{R}^{d})$ if $fv\in
L^{p}(\mathbb{R}^{d})$, $1\leq p\leq\infty$, where $v$ is a weight function,
see below.

###  2.2. Weights

Function $v$ is called a weight function, or simly a weight, if it is locally
bounded, non-negative, even and continuous on $\mathbb{R}^{d}$. Recall,

  * a)

$v$ is submultiplicative if, $v(x+y)\lesssim v(x)v(y)$,
$x,y\in\mathbb{R}^{d}$,

  * b)

$v$ is subconvolutive if $v^{-1}\in L^{1}$ and $(v^{-1}*v^{-1})(x)\lesssim
v^{-1}(x)$, $x\in\mathbb{R}^{d}$.

  * c)

$m$ is moderate if there exists submultiplicative weight $v$ such that,

$m(x+y)\lesssim v(x)m(y),\qquad x,y\in\mathbb{R}^{d}.$

Clearly, if $m$ is submultiplicative than it is also moderate. By
${\mathcal{M}}_{v}$ we denote the set of all $v-$ moderate weights.

######  Example 2.1.

Standard examples of weight functions are given by

$m_{a,b,c,t}(x)=e^{a|x|^{b}}(1+|x|)^{c}(\log(e+|x|))^{t},\quad
a,c,t\in\mathbb{R},\,\,b\geq 0,\,\,x\in\mathbb{R}^{d}.$ (4)

Properties of $m_{a,b,c,t}$ are given in the following Lemma from [16] (see
also [7]).

######  Lemma 2.1.

If $m=m_{a,b,c,t}$ is given by (4) then

  * a)

$m$ is submultiplicative if $a,c,t\geq 0$ and $0\leq b\leq 1$,

  * b)

$m$ is subconvolutive if $a>0,c,t\in\mathbb{R}$ and $0<b<1$,

  * c)

$m$ is moderate if $a,c,t\in\mathbb{R}$ and $0\leq b\leq 1$.

In addition, if $v$ is arbitrary submultiplicative weight then there exists
$r>0$ such that $\displaystyle v(x)\lesssim e^{r|x|}$, $x\in\mathbb{R}^{d}$.

In this paper we consider weights of the form

$m_{r}^{s}(x)=e^{r|x|^{1/s}},\quad r>0,\quad s>0\quad x\in\mathbb{R}^{d}.$ (5)

By Lemma 2.1 it follows that $m^{s}_{r}=m_{r,1/s,0,0}$ is submultiplicative
and subconvolutive if $r>0$ and $s>1$.

Let us briefly discuss subconvolutivity of $m^{s}_{r}$, since it will be used
later on. Clearly, $(m_{r}^{s})^{-1}=m_{-r}^{s}\in L^{1}(\mathbb{R}^{d})$. For
the case $r>0$ and $0<s\leq 1$ [5, Lemma 1.3.5] gives

$m_{-r}^{s}*m_{-r}^{s}(x)\leq Cm_{-2^{-1/s}r}^{s}(x),\quad
x\in\mathbb{R}^{d}.$ (6)

Indeed, note that simple inequality $\displaystyle|x+y|^{1/s}\leq
2^{\frac{1-s}{s}}(|x|^{1/s}+|y|^{1/s}),\,x,y\in\mathbb{R}^{d}$ implies that

$m_{r}^{s}(x+y)\leq
m_{2^{\frac{1-s}{s}}r}^{s}(x)m_{2^{\frac{1-s}{s}}r}^{s}(y),\quad
x,y\in\mathbb{R}^{d}.$

or equivalently

$m_{-r}^{s}(x-y)m_{-r}^{s}(y)\leq m_{-2^{\frac{s-1}{s}}r}^{s}(x),\quad
x,y\in\mathbb{R}^{d}.$

Then we obtain

$m_{-r}^{s}*m_{-r}^{s}(x)=\int_{\mathbb{R}^{d}}m_{-r}^{s}(x-y)m_{-r}^{s}(y)dy\\\
\leq\int_{\mathbb{R}^{d}}m_{-r/2}^{s}(x-y)\,m_{-r/2}^{s}(y)\,m_{-r/2}^{s}(y)dy\lesssim
m_{-2^{-1/s}r}^{s}(x),\quad x\in\mathbb{R}^{d}.$ (7)

Function $f$ belong to a weighted space $L^{p}_{v}(\mathbb{R}^{d})$ if $fv\in
L^{p}(\mathbb{R}^{d})$.

###  2.3. STFT and modulation spaces

By $(f,\varphi)$ we denote the dual pairing between
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and
$\varphi\in\mathcal{S}(\mathbb{R}^{d})$ and the dual pairing in the context of
Gelfand-Shilov type spaces in Section 4.

Let

$\pi(Z)g(t)=M_{\xi}T_{x}g=e^{2\pi
it\cdot\xi}g(t-x),\;\;g\in\mathcal{S}\left(\mathbb{R}^{d}\right)\backslash\\{0\\},\;\;Z=(x,\xi)\in\mathbb{R}^{2d}.$

The short-time Fourier transform (STFT) of
$f\in\mathcal{S}^{\prime}\left(\mathbb{R}^{d}\right)$ with respect to a given
window $g\in\mathcal{S}\left(\mathbb{R}^{d}\right)\backslash\\{0\\}$ is
defined as

$V_{g}f(x,\xi)=\langle
f,\pi(Z)g\rangle=\int_{\mathbb{R}^{d}}f(t)\overline{g(t-x)}e^{-2\pi it\xi}dt.$

The same formula over $\mathbb{R}^{2d}$ is given by

$\mathcal{V}_{g}f(X,\Xi)=\int_{\mathbb{R}^{2d}}f(t_{1},t_{2})\overline{g((t_{1},t_{2})-(x_{1},x_{2}))}e^{-2\pi
i(t_{1},t_{2})\cdot(\xi_{1},\xi_{2})}dt_{1}dt_{2},$ (8)

where $X=(x_{1},x_{2})$ and $\Xi=(\xi_{1},\xi_{2})$.

The (cross-)Wigner distribution is given by

$W(f,g)(x,\xi)=\int_{\mathbb{R}^{d}}f(x-\frac{t}{2})\overline{g(x+\frac{t}{2})}e^{-2\pi
i\xi t}dt,\qquad f,g\in L^{2}(\mathbb{R}^{d}),$ (9)

and when $g\in\mathcal{S}\left(\mathbb{R}^{d}\right)$ it extends to
$f\in\mathcal{S}^{\prime}\left(\mathbb{R}^{d}\right)$ by duality.

If $X=(x_{1},x_{2})$ and $\Xi=(\xi_{1},\xi_{2})$ belong to $\mathbb{R}^{2d}$,
we write

${\mathcal{W}}(f,g)(X,\Xi)\\\
=\int_{\mathbb{R}^{2d}}f((x_{1},x_{2})-\frac{(t_{1},t_{2})}{2})\overline{g((x_{1},x_{2})+\frac{(t_{1},t_{2})}{2})}e^{-2\pi
i(t_{1},t_{2})\cdot(\xi_{1},\xi_{2})}dt_{1}dt_{2}.$

We recall that for $f\in\mathcal{S}^{\prime}\left(\mathbb{R}^{d}\right)$ and
$g\in\mathcal{S}\left(\mathbb{R}^{d}\right)$ it holds

$W(f,g)(x,\xi)=2^{d}e^{4\pi ix\xi}\,V_{g^{*}}f(2x,2\xi),\quad
x,\xi\in\mathbb{R}^{d},$

where $g^{*}(x)=g(-x)$ (see [12]).

Modulation spaces are introduced by imposing mixed Lebesgue spaces norm to the
STFT as follows.

Let $1\leq p,q\leq\infty$ and $m\in{\mathcal{M}}_{v}$, and let
$g\in\mathcal{S}\left(\mathbb{R}^{d}\right)\backslash\\{0\\}$. Then the
weighted modulation space $M_{m}^{p,q}\left(\mathbb{R}^{d}\right)$ consists of
$f\in\mathcal{S}^{\prime}\left(\mathbb{R}^{d}\right)$ with the property

$\|f\|_{M_{m}^{p,q}}=\left(\int_{\mathbb{R}^{d}}\Big{(}\int_{\mathbb{R}^{d}}\left|V_{g}f(x,\xi)\right|^{p}\left|m(x,\xi)\right|^{p}dx\Big{)}^{q/p}d\xi\right)^{1/q}<\infty,$
(10)

with obvious changes if $p,q=\infty$. It is a Banach space with the norm
$\|\cdot\|_{M_{m}^{p,q}}$, and it is well known that the definition does not
depend on the choice of the window
$g\in\mathcal{S}\left(\mathbb{R}^{d}\right)\backslash\\{0\\}$ in the sense
that different Schwartz functions yield equivalent norms.

The original source for modulation spaces is [9], see also [12], and the
recent monograph [5].

###  2.4. Pseudo-differential operators

We end this section with some remarks on the Weyl-Hörmander pseudo-
differential operators given by (3). By a straightforward calculation one can
show that for $f\in\mathcal{S}^{\prime}\left(\mathbb{R}^{d}\right)$ formula
(3) is (in the weak sense) equivalent to

$\left\langle a^{w}f,g\right\rangle=\langle a,W(g,f)\rangle\qquad
g\in\mathcal{S}\left(\mathbb{R}^{d}\right),$

where $W$ is the Wigner distribution given by (9). It is well known that the
mapping

$a^{w}:\mathcal{S}(\mathbb{R}^{d})\rightarrow\mathcal{S}^{\prime}(\mathbb{R}^{d})$

is continuous.

The relation between $a^{w}$ and the STFT is given by the use of the
symplectic structure on $\mathbb{R}^{d}$. This is an important observation
when considering metrics different than the Euclidean ones, see Section 5.

For $(\xi,\eta)\in\mathbb{R}^{2d}$ we denote $j(\xi,\eta)=(\eta,-\xi)$, Let
$g\in\mathcal{S}\left(\mathbb{R}^{d}\right)\setminus\\{0\\}$. Recall that
Lemma [12, Lemma 14.5.1] implies the following formula:

$\left|\left(a^{w}\pi(X)g,\pi(Y)g\right)\right|=\left|\mathcal{V}_{\Phi}a\left(\frac{X+Y}{2},j(Y-X)\right)\right|,\quad
X,Y\in\mathbb{R}^{2d},$ (11)

where $\Phi=W(g,g)\in\mathcal{S}\left(\mathbb{R}^{2d}\right)$. By the change
of variables we obtain

$\left|\mathcal{V}_{\Phi}\sigma(U,V)\right|=\left|\left\langle\sigma^{w}\pi\left(U-\frac{j^{-1}(V)}{2}\right)g,\pi\left(U+\frac{j^{-1}(V)}{2}\right)g\right\rangle\right|,$
(12)

$U,V\in\mathbb{R}^{2d}$. Note that the standard symplectic form on
$\mathbb{R}^{2d}$ is related to $j$ by

$[(x,\xi),(y,\eta)]=\langle
j(x,\xi),(y,\eta)\rangle=\langle\xi,y\rangle-\langle x,\eta\rangle,\quad
x,y,\xi,\eta\in\mathbb{R}^{d}.$

Also,

$[(x,\xi),(y,\eta)]=\begin{bmatrix}y&\eta\\\
\end{bmatrix}\cdot\begin{bmatrix}0&I\\\
-I&0\end{bmatrix}\cdot\begin{bmatrix}x\\\ \xi\end{bmatrix},\quad
x,y,\xi,\eta\in\mathbb{R}^{d},$

where the standart symplectic matrix $\displaystyle\begin{bmatrix}0&I\\\
-I&0\end{bmatrix}$ is $2d\times 2d$ block matrix and $I$ is $d\times d$
identity matrix.

##  3\. Novel spaces for symbols of pseudo-differential operators

To extend results from [23] into the framework of Gelfand–Shilov spaces given
in [5] (see Section 4) we need a careful preparation. This section contains
original material, namely the construction of particular Wiener-amalgam spaces
$W\left(L^{\infty},\mathcal{A}_{r}^{s}\right)$ which are used in the
definition of new modulation spaces $\widetilde{M}_{proj,s}^{\infty}$ and
$\widetilde{M}_{ind,s}^{\infty}$. Then the symbols of pseudo-differential
operators are distributions from $\widetilde{M}_{proj,s}^{\infty}$ or
$\widetilde{M}_{ind,s}^{\infty}$.

###  3.1. Spaces of sequences

We introduce spaces of sequences which are convenient for our investigations.
We note that these sequences are considered in [26] in the context of mapping
properties of the Bargmann transforms (cf. [26, Definition 3.1]).

Let $\Lambda$ be a discrete subgroup of $\mathbb{R}^{d}$, and

$m^{s}_{r}(\cdot)=e^{r|\cdot|^{1/s}},\qquad r,s>0.$ (13)

Then the sequence $\mathbf{a}=(a_{\lambda})_{\lambda\in\Lambda}$ belongs to
the Banach space $l_{m^{s}_{r}}^{\infty}(\Lambda)$ if

$||\mathbf{a}||_{r,s}=\|\mathbf{a}\,m^{s}_{r}\|_{\infty}=\sup_{\lambda\in\Lambda}|a_{\lambda}|e^{r|\lambda|^{1/s}}<\infty.$

We will use the notation
$\mathcal{A}_{r}^{s}=l_{m^{s}_{r}}^{\infty}(\Lambda)$.

It is well known that $l_{v}^{\infty}(\Lambda)$ is Banach algebra with respect
to convolution if and only if $v$ is subconvolutive (see [7]). Since
$m^{s}_{r}$ is not subconvolutive when $0<s\leq 1$ (see (6)), it follows that
$\mathcal{A}_{r}^{s}$ is not a convolution algebra.

We collect some of the basic properties of $\mathcal{A}_{r}^{s}$ in the
following Lemma.

######  Lemma 3.1.

Let $s,r>0$.

  * i)

If $0<r_{1}<r_{2}$ then
$\mathcal{A}_{r_{2}}^{s}\hookrightarrow\mathcal{A}_{r_{1}}^{s}\hookrightarrow
l^{1}$ where $\hookrightarrow$ denotes continuous embedding. Moreover, this
embedding is compact.

  * ii)

$\mathcal{A}_{r}^{s}$ is involutive, i.e., if
$\mathbf{a}\in\mathcal{A}_{r}^{s}$ then
$\mathbf{a}^{*}=(a_{-\lambda})_{\lambda\in\Lambda}\in\mathcal{A}_{r}^{s}$ and
$||\mathbf{a}^{*}||_{r,s}=||\mathbf{a}||_{r,s}$.

  * iii)

$\mathcal{A}_{r}^{s}$, is solid, i.e., if $\mathbf{b}\in\mathcal{A}_{r}^{s}$
and $a_{\lambda}\leq b_{\lambda}$ for all $\lambda\in\Lambda$ then
$\mathbf{a}\in\mathcal{A}_{r}^{s}$ and
$||\mathbf{a}||_{r,s}\leq||\mathbf{b}||_{r,s}$.

  * iv)

Let $\mathbf{a},\mathbf{b}\in\mathcal{A}_{r}^{s}$ and
$\mathbf{c}=\mathbf{a}*\mathbf{b}$. Then there exists $c_{s}=c\in(0,1]$ such
that

$\|\mathbf{c}\|_{cr,s}\lesssim\|\mathbf{a}\|_{r,s}\|\mathbf{b}\|_{r,s}.$ (14)

###### Proof.

We will only prove $i)$ and $iv)$ since $ii)$ and $iii)$ are straightforward.

$i)$ The compactness of the embedding is the consequence of the Köthe theory
of sequence spaces since the weights satisfy
$e^{(r_{1}-r_{2})|\lambda|^{1/s}}\to 0$ as $|\lambda|\to\infty$.

$iv)$ Note that Lemma 2.1 and (6) implies that there exists $c\in(0,1]$ ($c=1$
for $s>1$ or $c=2^{-1/s}$ for $0<s\leq 1$) such that

$\displaystyle m_{-r}^{s}*m_{-r}^{s}(x)\lesssim m_{-cr}^{s}(x)\quad
x\in\mathbb{R}^{d}.$ (15)

Therefore, if $\mathbf{c}=(c_{\lambda})_{\lambda\in\Lambda}$ we obtain

$|\displaystyle
c_{\lambda}|\leq\sum_{\mu\in\Lambda}|a_{\lambda-\mu}||b_{\mu}|\leq\|\mathbf{a}\|_{r,s}\|\mathbf{b}\|_{r,s}(m_{-r}^{s}*m_{-r}^{s}(\lambda))\\\
\lesssim\|\mathbf{a}\|_{r,s}\|\mathbf{b}\|_{r,s}m_{-cr}^{s}(\lambda),\quad\lambda\in\Lambda,$
(16)

which implies $\displaystyle|\displaystyle
c_{\lambda}|\,m_{cr}^{s}(\lambda)\lesssim\|\mathbf{a}\|_{r,s}\|\mathbf{b}\|_{r,s}$,
$\lambda\in\Lambda$. The claim follows by taking the supremum over $\lambda$.

∎

We introduce the Frechét space ($FS-$ space) and the dual Frechet space
($DFS-$ space) of sequences, respectively, by taking the projective and
inductive limit topologies:

$\mathcal{A}_{proj}^{s}=\varprojlim_{r\to\infty}\mathcal{A}_{r}^{s},\quad\mathcal{A}_{ind}^{s}=\varinjlim_{r\to
0}\mathcal{A}_{r}^{s},\quad s>0.$

These spaces are nuclear, and by Lemma 3.1 it follows that the spaces
$\mathcal{A}_{proj}^{s}$ and $\mathcal{A}_{ind}^{s}$ are closed under
convolution for any $s>0$. Moreover, for a given $r_{0}>0$ we have

$\mathcal{A}_{proj}^{s}\hookrightarrow\mathcal{A}_{r_{0}}^{s}\hookrightarrow\mathcal{A}_{ind}^{s}\hookrightarrow
l^{1},\quad s>0.$

###  3.2. Wiener amalgam spaces related to $\mathcal{A}_{r}^{s}$

Next we introduce the Wiener-Amalgam space related to $\mathcal{A}_{r}^{s}$
when $r,s>0$.

Let $C$ be an open relatively compact set in $\mathbb{R}^{2d}$ that contains
the origin and $\Lambda$ be a lattice such that
$\mathbb{R}^{2d}\subset\displaystyle\bigcup_{\lambda\in\Lambda}(\lambda+C)$,
and $\Lambda$ be a lattice in $\mathbb{R}^{2d}$. For a locally bounded
function $F$ on $\mathbb{R}^{2d}$ we set

$F_{\lambda}=\sup_{Y\in\lambda+C}|F(Y)|,\quad\lambda\in\Lambda.$ (17)

Then $F$ belongs to the Banach space
$W\left(L^{\infty},\mathcal{A}_{r}^{s}\right):=W^{s}_{r}$ if
${\mathbf{F}}=\left(F_{\lambda}\right)_{\lambda}\in\mathcal{A}_{r}^{s}(\Lambda)$,
and The norm in $W^{s}_{r}$ is given by $\|\displaystyle
F\|_{W^{s}_{r}}=\|F_{\lambda}\|_{r,s}$. In particular, $W^{s}_{r}$ contains
locally bounded functions whose decay rate at infinity is bounded by
$m^{s}_{-r}(\cdot)=e^{-r|\cdot|^{1/s}}$.

######  Remark 3.1.

Note that $\displaystyle
W^{s}_{r}=W\left(L^{\infty},\mathcal{A}_{r}^{s}\right)=W(L^{\infty},l^{\infty}_{m^{s}_{r}}(\Lambda))=L_{m^{s}_{r}}^{\infty}(\mathbb{R}^{2d}).$
Moreover, by Lemma 3.1 it follows that
$\displaystyle\mathcal{A}_{r}^{s}\hookrightarrow l^{1}$, and therefore

$W^{s}_{r}\hookrightarrow W(L^{\infty},{l^{1}}):=W(l^{1}),$

where $W(l^{1})$ is the Wiener space defined by
$(F_{\lambda})_{\lambda\in\Lambda}\in l^{1}$ (see (17)).

If $g\in W(l^{1})$ and the lattice $\Lambda$ is sufficiently dense in
$\mathbb{R}^{2d}$, then ${\mathcal{G}}(g,\Lambda)$ is a frame for
$L^{2}(\mathbb{R}^{d})$ (see [12] for details).

By taking the projective and inductive limits we obtain

$W^{s}_{proj}=\varprojlim_{r\to\infty}W^{s}_{r}=W\left(L^{\infty},\mathcal{A}_{proj}^{s}\right)\;\;{\rm
and}\;\;W^{s}_{ind}=\varinjlim_{r\to
0}W^{s}_{r}=W\left(L^{\infty},\mathcal{A}_{ind}^{s}\right).$

We can prove the following Lemma.

######  Lemma 3.2.

Let $s>0$. $W^{s}_{proj}$ and $W^{s}_{ind}$ are involutive algebras with
respect to convolution. In particular,

$\|F*G\|_{W^{s}_{cr}}\lesssim\|F\|_{W^{s}_{r}}\|G\|_{W^{s}_{r}},\quad r>0.$
(18)

where $c\in(0,1]$ as in part $iv)$ of the Lemma 3.1

###### Proof.

Directly form the definition it follows that $\displaystyle W^{s}_{r}$ is an
involutive algebra.

Note that part $iv)$ of Lemma 3.1 implies that
$\displaystyle\mathcal{A}_{r}^{s}*\mathcal{A}_{r}^{s}\hookrightarrow\mathcal{A}_{cr}^{s}$.
Then [8, Theorem 3] implies that $W^{s}_{r}*W^{s}_{r}\hookrightarrow
W^{s}_{cr}$ and the statement follows. ∎

###  3.3. Modulation spaces related to $\mathcal{A}_{r}^{s}$

We end the section by introducing modulation spaces with respect to
$\mathcal{A}^{s}_{r}$. For a symbol $a$ on $\mathbb{R}^{2d}$ let we define

$\mathcal{G}(a)(Y)=\sup_{X\in\mathbb{R}^{2d}}|\mathcal{V}_{\Phi}(a)(X,Y)|,\quad
Y\in\mathbb{R}^{2d},$

where $\Phi=W(g,g),$ for a given window
$g\in\mathcal{S}(\mathbb{R}^{d})\setminus\\{0\\}.$

We say that $a\in\widetilde{M}^{\infty,\mathcal{A}^{s}_{r}}(\mathbb{R}^{2d})$
if $G(a)\circ j\in W^{s}_{r}$, where $j(\xi,\eta)=(\eta,-\xi)$,
$(\xi,\eta)\in\mathbb{R}^{2d}$.

The space $\widetilde{M}^{\infty,\mathcal{A}^{s}_{r}}(\mathbb{R}^{2d})$ is a
Banach space with the norm given by

$\displaystyle\|\sigma\|_{\widetilde{M}^{\infty,\mathcal{A}^{s}_{r}}}=\|G(\sigma)\circ
j\|_{W^{s}_{r}}.$

To shorten the notation we will write
$\displaystyle\widetilde{M}^{\infty,\mathcal{A}^{s}_{r}}(\mathbb{R}^{2d}):=\widetilde{M}_{r,s}^{\infty}$.

Again, we consider the corresponding projective and inductive limit spaces

$\widetilde{M}_{proj,s}^{\infty}=\varprojlim_{r\to\infty}\widetilde{M}_{r,s}^{\infty}\quad\text{
and}\quad\widetilde{M}_{ind,s}^{\infty}=\varinjlim_{r\to
0}\widetilde{M}_{r,s}^{\infty}.$

It can be proved that these spaces are related to the standard modulation
spaces as given in (10) in the following way.

######  Proposition 3.1.

Let $g\in{\mathcal{S}}(\mathbb{R}^{d})\backslash\\{0\\}$, and $\Phi=W(g,g)$.
Then

$\displaystyle\widetilde{M}_{proj,s}^{\infty}=\bigcap_{r>0}M_{1\otimes
m^{s}_{r}}^{\infty,\infty}\quad{and}\quad\displaystyle\widetilde{M}_{ind,s}^{\infty}=\bigcup_{r>0}M_{1\otimes
m^{s}_{r}}^{\infty,\infty},$

where

$M_{1\otimes
m^{s}_{r}}^{\infty,\infty}=\\{f\in\mathcal{S}^{\prime}(\mathbb{R}^{2d})\,|\,\sup_{X,\Xi\in\mathbb{R}^{2d}}|\mathcal{V}_{\Phi}f(X,\Xi)|e^{r|\Xi|^{1/s}}<\infty\\}.$

The class of symbols
$\widetilde{M}^{\infty,\mathcal{A}^{s}_{r}}(\mathbb{R}^{2d})$ is an
appropriate choice when dealing with pseudo-differential operators on Gelfand-
Shilov type spaces with Hörmander metrics.

##  4\. Approximate diagonalization in Gelfand-Shilov spaces

In this section we first recall the definition of standard Gelfand-Shilov
spaces, and then recall the results from [4] and [5] which are necessary when
extending results from Section 5 to weights that decays at infinity faster
then the inverse of any polynomial. Basic facts of Gelfand-Shilov spaces can
be found in the original source [11].

Let $s,t,A,B>0$. We start with the Banach space
$S_{t,B}^{s,A}\left(\mathbb{R}^{d}\right)$ consisting of functions $f\in
C^{\infty}(\mathbb{R}^{d})$ with the finite norm

$||f||_{\mathcal{S}^{s,A}_{t,B}}=\sup_{\alpha,\beta\in\mathbb{N}^{d}_{0},x\in\mathbb{R}^{d}}\frac{\left|x^{\alpha}\partial^{\beta}f(x)\right|}{A^{|\alpha|}B^{|\beta|}\alpha!^{t}\beta!^{s}}.$

Then, by taking projective and inductive limits we obtain

$\Sigma_{t}^{s}=\varprojlim_{A>0,B>0}S_{t,B}^{s,A};\quad
S_{t}^{s}=\varinjlim_{A>0,B>0}S_{t,B}^{S,A}\text{. }$

Let us denote $S_{t,*}^{s}$ for $\Sigma_{t}^{s}$ or $S_{t}^{s}$. Also, the
usual notation for isotropic spaces is $\mathcal{S}^{s}(\mathbb{R}^{d})$ for
$\mathcal{S}^{s}_{s}(\mathbb{R}^{d}),s\geq 1/2,$ and
$\Sigma^{s}(\mathbb{R}^{d})$ for $\Sigma^{s}_{s}(\mathbb{R}^{d}),s>1/2.$

The following results from [5] give rise to approximate diagonalization when
considering the Euclidean metrics.

######  Theorem 4.1.

([5, Theorem 5.2.7]) Let
$s>0,m\in\mathcal{M}_{v}\left(\mathbb{R}^{d}\right),g\in M_{v\otimes
1}^{1}\left(\mathbb{R}^{d}\right)\backslash\\{0\\}$ such that

$\left\|\partial^{\alpha}g\right\|_{L_{v}^{1}\left(\mathbb{R}^{d}\right)}\lesssim
C^{|\alpha|}(\alpha!)^{s},\quad\alpha\in\mathbb{N}^{d},$

for some $C>0$. If $f\in C^{\infty}\left(\mathbb{R}^{d}\right)$ the following
conditions are equivalent:

  * (i)

There exists a constant $C>0$ such that

$\left|\partial^{\alpha}f(x)\right|\lesssim
m(x)C^{|\alpha|}(\alpha!)^{s},\quad
x\in\mathbb{R}^{d},\alpha\in\mathbb{N}^{d}.$

  * (ii)

There exists a constant $\epsilon>0$ such that

$\left|V_{g}f(x,\xi)\right|\lesssim m(x)e^{-\epsilon|\xi|^{\frac{1}{s}}},\quad
x,\xi\in\mathbb{R}^{2d},\alpha\in\mathbb{N}^{d}.$

######  Theorem 4.2.

([5, Theorem 5.2.10]) Let $s\geq 1/2$ and
$m\in\mathcal{M}_{v}\left(\mathbb{R}^{2d}\right)$.

  * $a)$

If $1/2\leq s$ let $g\in S_{s}^{s}\left(\mathbb{R}^{d}\right)$.

  * $b)$

If $1/2<s$ let $g\in\Sigma_{s}^{s}\left(\mathbb{R}^{d}\right)$.

Assume the following growth condition on the weight $v$ :

$v(z)\lesssim e^{\epsilon|z|^{1/s}},\quad z\in\mathbb{R}^{2d},$

for every $\epsilon>0$.

Let $a\in C^{\infty}\left(\mathbb{R}^{2d}\right)$. Then the following is
equivalent:

  * (i)

The symbol $a$ satisfies

$\left|\partial^{\alpha}a(z)\right|\lesssim
m(z)C^{|\alpha|}(\alpha!)^{s},\quad
z\in\mathbb{R}^{2d},\alpha\in\mathbb{N}^{2d}.$

  * (ii)

There exists $\epsilon>0$ such that

$\left|\left\langle a^{w}\pi(z)g,\pi(w)g\right\rangle\right|\lesssim
m\left(\frac{w+z}{2}\right)e^{-\epsilon|w-z|^{\frac{1}{s}}},\quad
z,w\in\mathbb{R}^{2d}.$ (19)

######  Theorem 4.3.

([5, Theorem 5.2.12]) Let $s\geq 1/2$, $g\in
S_{s}^{s}\left(\mathbb{R}^{d}\right)$,
$m\in\mathcal{M}_{v}\left(\mathbb{R}^{2d}\right)$, and let
$\mathcal{G}(g,\Lambda)$ be a Gabor superframe for
$L^{2}\left(\mathbb{R}^{d}\right)$. If
$a\in{C}^{\infty}\left(\mathbb{R}^{2d}\right)$, thne the following properties
are equivalent:

  * (i)

There exists $\epsilon>0$ such that the estimate (19) holds.

  * (ii)

There exists $\epsilon>0$ such that

$\left|\left\langle a^{w}\pi(\mu)g,\pi(\lambda)g\right\rangle\right|\lesssim
m\left(\frac{\lambda+\mu}{2}\right)e^{-\epsilon|\lambda-\mu|^{\frac{1}{s}}},\quad\lambda,\mu\in\Lambda.$

Now we state a result on approximate diagonalization in the spirit of [14].
The proof follows from a carefuli inspection of the proofs of Theorems 4.1,
4.2, and 4.3, and will be given elsewhere.

Here below $\displaystyle\mathcal{A}_{*}^{s}$ denotes $\mathcal{A}_{proj}^{s}$
or $\mathcal{A}_{ind}^{s}$, $W^{s}_{*}$ stands for $W^{s}_{proj}$ or
$W^{s}_{ind}$, and by $\displaystyle\widetilde{M}_{*,s}^{\infty}$ we denote
$\widetilde{M}_{proj,s}^{\infty}$ or $\widetilde{M}_{ind,s}^{\infty}$.

######  Theorem 4.4.

([5, Theorem 5.2.12]) Let $s\geq 1/2$ and
$g\in{\mathcal{S}}_{s}^{s}(\mathbb{R}^{d})$. Then the following is equivalent:

  * a)

$a\in\widetilde{M}_{*,s}^{\infty}$,

  * b)

there exists $H\in W^{s}_{*}$ such that

$|\langle a^{w}\pi(X)g,\pi(Y)g\rangle|\leq H(Y-X),\quad
X,Y\in\mathbb{R}^{2d},$ (20)

  * c)

There exists a sequence $h\in\mathcal{A}_{*}^{s}(\Lambda)$ such that

$|\langle a^{w}\pi(\mu)g,\pi(\nu)\gamma\rangle|\leq
h(\nu-\mu),\quad\mu,\nu\in\Lambda.$

##  5\. Extensions on the spaces with the Hörmander metric

In this section we present some general results from [23] in the context of
tempered distributions, and symbol classes $S(M,g)$, see (21). Extension of
these results to Gelfand-Shilov spaces and their dual spaces of tempered
ultra-distributions is a highly nontrivial task and will be the subject of our
future investigations.

Before explaining the main results, we need to recall necessary notions
related to the geometry in the observed spaces. We refer to [20] for more
details on the subject.

We assume that a Riemannian metric $g$ on $\mathbb{R}^{2d}$ is a Borel
measurable section of the 2-covariant tensor bundle
$T^{2}T^{*}\mathbb{R}^{2d}$ that is symmetric and positive-definite at every
point. The corresponding quadratic forms are denoted by the same symbol:
$g_{X}(T):=g_{X}(T,T)$, $T\in T_{X}\mathbb{R}^{2d}$. For each
$X\in\mathbb{R}^{2d}$, we identify $\mathbb{R}^{2d}$ with
$T_{X}\mathbb{R}^{2d}$ that sends every $Y\in\mathbb{R}^{2d}$ to the
directional derivative in direction $Y$ at $X$. Let
$T\in\mathbb{R}^{2d}\backslash\\{0\\}$. We denote by $\partial_{T}$ the vector
field on $\mathbb{R}^{2d}$ given by the directional derivative in direction
$T$ at every point $X\in\mathbb{R}^{2d}$. We denote by
$\sigma:\mathbb{R}^{2d}\rightarrow\mathbb{R}^{2d}$ the isomorphism induced by
the symplectic form; notice that ${}^{t}\sigma=-\sigma$, $\sigma_{X}\in
L(\mathbb{R}^{2d},\mathbb{R})$, $\sigma_{X}(T)=[X,T]$, $T\in\mathbb{R}^{2d}$.
Let $g$ be a Riemannian metric on $\mathbb{R}^{2d}$ and, for
$X\in\mathbb{R}^{2d}$, denote by
$Q_{X}:\mathbb{R}^{2d}\rightarrow\mathbb{R}^{2d}$ the isomorphism induced by
$g_{X}$. In particular, $Q_{X}$ is $2d\times 2d$ diagonal matrix with elements
$q_{X}(E_{j},E_{j})$ where $\\{E_{j}\\}_{j=1}^{2d}$ is the basis of
$\mathbb{R}^{2d}$.

For $X\in\mathbb{R}^{2d}$, set $Q^{\sigma}_{X}:={}^{t}\sigma
Q_{X}^{-1}\sigma:\mathbb{R}^{2d}\rightarrow\mathbb{R}^{2d}$ and let
$g^{\sigma}_{X}(T,S):=\langle Q^{\sigma}_{X}T,S\rangle$,
$T,S\in\mathbb{R}^{2d}$. Then $g^{\sigma}$ is again a Riemannian metric on
$\mathbb{R}^{2d}$ called the symplectic dual of $g$; it is also given by

$g^{\sigma}_{X}(T)=\sup_{S\in\mathbb{R}^{2d}\backslash\\{0\\}}[T,S]^{2}/g_{X}(S).$

The Riemannian metric $g$ is said to be a Hörmander metric [18] (i.e., an
admissible metric in the terminology of [2, 20]) if the following three
conditions are satisfied:

  * $(i)$

(slow variation) there exist $C_{0}\geq 1$ and $r_{0}>0$ such that

$g_{X}(X-Y)\leq r^{2}_{0}\Rightarrow C^{-1}_{0}g_{Y}(T)\leq g_{X}(T)\leq
C_{0}g_{Y}(T),$

for all $X,Y,T\in\mathbb{R}^{2d}$;

  * $(ii)$

(temperance) there exist $C_{0}\geq 1$ and $N_{0}\geq 0$ such that

$\left(g_{X}(T)/g_{Y}(T)\right)^{\pm 1}\leq
C_{0}(1+g^{\sigma}_{X}(X-Y))^{N},\quad\mbox{for
all}\,\,X,Y,T\in\mathbb{R}^{2d};$

  * $(iii)$

(the uncertainty principle) $g_{X}(T)\leq g^{\sigma}_{X}(T)$, for all
$X,T\in\mathbb{R}^{2d}$.

Next we need an admissibility condition. For a given metric $g$, a positive
Borel measurable function $M$ on $\mathbb{R}^{2d}$ is said to be
$g$-admissible if there are $C\geq 1$, $r>0$ and $N\geq 0$ such that

$\displaystyle g_{X}(X-Y)\leq r^{2}\Rightarrow C^{-1}M(Y)\leq M(X)\leq
CM(Y),\quad\text{and}$ $\displaystyle\left(M(X)/M(Y)\right)^{\pm 1}\leq
C(1+g^{\sigma}_{X}(X-Y))^{N},\quad\mbox{for all}\,\,X,Y\in\mathbb{R}^{2d}.$

All the constants mentioned above are called admissibility constants.

Given a $g$-admissible weight $M$, the space of symbols $S(M,g)$ is defined as
the space of all $a\in\mathcal{C}^{\infty}(\mathbb{R}^{2d})$ for which

$\|a\|^{(k)}_{S(M,g)}=\sup_{l\leq
k}\sup_{\begin{subarray}{c}X\in\mathbb{R}^{2d}\\\
T_{1},\ldots,T_{l}\in\mathbb{R}^{2d}\backslash\\{0\\}\end{subarray}}\frac{|a^{(l)}(X;T_{1},\ldots,T_{l})|}{M(X)\prod_{j=1}^{l}g_{X}(T_{j})^{1/2}}<\infty,\quad\forall
k\in\mathbb{N}.$ (21)

With this system of seminorms, $S(M,g)$ becomes a Fréchet space. One can
always regularize the metric making it smooth without changing the notion of
$g$-admissibility of a weight and the space $S(M,g)$; the same can be done for
any $g$-admissible weight (see [18]).

For any symbol $a\in\mathcal{S}(\mathbb{R}^{2d})$, we consider the Weyl
quantization $a^{w}$, see (3) This correspondence extends to symbols in
$\mathcal{S}^{\prime}(\mathbb{R}^{2d})$ in a usual manner, and in this case
$a^{w}:\mathcal{S}(\mathbb{R}^{2d})\rightarrow\mathcal{S}^{\prime}(\mathbb{R}^{2d})$
is a continuous mapping. For any $a\in S(M,g)$ with a $g$-admissible weight
$M$, the operator $a^{w}$ is continuous on $\mathcal{S}(\mathbb{R}^{2d})$ and
uniquely extends to an operator on $\mathcal{S}^{\prime}(\mathbb{R}^{2d})$
(cf. [18]).

###  5.1. The symplectic short-time Fourier transform

We extended in [22] the short time Fourier transform to the spaces of
functions over the ground space equipped with the Hörmader metrics with the
aim to analyze the almost diagonalization of a class of pseudo differential
operators that is, the estimate of the action of a pseudo differential
operator on wave packages accommodated to the involved metrics.

We denote by $\mathcal{F}_{\sigma}$ the symplectic Fourier transform on
$\mathbb{R}^{2d}$:

$\mathcal{F}_{\sigma}f(X)=\int_{\mathbb{R}^{2d}}e^{-2\pi i[X,Y]}f(Y)dY,\quad
f\in L^{1}(\mathbb{R}^{2d});$

recall that $\mathcal{F}_{\sigma}\mathcal{F}_{\sigma}=\operatorname{Id}$.
($i[X,Y]$ is the symplectic product)
Let $\mathbb{\varphi}\in\mathcal{S}(\mathbb{R}^{2d})$ and set
$\varphi_{X}:=\mathbb{\varphi}(X)$, $X\in\mathbb{R}^{2d}$. We define the
symplectic short-time Fourier transform $\mathcal{V}_{\mathbb{\varphi}}f$ of
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{2d})$ with respect to $\mathbb{\varphi}$
as

$\mathcal{V}_{\mathbb{\varphi}}f(X,\Xi):=\mathcal{F}_{\sigma}(f\overline{\varphi_{X}})(\Xi)=\langle
f,e^{-2\pi i[\Xi,\cdot]}\overline{\varphi_{X}}\rangle,\quad
X,\Xi\in\mathbb{R}^{2d}.$

When $f\in L^{1}_{(1+|\cdot|)^{-s}}(\mathbb{R}^{2d})$ for some $s\geq 0$
($|\cdot|$ is (any) norm on $\mathbb{R}^{2d}$), we have

$\mathcal{V}_{\mathbb{\varphi}}f(X,\Xi)=\int_{\mathbb{R}^{2d}}e^{-2\pi
i[\Xi,Y]}f(Y)\overline{\varphi_{X}(Y)}dY.$

The mapping $\Xi\mapsto e^{-2\pi i[\Xi,\cdot]}$, from $\mathbb{R}^{2d}$ to the
space of $\phi$ such that
$\|\phi^{(\alpha)}(\cdot)(1+|\cdot|)^{-1}\|_{L^{\infty}}<\infty$, for every
$\alpha\in\mathbb{N}_{0}^{d}$, is well-defined and smooth. Therefore the
mapping

$(X,\Xi)\mapsto e^{-2\pi
i[\Xi,\cdot]}\overline{\varphi}_{X},\quad\text{from}\quad\mathbb{R}^{2d}\times\mathbb{R}^{2d}\rightarrow\mathcal{S}(\mathbb{R}^{2d}),$

is strongly Borel measurable. Consequently, the function

$(X,\Xi)\mapsto\mathcal{V}_{\mathbb{\varphi}}f(X,\Xi)\quad\text{from}\quad\mathbb{R}^{2d}\times\mathbb{R}^{2d}\rightarrow\mathbb{C},$

is always Borel measurable, and if $\mathbb{\varphi}$ is of class
$\mathcal{C}^{k}$, $0\leq k\leq\infty$, then

$\mathcal{V}_{\mathbb{\varphi}}f\in\mathcal{C}^{k}(\mathbb{R}^{2d}\times\mathbb{R}^{2d}),\quad
0\leq k\leq\infty.$

For the study of $\mathcal{V}_{\mathbb{\varphi}}$, we consider the Fréchet
space

$\displaystyle\lim_{\begin{subarray}{c}\longleftarrow\\\
s\rightarrow\infty\end{subarray}}L^{\infty}_{(1+|\cdot|)^{s}}(\mathbb{R}^{2d}\times\mathbb{R}^{2d}),$

and the inductive limit $(LB)$-spaces,

$\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
s\rightarrow\infty\end{subarray}}L^{\infty}_{(1+|\cdot|)^{-s}}(\mathbb{R}^{2d}\times\mathbb{R}^{2d}),\quad\text{
and }\quad\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
s\rightarrow\infty\end{subarray}}L^{1}_{(1+|\cdot|)^{-s}}(\mathbb{R}^{2d}\times\mathbb{R}^{2d}).$

We note that $|\cdot|$ is any norm on $\mathbb{R}^{2d}\times\mathbb{R}^{2d}$
and none of these spaces depends on the particular choice of $|\cdot|$. We
have the following embeddings:

$\mathcal{S}(\mathbb{R}^{2d}\times\mathbb{R}^{2d})\hookrightarrow\lim_{\begin{subarray}{c}\longleftarrow\\\
s\rightarrow\infty\end{subarray}}L^{\infty}_{(1+|\cdot|)^{s}}(\mathbb{R}^{2d}\times\mathbb{R}^{2d})\hookrightarrow\lim_{\begin{subarray}{c}\longrightarrow\\\
s\rightarrow\infty\end{subarray}}L^{\infty}_{(1+|\cdot|)^{-s}}(\mathbb{R}^{2d}\times\mathbb{R}^{2d})\\\
\hookrightarrow\lim_{\begin{subarray}{c}\longrightarrow\\\
s\rightarrow\infty\end{subarray}}L^{1}_{(1+|\cdot|)^{-s}}(\mathbb{R}^{2d}\times\mathbb{R}^{2d})\hookrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{2d}\times\mathbb{R}^{2d}).$
(22)

The following assertions are proved in [22]:

  * $(i)$

The sesquilinear mapping

$\mathcal{S}^{\prime}(\mathbb{R}^{2d})\times\mathcal{S}(\mathbb{R}^{2d})\rightarrow\lim_{\begin{subarray}{c}\longrightarrow\\\
s\rightarrow\infty\end{subarray}}L^{\infty}_{(1+|\cdot|)^{-s}}(\mathbb{R}^{2d}\times\mathbb{R}^{2d}),\quad(f,\mathbb{\varphi})\mapsto\mathcal{V}_{\mathbb{\varphi}}f,$
(23)

is well-defined and hypocontinuous. Furthermore, for any bounded subset $B$ of
$\mathcal{S}^{\prime}(\mathbb{R}^{2d})$ there is $s>0$ such that
$\mathcal{V}_{\mathbb{\varphi}}f\in
L^{\infty}_{(1+|\cdot|)^{-s}}(\mathbb{R}^{2d})$, for all $f\in B$,
$\mathbb{\varphi}\in\mathcal{S}(\mathbb{R}^{2d})$, and the set of linear
mappings

$\mathcal{S}(\mathbb{R}^{d})\rightarrow
L^{\infty}_{(1+|\cdot|)^{-s}}(\mathbb{R}^{2d}\times\mathbb{R}^{2d}),\quad\mathbb{\varphi}\mapsto\mathcal{V}_{\overline{\mathbb{\varphi}}}f,\qquad
f\in B,$

is equicontinuous subset of
$\mathcal{L}(\mathcal{S}(\mathbb{R}^{2d})),L^{\infty}_{(1+|\cdot|)^{-s}}(\mathbb{R}^{2d}\times\mathbb{R}^{2d}))$.

  * $(ii)$

The sesquilinear mapping

$\mathcal{S}(\mathbb{R}^{2d}\times\mathbb{R}^{2d})\rightarrow\lim_{\begin{subarray}{c}\longleftarrow\\\
s\rightarrow\infty\end{subarray}}L^{\infty}_{(1+|\cdot|)^{s}}(\mathbb{R}^{2d}\times\mathbb{R}^{2d}),\quad(\psi,\mathbb{\varphi})\mapsto\mathcal{V}_{\mathbb{\varphi}}\psi,$
(24)

is well-defined and continuous.

  * $(iii)$

Under a suitable (geometric) condition, and if
$\mathbb{\varphi}\in\mathcal{S}(\mathbb{R}^{2d})$, then the conjugate-linear
mapping
$\mathcal{S}(\mathbb{R}^{2d})\rightarrow\mathcal{S}(\mathbb{R}^{2d}\times\mathbb{R}^{2d})$,
$\psi\mapsto\mathcal{V}_{\mathbb{\varphi}}\psi$, is well-defined and
continuous.

Our main result in [22] is devoted to the almost diagonalization of $a\in
S(M,g)$. We present a result which is a consequence of the main theorem in
[22]. Recall, $\mathbb{R}^{2d}$ is occupied with the symplectic structure
$[(x,\xi),(y,\eta)]=\langle\xi,y\rangle-\langle\eta,x\rangle$.

We denote by $\Psi^{g,L}_{X}$ the topological isomorphism over the Schwartz
space ${\mathcal{S}}(\mathbb{R}^{2d})$,

$\Psi^{g}_{X}:\mathcal{S}(\mathbb{R}^{2d})\rightarrow\mathcal{S}(\mathbb{R}^{2d}),\;\;(\Psi^{g,L}_{X}\varphi)(Y)=\varphi({\widetilde{Q}_{X}}^{-1/2}Y),\,Y\in\mathbb{R}^{2d},$

and we extend it, by duality, to the topological isomorphism

$\Psi^{g,L}_{X}:\mathcal{S}^{\prime}(\mathbb{R}^{2d})\rightarrow\mathcal{S}^{\prime}(\mathbb{R}^{2d}),\;\;\langle\Psi^{g}_{X}f,\varphi\rangle=|\det\widetilde{Q}|^{1/2}\langle
f,\varphi\circ\widetilde{Q}^{1/2}\rangle.$ (25)

In the simplest case when the metric $g$ is symplectic ($g=g^{\sigma}$, see
below), then our main Theorem in [22] has the following form: for each
$N\in\mathbb{N}$,

$\mathbb{R}^{2d}\times\mathbb{R}^{2d}\ni(X,\Xi)\mapsto
M((X+\Xi)/2)^{-1}(1+g_{\frac{X+\Xi}{2}}(X-\Xi))^{N}$ (26)
$\times\left|\left\langle\left(\Psi^{g}_{\frac{X+\Xi}{2}}a\right)^{w}\pi\left((Q_{\frac{X+\Xi}{2}})^{1/2}X\right)\chi,\overline{\pi\left((Q_{\frac{X+\Xi}{2}})^{1/2}\Xi\right)\chi}\right\rangle\right|\in
L^{\infty}(\mathbb{R}^{2d}\times\mathbb{R}^{2d}).$

This is a generalization of the result of Gröchenig and Rzeszotnik [14,
Theorem 4.2 $(i)$-$(ii)$], since when $g$ is the Euclidean metric on
$\mathbb{R}^{2d}$ and $M(X)=1$ (which corresponds to the Hörmander class
$S^{0}_{0,0}$), then $\Psi^{g,L}_{X}=\operatorname{Id}$,
$\tilde{Q}_{X}=\operatorname{Id}$, $\forall X\in\mathbb{R}^{2d}$ and

$(X,\Xi)\mapsto\langle X-\Xi\rangle^{N}|\langle
a^{w}\pi(X)\chi,\overline{\pi(\Xi)\chi}\rangle|\in
L^{\infty}(\mathbb{R}^{4n}).$

The right hand side of (26) can be explained by the use of simplectic and
metaplectic transformations. Denote by $Mp(\mathbb{R}^{2d})$ and
$Sp(\mathbb{R}^{2d})$ spaces of metaplectic operators and of symplectic
transformations over $\mathbb{R}^{2d}$ (cf. [10], [20]). Note that thew
interesting approach to the analysis of metaplectic transforms are given in
[6] as well as in [10]. The so called lifting theorems using the Hörmander
metric are considered in [1] and [17].

Let $\Pi$ be the surjective homomorphism
$\Pi:\operatorname{Mp}(\mathbb{R}^{2d})\rightarrow\operatorname{Sp}(\mathbb{R}^{2d})$.
Since ${Q}^{1/2}_{X}\in\operatorname{Sp}(\mathbb{R}^{2d})$ for each
$X\in\mathbb{R}^{2d}$, there exists
$\Phi^{g}_{X}\in\operatorname{Mp}(\mathbb{R}^{2d})$ such that

$\Pi(\Phi^{g}_{X})={Q}^{-1/2}_{X}\mbox{\quad}\text{and}\quad(\Psi^{g}_{X}a)^{w}=(\Phi^{g}_{X})^{*}a^{w}\Phi^{g}_{X},\quad
a\in\mathcal{S}^{\prime}(\mathbb{R}^{2d}),\;X\in\mathbb{R}^{2d}.$

Let $\tau_{X}$, $X\in\mathbb{R}^{2d}$, be a metaplectic operator

$\tau_{(x,\xi)}\kappa=e^{2\pi i\langle
y-x/2,\xi\rangle}\chi(\cdot-x),\quad\kappa(X,\Xi)\in\mathcal{S}(\mathbb{R}^{2d}).$

Clearly, $\tau_{(x,\xi)}=e^{-\pi i\langle x,\xi\rangle}\pi(x,\xi)$. Then,
since

$\Omega\tau_{X}\Omega^{*}=\tau_{\Pi(\Omega)X}\mbox{ for all
}\Omega\in\operatorname{Mp}(\mathbb{R}^{2d}),X\in\mathbb{R}^{2d},$

(see [10, Theorem 7.13, p. 205]) we infer that for each
$X,Y\in\mathbb{R}^{2d}$, $\pi({Q}^{1/2}_{Y}X)$ is equal to
$(\Phi^{g}_{Y})^{*}\pi(X)\Phi^{g}_{Y}$ up to a constant of modulus $1$. Thus,
(26) is equivalent to the following: for each $N\in\mathbb{N}$,

$(X,\Xi)\mapsto M((X+\Xi)/2)^{-1}(1+g_{\frac{X+\Xi}{2}}(X-\Xi))^{N}\\\
\times\left|\left\langle
a^{w}\pi(X)\Phi^{g}_{\frac{X+\Xi}{2}}\chi,\overline{\pi(\Xi)\Phi^{g}_{\frac{X+\Xi}{2}}\chi}\right\rangle\right|\in
L^{\infty}(W\times\mathbb{R}^{2d}).$

## Acknowledgements

S. Pilipović is supported by the Serbian Academy of Sciences and Arts, project
F10. N. Teofanov and F.Tomić were supported by the Science Fund of the
Republic of Serbia, $\\#$GRANT No 2727, _Global and local analysis of
operators and distributions_ \- GOALS. N. Teofanov gratefully acknowledges the
financial support of the Ministry of Science, Technological Development and
Innovation of the Republic of Serbia (Grants No. 451-03-66/2024-03/ 200125
$\&$ 451-03-65/2024-03/200125).

## References

  * [1] A. Abdeljawad, S. Coriasco,J. Toft, Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey-type pseudo-differential operators Anal. Appl. (Singap.) 18 (2020), 523–583.
  * [2] J. M. Bony, On the Characterization of Pseudodifferential Operators (old and new), in Studies in Phase Space Analysis with Applications to PDEs, pp. 21-34, Birkhäuser, New York, NY 2013.
  * [3] E. Cordero, K. Grochenig, F. Nikola, L. Rodino, Wiener algebras of Fourier integral Operators, J. Math Pures Appl., 99(2013), 219-233.
  * [4] E. Cordero, F. Nicola, L. Rodino, Gabor representations of evolution operators, Trans. Am. Math. Soc., 367 (2015), 7639–7663.
  * [5] E. Cordero, L. Rodino, Time-Frequency Analysis of Operators, De Gruyter Studies in Mathematics, 75 De Gruyter, 2020.
  * [6] E. Cordero, K. Gröchenig, F. Nicola, L. Rodino, Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class, J. Math. Phys. 55 (2014), no. 8, 081506, 17 pp.
  * [7] H. G. Feichtinger, Gewichtsfunktionen auf lokalkompakten Gruppen, Ōsterreich. Akad. Wiss. Math.-Natur. K1. Sitzungsber. II 188 (1979), no. 8-10, p. 451-471.
  * [8] H. G. Feichtinger, Banach convolution algebras of Wiener type, Colloquia mathematica Societatis János Bolyai, 35 1980, 509-523.
  * [9] H. G. Feichtinger, Modulation spaces on locally compact abelian groups Technical report, University of Vienna, 1983.
  * [10] M. de Gosson, Symplectic geometry and quantum mechanics, Springer Science & Business Media, 2006.
  * [11] I. M. Gelfand, G. E. Shilov, Generalized Functions II, Academic Press, New York, 1968.
  * [12] K. Gröchenig, Foundation of Time-Frequency Analysis, Birkhäuser, Boston, 2001.
  * [13] K. Gröchenig, Time-frequency analysis of Sjöstrand’s class, Revista Matemática Iberoamericana, 22(2) (2006), 703-724.
  * [14] K. Gröchenig, Z. Rzeszotnik, Banach algebras of pseudodifferential operators and their almost diagonalization, Ann. Inst. Fourier (Grenoble) 58 (2008), 2279–2314.
  * [15] K. Gröchenig, Yu. Lyubarskii, Gabor (super)frames with Hermite functions. Math. Ann. 345 (2009), 267–286.
  * [16] K. Grc̈henig, Weight functions in time-frequency analysis in ”Pseudodifferential Operators: Partial Differential Equations and Time-Frequency Analysis”, L. Rodino et al., eds., Fields Institute Comm. 52, 343 – 366, 2007.
  * [17] K. Gröchenig, J. Toft, The range of localization operators and lifting theorems for modulation and Bargmann-Fock spaces, Trans. Amer. Math. Soc. 365 (2013), 4475–4496.
  * [18] L. Hörmander, The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32(3) (1979), 359-443.
  * [19] L. Hörmander, The analysis of linear partial differential operators. III. Pseudo-differential operators, Reprint of the 1994 edition. Classics in Mathematics. Springer, Berlin, 2007.
  * [20] N. Lerner, Metrics on the phase space and non-selfadjoint pseudo-differential operators, Vol. 3. Springer Science & Business Media, 2011.
  * [21] Yu. Lyubarski, Frames in the Bargmann space of entire functions, Entire and subharmonic functions, 167-180, Adv. Soviet Math., 11, Amer. Math. Soc., Providence, RI, 1992.
  * [22] S. Pilipović, B. Prangoski, Anti-Wick and Weyl quantization on ultradistribution spaces, J. Math. Pures Appl. 103(2) (2015) 472–503.
  * [23] S. Pilipović, B. Prangoski, Characterisation of the Weyl-Hörmander classes by time-frequency shifts, Advances in Mathematics, Volume 410, Part B, 2022
  * [24] S. Pilipović, N. Teofanov, Pseudodifferential operators on ultra-modulation spaces, J. Funct. Anal., 208(2004), 194-228
  * [25] K. Seip, R. Wallsten, Density theorems for sampling and interpolation in the BargmannFock space. II, J. Reine Angew. Math., 429 (1992), 107?113.
  * [26] J. Toft _Images of function and distribution spaces under the Bargmann transform_ , J. Pseudo-Differ. Oper. Appl. 8 (2017), 83–139.