# GraphMuse: A Library for
Symbolic Music Graph Processing

###### Abstract

Graph Neural Networks (GNNs) have recently gained traction in symbolic music
tasks, yet a lack of a unified framework impedes progress. Addressing this
gap, we present GraphMuse, a graph processing framework and library that
facilitates efficient music graph processing and GNN training for symbolic
music tasks. Central to our contribution is a new neighbor sampling technique
specifically targeted toward meaningful behavior in musical scores.
Additionally, GraphMuse integrates hierarchical modeling elements that augment
the expressivity and capabilities of graph networks for musical tasks.
Experiments with two specific musical prediction tasks – pitch spelling and
cadence detection – demonstrate significant performance improvement over
previous methods. Our hope is that GraphMuse will lead to a boost in, and
standardization of, symbolic music processing based on graph representations.
The library is available at https://github.com/manoskary/graphmuse

##  1 Introduction

Symbolic music processing entails the manipulation of digital music scores,
encompassing various formats such as MusicXML, MEI, Humdrum, **kern, and MIDI.
Unlike audio-based representations, symbolic formats offer granular
information on note elements, including onset, pitch, duration, and other
musical attributes like bars and time signatures.

While prior research in symbolic music processing often adopted techniques
from the image processing [1, 2, 3] or natural language processing [4, 5, 6]
domains, recent attention has shifted towards graph-based models, which could
presumably better capture the dual sequential and hierarchical nature of
music. Graph Neural Networks (GNNs) have been showcased as potent tools for
diverse symbolic music tasks, including cadence detection[7], optical music
recognition [8], music generation [9], Roman numeral analysis [10], composer
classification [11], voice separation[12], and expressive performance
rendering[13]. However, a standardized framework for constructing and
processing music graphs has not yet been introduced to the field. To address
this challenge, we developed GraphMuse, a Python-based framework to
efficiently and effectively process information from musical scores, construct
musically meaningful graphs, and facilitate the training of graph-based models
for symbolic music tasks.

A key innovation of our work lies in the introduction of a new sampling
technique tailored to specific properties of music while maintaining efficient
and robust training of GNNs. Additionally, GraphMuse integrates within the
graphs and models hierarchical elements that augment the capabilities of graph
networks for musical tasks.

We evaluate our framework on pitch spelling and cadence detection tasks,
comparing it against existing state-of-the-art methods. Through the
synergistic utilization of our framework’s components, we achieve a
significant performance increase compared to the previous methods. Our
overarching objective is to establish a standardized framework for graph
processing in symbolic music analysis, thus catalyzing further progress in the
field.

Altogether, our contributions are three-fold: i) We provide a structured,
generic, and flexible framework for graph-based music processing; ii) we
release an open source Python library that comes with it; iii) we achieve
performance improvements in a principled way by focusing on the design of the
individual parts of the framework.

##  2 Processing Music Scores with GNNs

Figure 1: The general graph processing/training pipeline for symbolic music
scores involves several steps: i) Preprocess the database of scores to
generate input graphs; ii) Sample the input graphs to create memory-efficient
batches; iii) Form a batch as a new graph with nodes and edges from various
input graphs; iv) Sample a subset of nodes (target nodes) and their neighbors
from the input graphs; v) Update the target nodes’ representations through
graph convolution to create node embeddings; vi) Use these embeddings for
task-specific applications. Note that target nodes may include all or a subset
of batch nodes depending on the sampling strategy.

In this section, we describe existing graph modeling approaches for musical
scores. They all have a common pipeline which involves building a graph from a
given musical score (see Figure 1) and using a series of convolutional blocks
to produce context-aware hidden representations for each node. We start by
describing the graph-building procedure and a generic graph convolutional
block; we then take a detailed look at the problem of graph sampling, which
will motivate a new sampling procedure that will be presented in the next
section.

###  2.1 Preprocessing: Constructing Graphs from Scores

A score graph can be represented as a heterogenous attributed graph. A
heterogeneous graph has a type associated with each node and edge in the graph
[14]. An attributed graph has an associated feature vector for each node in
the graph [15]. Therefore, a heterogenous attributed graph is defined by a
quintuple $G=(V,E,X,\mathcal{A},\mathcal{R})$, together with the mappings
$\phi:V\to\mathcal{A}$ and $\psi:E\to\mathcal{R}$, where $V$ is the set of
nodes, $E$ is the set of edges, $X\in V\times\mathbb{R}^{k}$ the feature
matrix $\mathcal{A}$ is the node types and $\mathcal{R}$ is the edge types.
$\phi$ maps each node to its type and $\psi$ maps its each edge to its
corresponding type.

We create such a graph from a musical score by following previous work [12,
10, 13, 11]. Each node $v\in V$ corresponds to one and only one note in the
musical score. $\mathcal{R}$ includes 4 types of relations: onset, during,
follow, and silence, corresponding, respectively, to two notes starting at the
same time, a note starting while the other is sounding, a note starting when
the other ends, and a note starting after a time when no note is sounding. The
inverse edges for during, follows, and silence relations are also created.

Formally, let us consider three functions $on(v)$, $dur(v)$, and $pitch(v)$
defined on a note $v\in V$ that extract the onset time, duration, and pitch,
respectively. A typed edge $(u,r,v)$ of type $r\in\mathcal{R}$ between two
notes $u,v\in V$ belongs to $E$ if the following conditions are met:

  * •

$on(u)=on(v)\rightarrow{}r=\textrm{onset}$

  * •

$on(u)>on(v)\land on(u)\leq on(v)+dur(v)\rightarrow{}r=\textrm{during}$

  * •

$on(u)+dur(u)=on(v)\rightarrow{}r=\textrm{follow}$

  * •

$on(u)+dur(u)<on(v)\land\nexists v^{\prime}\in V,\;on(v^{\prime})<on(v)\land
on(v^{\prime})>on(u)+dur(u)\rightarrow{}r=\textrm{silence}$

$\mathcal{A}$ in the literature usually only includes a single type, i.e. the
note type $\nu$. However, we extend this definition in Section 3.1.

###  2.2 Encoding: Graph Convolution

Graph convolution and message passing are core operations in graph neural
networks (GNNs) for learning node representations. In graph convolution, in
its simplest form, each node aggregates messages from its immediate neighbors
by computing a weighted sum of their features:

$\mathbf{h}_{v}^{(l+1)}=\sigma\left(\left(\sum_{u\in\mathcal{N}(v)}\mathbf{W}^{(l)}\mathbf{h}_{u}^{(l)}\right)+\mathbf{h}_{v}^{(l)}\right)$
(1)

where $\mathbf{h}_{v}^{(l)}$ is the representation of node $v$ at layer $l$,
$\mathcal{N}(v)$ denotes the neighbors of node $v$, $\mathbf{W}^{(l)}$ is a
learnable weight, and $\sigma$ is a non-linear activation function. Through
successive iterations of message passing and aggregation, each node refines
its representation by incorporating information from increasingly distant
nodes in the graph, ultimately enabling the network to capture complex
relational patterns and dependencies within the graph data.

In the context of music, graph convolution can be understood as a method for
defining a note not only by its own characteristics and properties but by also
considering the characteristics of its neighboring notes within the musical
graph. In this work, as well as previous graph-based work on music [7, 10, 11]
the preferred graph convolutional block is SageConv taken from one of the
first and fundamental works in graph deep learning [16].

###  2.3 Sampling: Handling Graph Data for Training

In an ideal world without computing resource considerations, we can imagine a
training pipeline that receives an entire graph as input to a graph
convolutional model. Assuming that we have the resources and time to perform
such a task the process is easy to grasp. All nodes of the graph are updated
in a single step based on their neighbors as described in the previous
section.

However, the graph world presents us with several complexity issues. Graph
datasets in the wild typically come in two forms: i) a (possibly large)
collection of small graphs, each containing maybe fewer than 50 nodes[15]; ii)
a single large-scale graph such as a social network [17], a recommender system
[18], or a knowledge graph [19]. The previous naive scenario presents a time-
efficiency and computation waste bottleneck for the former and a memory
insufficiency issue for the latter. To mitigate these issues, in the former
case one can batch many small graphs together to maximize the available
resources and reduce the computation time, then the full graphs can be updated
during convolution within each batch.

Training Graph Convolutional Networks (GCNs) for large-scale graphs is a bit
more complicated. Such graphs can be exceptionally large – for example, the
2019 Facebook social network boasted 3.51 billion
users111https://zephoria.com/top-15-valuable-facebook-statistics. To train
models with such graphs we need to devise a sampling algorithm to derive
subgraphs in steps [16, 20, 21, 22]. Such an algorithm may, for example,
choose a subset of nodes across the graph and perform random walks to fetch a
subset of the k-hop neighbors for the sampled nodes [16]. This process, called
neighbor sampling or node-wise sampling, is shown in Figure 2 and compared to
the full-graph process.

Figure 2: Full graph vs neighbor sampling. The pink-colored nodes are selected
for convolution by message passing. With neighbor sampling, the pink node is
the one whose representation is ultimately updated after convolution (however,
for the blue nodes also take part in the convolution process as its context).

Musical score graphs fall in between the two scenarios, varying notably in
size. For instance, a Bach Chorale might contain 100 notes, while a Beethoven
Sonata could exceed 5000 notes, with each note corresponding to a graph node.
Furthermore, a musical dataset may contain many such graphs. Therefore the
question arises how to efficiently train models on music graph datasets.

Since music graphs are not uniform enough to be batched together like small
graph datasets, we investigate the suitability of neighbor sampling methods
for music graph processing, taking into account special properties relevant in
music. Standard neighborhood sampling would sample notes across different
scores and fetch neighbors for those notes, creating a subgraph that can
maximize the use of the available resources during training.

However, music has a specific coherence, in both the horizontal (time) and
vertical (chords, harmonies) dimensions, which makes sampling approaches from
the literature[22] not appropriate for music. Specifically, sampling and
updating/encoding single notes without simultaneously doing so also to notes
in their local context makes it difficult to learn properties that persist in
time (such as local key or a harmonic function). In this work, we address this
issue by presenting a simple and musically intuitive sampling process for
graphs that efficiently creates batches containing musically related notes
which, as experiments will show, can notably improve the learning results.

###  2.4 Task-specific Modeling

Finally, the node embeddings created by the graph convolutional encoder serve
as input to task-specific models that solve some specific prediction or
recognition task. In a graph context, we distinguish, at an abstract level,
between node classification, link prediction, and entire graph classification
tasks. Examples of node classification tasks can be found in [7] which takes
the embeddings from the GCN encoder and employs an edge decoder coupled with a
graph convolution classifier for cadence prediction labels; and in [10], which
forwards the embeddings to sequential layer and then MLP classifiers to
perform Roman Numeral Analysis. In [12], musical voice separation is framed as
a link prediction task; the node embeddings are input to a pairwise edge
similarity encoder to predict link probabilities between notes in the same
voice. An example of a graph classification task can be found in [11] where
the embeddings are aggregated and passed through a shallow MLP for composer
classification.

Naturally, task-specific models will not be part of the generic graph
processing pipeline and library which we publish with this paper.

##  3 Methodology

Figure 3: Sampling process per score. Top: sampled notes and their neighbors;
middle: score graph and sampling process; bottom: sampling process for beats
and measures. A randomly selected note (in yellow) is first sampled. The
boundaries of the target notes are then computed with a budget of 15 notes in
this example (pink and yellow notes). Then the k-hop neighbors are fetched for
the targets (light blue for 1-hop and darker blue for 2-hop). The k-hop
neighbors are computed with respect to the input graph (depicted with colored
edges connecting noteheads in the figure). We can also extend the sampling
process for the beat and measure elements (introduced in Section 3.1). Note
that the k-hop neighbors need not be strictly related to a time window.

In this section, we discuss our approach to addressing the different
components of the pipeline shown in Figure 1. In particular, we explain the
preprocessing procedure for creating score graphs, we detail our strategy for
musically intuitive graph sampling, and finally, we discuss model variants
that are made possible by the previous steps of the pipeline.

###  3.1 Preprocessing

The central activity in the preprocessing step is the creation of graphs from
musical scores. In our library, we extend the conventional graph creation
process by introducing hierarchical musical dimensions (beats and measures),
in order to enhance the score graphs’ representational capacity. More
specifically, we enrich the node type set $\mathcal{A}$ (defined in Section
2.1) with two additional types $\beta$ and $\mu$ for beats and measures
respectively. The process involves detecting beats and measures within the
musical score, generating edges (of type connect to every beat from each note
falling within its temporal boundaries, and repeating this process for
measures. Additional edges of type next are drawn between consecutive beats
and measures to enrich the connectivity and contextual understanding within
the graph. Furthermore, we aggregate features from constituent notes through
the connect edges via message passing to equip each beat and measure with
informative attributes by computing the mean vector of their note features.

The inclusion of beat and measure node elements, as well as the creation of
inverse edges, are made optional, ensuring compatibility with diverse research
needs and avoiding imposing rigid structures onto the graph-based music
processing framework.

We prioritize the efficiency and speed of the graph creation process by
transitioning the graph creation implementation to C code, leveraging its
performance benefits, and establishing a Python binding for seamless
integration into our workflow. Recognizing the temporal nature of musical
elements, such as notes, beats, and measures, we refine our neighbor searching
windows accordingly, optimizing computational efficiency.

###  3.2 Sampling

We discussed general neighbor sampling for large-scale graphs in Section 2.3
and some problems related to graph-structured music data. In this section, we
elaborate on our musically informed sampling process for music graphs, which
enables the training of the models outlined in the subsequent sections. In
this process, we aim to sample sections of scores and employ neighbor sampling
to fetch the neighbors of notes within those sections.

Indeed while our nodes could be ordered in various ways, the most perceptually
significant aspect is time organization. Recognizably, individuals can still
identify a musical piece when segmented along the time axis, whereas focusing
solely on pitch intervals may be challenging. Moreover, perceptual research
indicates that the commencement time of a note holds greater salience than its
offset time, particularly for percussive instruments like the piano, where the
sound naturally fades over time [23]. Hence, when constructing graphs from
musical scores, we prioritize node arrangement based on absolute onset time
followed by pitch.

Our initial limitations are mostly related to memory usage. To limit our
memory we need to predefine three initial arguments: i) the size of each
target subgraph $S$ from every score, ii) the number $B$ of scores in each
batch, and iii) the number of hops and neighbors for each hop (similar to
node-wise sampling techniques). In each batch, we update the representation of
our target nodes which is essentially the size of $S\times B$.

Once the ordering is set and the three arguments are defined we can initiate
the process of sampling a subgraph, as shown in Figure 3. First, we sample a
random note from the graph of each score. Next, we correct the position of the
note by searching for any vertical neighbors (same onset value notes and
potentially different pitch). Then we extend to $S$ notes to the right where
$S$ indicates a predefined maximum subgraph size. We also correct the
rightmost boundaries to include or exclude vertical neighbors for the last
onset always respecting the aforementioned size $S$. Once this process is
completed we obtain the target nodes per score within the batch. These are the
nodes whose representation we want to update at the end of the graph
convolutional process.

However, since graph convolution is performed recursively we need to fetch the
$k$-hop neighbors for each one of the target nodes where $k$ indicates the
depth of the GCN. For this step, we can consult the literature [16] and
perform neighbor sampling to fetch the $k$-hop neighbors. This process is
repeated for $B$ different scores. Finally, the $B$ score subgraphs of size at
most $S$ each are first joined together and then fed to the model.

During this process, we can keep information about the target nodes and the
size of each score subgraph, which could allow us to design more creative
models that can exploit this information. Such models are presented in the
next section. Moreover, we adopt a potential approach for hierarchical graphs
by also extending the sampling for beat and measure nodes as shown in Figure
3.

###  3.3 Model Designs

In this section, we explore various model designs for the graph-based encoder
in our processing pipeline (Figure 1). Designing such an encoder involves
addressing two fundamental questions: the selection of graph convolutional
blocks and the selective exploitation of information from the input graph.

The first question, regarding graph convolutional blocks, remains open-ended,
offering numerous possibilities for exploration and customization. In its
current version, GraphMuse offers the options of convolutional blocks on a
per-node or per-edge type basis. We suggest that graph-attention networks may
offer promising avenues, particularly for hierarchical elements such as beats
or measures.

In response to the second question, we devise a series of models by
selectively incorporating or excluding elements from the input graph. Our
foundational model, termed NoteGNN, exclusively utilizes note elements and
their corresponding edges. This model serves as the basis for further
extension. For instance, we expand upon NoteGNN to construct BeatGNN, which
incorporates beat elements (see Section 3.1 above) alongside notes. Similarly,
we develop MeasureGNN by integrating measures into the encoding process. When
all note, measure, and beat elements are included, the resultant model is
denoted as MetricalGNN.

Furthermore, we explore the possibility of hybridizing model types, such as
combining GNNs with sequential models. This hybridization is facilitated by
the sampling process that organizes notes in onset order, allowing for the
batch to be unfolded by score. Consequently, the same batch can be processed
through both GNN and sequential models simultaneously. Specifically, we employ
a graph encoder and a sequential encoder in parallel – in our case we use a
stack of 2 bidirectional GRU layers. The GRU stack receives the unfolded batch
of size $(B,S,K)$ where $B$ is the number of scores within the batch, $S$ is
the number of sampled target nodes for each score order by onset and then by
pitch, and $K$ is the number of node features. The embeddings of both encoders
are concatenated together and an additional linear layer is applied to project
them to the required dimension.

This architecture, which we call HybridGNN in our experiments, combines the
strengths of both GNNs and sequential models, resulting in better performance
as shown in our experiments.

###  3.4 The Library

The components discussed in the preceding section have been implemented and
made available in an open-source Python library called GraphMuse. This library
follows a similar philosophy as PyTorch and PyTorch Geometric, comprising
models and graph convolutional blocks, loader pipelines, data pipelines, and
related utilities. GraphMuse is built upon and thus requires PyTorch and
PyTorch Geometric. The loaders and models provided by GraphMuse are fully
compatible with those of PyTorch Geometric. For musical input and output,
GraphMuse is compatible with Partitura [24], a Python library for symbolic
music processing, allowing it to work with a variety of input formats such as
MusicXML, MEI, Humdrum **kern, and MIDI.

##  4 Evaluation

To evaluate our framework, we perform experiments on two tasks, cadence
detection and pitch spelling. We put to the test both the models discussed as
well as the sampling process. For pitch spelling, we compare our models to the
previous sequential state-of-the-art model, PKSpell [25] and the GraphSAGE
variant of our note-level model. For cadence detection, we compare our models
to the previous state-of-the-art model by Karystinaios and Widmer [7] which is
also graph-based and follows a GraphSAGE sampling strategy. For both tasks, we
perform ablations by removing the hierarchical elements such as beat and
measure nodes and edges, or incorporating hybrid models. This work focuses on
the application of the GraphMuse library therefore, a detailed comparison of
various input encodings and architectures, as conducted by [11], is beyond the
scope of this paper.

###  4.1 Pitch Spelling

Previous work on Pitch Spelling set the state-of-the-art by using a sequential
model [25]. The task of pitch spelling tackles in parallel key signature
estimation and pitch spelling estimation per note, however, the key signature
is a global attribute usually set for the whole piece although it can
sometimes change midway. The previous architecture uses a GRU encoder for
pitch spelling and then infuses the logits together with the latent
representation to another GRU layer for the key signature prediction.

For our approach, we use a GNN encoder as described in Section 3.3 followed by
two classification heads for key and pitch spelling respectively. We train and
evaluate all models on the ASAP dataset [26] using a random split with $15\%$
of the data for testing and the $85\%$ for train and validation as described
in [25].

###  4.2 Cadence Detection

For the cadence detection model, we chose to use a modified version of the
cadence detection model originally proposed in [7]. Our considerations were
based on a more efficient training process, and the integration of our
pipeline possibilities. The model was expanded to accept a heterogeneous score
graph as input, as described in Section 2.1. Additionally, we enhanced the
model’s predictive capabilities from binary (no-cad or PAC) to multiclass
cadence prediction, encompassing PAC, IAC, and HC labels. Furthermore, we
refined the architecture by incorporating an onset regularization module,
which aggregates the latent representations (post-GNN encoder) of all notes
occurring at a distinct onset within the score and assigns them to every note
sharing that onset.

In the training phase, the input graph first undergoes processing through the
graph encoder. The resulting node embeddings are then grouped based on onset
information extracted from the score, and their representations are averaged.
Subsequently, embedded SMOTE [27] is applied to balance the distribution of
cadence classes compared to the notes lacking cadence labels in the score.
However, during inference, this synthetic oversampling step is omitted.
Finally, the oversampled embeddings are fed into a shallow 2-layer MLP
classifier to predict the cadence type.

We trained our model with a joined corpus of cadence annotations from the DCML
corpora222https://github.com/DCMLab/dcml_corpora, the Bach fugues from the
well-tempered clavier Book No.1 [28], the annotated Mozart string quartets
[29], and the annotated Haydn string quartets [30]. Our joined corpus makes
for $590,149$ individual notes and $17,188$ cadence annotations. We use 80% of
the data for training and validation and test on 20% using a random split.
Note that these results cannot be directly compared with [7] since we use a
different (bigger) dataset and perform multiclass prediction.

###  4.3 Experiments

####  4.3.1 Configuration

The configuration for training pitch spelling graph models with our sampling
technique uses a batch size $B=300$, sampling from 300 scores at each step,
and target node size $S=300$. For cadence graph models, $B=200$ and $S=500$.
All graph models, including GraphSAGE, utilize three heterogeneous SageConv
layers with a hidden size of 256 and a dropout of 0.5. Neighbor sampling for
each layer fetches up to three neighbors per sampled node per relation. We
train all models with the Adam optimizer (learning rate $10^{-3}$, weight
decay $5\times 10^{-4}$) on a GTX 1080 Ti. Each experiment is repeated at
least four times with different random seeds, and statistical significance
testing is performed using the ASO method at a confidence level $\alpha=0.05$
[31] 333For the detailed experiments visit:
https://wandb.ai/melkisedeath/GraphMuse.

####  4.3.2 Results

Table 1 presents the results of experiments experiments conducted on the two
tasks. The metrics used for evaluation are Accuracy (A) for pitch spelling and
key recognition, and the macro F1 score (F1) for cadence detection. Note that
the model employed on the GraphSAGE methods and the model NoteGNN are
virtually the same apart from the sampling strategy with which they were
trained.

For the pitch spelling task, we can observe that the actual pitch spelling
accuracy (A-Pitch) of all proposed models surpasses both the PKSpell and
GraphSAGE methods. Across all models, the MetricalGNN achieves the highest
accuracy of $95.6\%$, closely followed by BeatGNN and MeasureGNN with
accuracies of $95.1\%$ and $95.4\%$, respectively. These results indicate the
benefits of incorporating hierarchical musical elements such as beats and
measures. However, it is worth noting that while MetricalGNN achieves the
highest accuracy, it is closely followed by the hybrid model, HybridGNN, which
achieves an accuracy of $95.4\%$, suggesting that competitive performance can
also be achieved by mixing model types.

Focusing on the key estimation subtask (A-Key) of pitch spelling we notice
that the PKSpell model achieves a very good key accuracy of $69.9\%$, closely
followed by the MeasureGNN model and only surpassed by the Hybrid model. We
attribute the effectiveness of key detection of a sequential model such as
PKSpell to the persistence of the key label across elements of the sequence.
Therefore, a hybrid model in this case seems to be able to adapt to the
diversity of labels for pitch spelling and uniformity of labels for key
estimation. We found our best model to be stochastically dominant over PKSpell
with $min\\_\epsilon=0.17$.

In the cadence detection task, we evaluate the results using the macro F1
score to account for the overwhelming presence of non-cadence nodes, as
instructed by [7]. We observe that GraphSAGE, the previously used technique
for training, obtains the lowest F1 score and it is surpassed by all the
proposed GNN-based models trained with the new sampling method.

Among our GNN models, BeatGNN and HybridGNN achieve the highest scores of
$57.4\%$ and $58.6\%$, respectively, closely followed by MeasureGNN. In this
case, the MetricalGNN model surprisingly does not achieve such a good score
even though it includes both measure and beat elements. However, it still
performs better than the NoteGNN and the GraphSAGE method.

Overall, the results demonstrate the efficacy of GNN-based models trained
using our new sampling method. Incorporating hierarchical elements such as
beats and measures improves both pitch spelling and cadence detection tasks.
Additionally, the hybrid approach of combining GNNs with sequential models
produces promising results.

Task | Pitch Spelling | Cadence
---|---|---
A-Pitch | A-Key | F1-Cad
PKSpell | $94.8\pm 0.5$ | $\underline{69.9}\pm 1.6$ | -
GraphSAGE | $93.6\pm 0.1$ | $43.3\pm 0.1$ | $53.5\pm 0.8$
NoteGNN | $94.9\pm 0.1$ | $69.3\pm 7.0$ | $55.3\pm 0.9$
BeatGNN | $95.1\pm 0.2$ | $68.7\pm 1.1$ | $\underline{57.4}\pm 1.2$
MeasureGNN | $\underline{95.4}\pm 0.3$ | $69.5\pm 7.2$ | $57.0\pm 1.0$
MetricalGNN | $\textbf{95.6}\pm 0.1$ | $64.4\pm 5.3$ | $55.8\pm 0.6$
HybridGNN | $\underline{95.4}\pm 0.2$ | $\mathbf{72.6}\pm 2.8$ | $\mathbf{58.6}\pm 0.7$
Table 1: Results on the two tasks, in terms of accuracy (A) and F1 score,
respectively. Values in bold are the best score per metric; underlined values
are the second best. All runs are repeated 4 times. $\pm$ indicates standard
deviation.

##  5 Conclusion

In this paper, we introduced GraphMuse, a framework and Python library for
symbolic graph music processing. We designed a specialized sampling process
for musical graphs and demonstrated our pipeline’s effectiveness through
experiments on pitch spelling and cadence detection. Our results show that
carefully designed GNN architectures, especially those incorporating
hierarchical elements like beats and measures, can lead to better performance.
Finally, hybrid models that integrate GNNs with sequential models yield
further performance improvements.

Future research will focus on refining GNN-based models in music processing,
adding more tasks, and exploring novel architectures. This includes
investigating advanced graph convolutional blocks, other sampling techniques,
and attention mechanisms to enhance model performance.

##  6 Acknowledgements

The authors would like to thank Nimrod Varga for his contribution to
accelerating graph creation by adapting it to C code. This work is supported
by the European Research Council (ERC) under the EU’s Horizon 2020 research &
innovation programme, grant agreement No. 101019375 (Whither Music?), and the
Federal State of Upper Austria (LIT AI Lab).

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11institutetext: University of Washington WA 98195, USA
11email: {rk1997<EMAIL_ADDRESS>
{lopeze7, kjoleson, tonywli<EMAIL_ADDRESS>
# Share with Me: A Study on a Social Robot Collecting Mental Health
Data††thanks: This study was funded in part by National Science Foundation:
National Robotics Initiative, SES: Award Abstract 1734100 - Design and
Development of a Social Robot for Gathering Ecological Momentary Stress Data
from Teens.

Raida Karim 11 0000-0002-2854-3985 Edgar Lopez 11 0000-0001-6835-855X Katelynn
Oleson 11 Tony Li 11 0000-0001-7552-6689 Elin A. Björling 11
0000-0002-0385-2562 Maya Cakmak 11 0000-0001-8457-6610

###### Abstract

Social robots have been used to assist with mental well-being in various ways
such as to help children with autism improve on their social skills and
executive functioning such as joint attention and bodily awareness. They are
also used to help older adults by reducing feelings of isolation and
loneliness, as well as supporting mental well-being of teens and children.
However, existing work in this sphere has only shown support for mental health
through social robots by responding interactively to human activity to help
them learn relevant skills. We hypothesize that humans can also get help from
social robots in mental well-being by releasing or sharing their mental health
data with the social robots. In this paper, we present a human-robot
interaction (HRI) study to evaluate this hypothesis. During the five-day
study, a total of fifty-five (n=55) participants shared their in-the-moment
mood and stress levels with a social robot. We saw a majority of positive
results indicating it is worth conducting future work in this direction, and
the potential of social robots to largely support mental well-being.

###### Keywords:

Social Robot Mental Health Data Sharing.

##  1 Introduction

Apart from genetic or birth related causes, any kind of mental health issues
in our daily lives are usually caused by some kind of trouble such as
stressful events, grief from accidents or deaths, and trauma from tragic or
fearful incidents [19]. Research shows that sharing these troubling
experiences helps gain insights from others which helps develop coping skills,
and makes one feel less alone in pain which helps to tackle trouble more
effectively [3][12]. Therefore, sharing about trouble can potentially help
treat or lessen mental health issues. However, people are not always willing
to share their mental health with others or publicly, even with their family
members, let alone outside community members due to social stigma, personal
beliefs or limitations [18]. Therefore, to tackle mental health issues, the
main source of support is usually individualized therapy, which is expensive
and thus inaccessible to many [7][17]. Digital therapy is a relatively more
accessible option compared to in-person therapy because of lower cost and
availability in any location [1][20]. However, digital therapy has drawbacks,
such as lack of face-to-face interaction, user disengagement, and software
incompatibility [8]. So, we hypothesize that sharing mental health with an
endearing social robot that overcomes these challenges of therapy options can
help support mental health.

##  2 Related Work

When people share their personally relevant emotions in social media like
Facebook, they experience satisfaction causing positive mental state [3].
People tend to be discreet about sharing their emotional data, as we see
Facebook users share more intense and negative emotions in private messages
[3]. Prior work that looked into sharing one’s emotions through technologies
like a virtual mood wall reported a positive relation between online emotional
sharing and comfort in negative emotions [12]. Social robots are often
perceived as friendly or pet-like companions by humans for their endearing
appearances (e.g., outfit, facial expressions) and capabilities that can
include haptics, sounds, and movements [16]. Such perceptions or relationships
boost user engagement with social robots [4]. Thus, social robots can better
foster emotional support with mental health data compared to other
technologies (e.g., smartphone apps, websites rendered by touch screen
devices) [6]. In a study conducted by Björling et al. [5], teens reported
social robots as a plausible source of emotional support. Additionally, social
robots have been studied in the context of assisting mental healthcare by
sharing its emotions with people [11], having behavioral models fulfilling
user needs [15], and providing companionship [9].
To the best of our knowledge, the existing literature in this domain does
indicate that social robots have not yet been used as a means for collecting
mental health data from humans, which makes this research direction a novel
one in HRI.

##  3 Sharing Mental Health Data with a Social Robot

###  3.1 Study Setup

We conduct a five-day study with some people showing to share their mental
health data with a social robot, and based on their reactions, we analyzed the
potential of such interactions in improving mental well-being. The goal of
this study is to evaluate our hypothesis as stated before. The research
questions are:
(A) How will participants feel about sharing their mental health data with a
social robot?
(B) What information can we get about a community’s mental health from its
participants’ shared mental health data with a social robot?
We plan to investigate these research questions with findings from the study
results as discussed later.

Figure 1: Participants sharing mental health data with a social robot during
the study.

###  3.2 Sample

We conducted the study with a total of fifty-five (n=55) participants in an
American university campus, specifically a top Computer Science & Engineering
(CSE) school. The location chosen was the main entrance atrium of the CSE
department easily accessible to students and staff to interact with the robot.
Participants interacted with a social robot shown in Fig. 1 to share their
mental health data in this five days’ study.

###  3.3 Procedure

We asked the participants to share their mood and stress levels as a means to
share their mental health data with a social robot. A previous study conducted
by our lead author [10] showed that rather than using English words like
“Good”, “Bad”, and “Fine” to express different mood and stress levels, using
an emoji Likert scale can enhance coherence and accessibility by reducing the
need to read text and increase dependency on globally understandable graphics
and symbols, especially in diverse communities like in the United States. Thus
we used an emoji likert scale with 3 buttons expressing 3 different levels of
mood or stress data.
Before performing the study in the CSE department, we conducted a pilot
study111A video of the pilot study of the mental health data sharing
interaction by one participant with a social robot can be found here:
https://youtu.be/J6srKDg6OE0 with a female participant in order to ensure that
the planned experimental interaction can help us find out more about the
participants’ shared mental health data as expected. The brief (about 20
seconds) interaction had 3 stages: greeting, mood and stress data sharing, and
exit. In the data sharing or second stage, participants were asked to input
their mood and stress data by clicking on the appropriate emoji as answers to
these two questions, respectively: “How do you feel today?” and “Are you
stressed today?” Their shared data responses were then stored in a secured
Firebase database 222Firebase: https://firebase.google.com/. We decorated the
robot with a colorful hat and body frame to make it look endearing (see Fig.
1). Apart from showing the questions in its belly screen, there were sounds
incorporated in the robot that asked the questions aloud in a female voice in
English language. The robot also had head and neck movements in different
stages of the interaction. The first three authors of this paper invited
passerby on different days verbally to participate in the study. After each
passerby completed the interaction, they were interviewed to respond either
verbally or by writing in post-its to a prompt as shown in Fig. 2.

###  3.4 Results

The results of the study include participants’ comments responding to the
prompt of “Sharing my mental health data makes me feel….” (see Fig. 2) and the
shared mood and stress level data through emoji counts. We observed in these
data that a positive mood was prevalent on Monday, Tuesday, Thursday, and
Friday. A neutral mood was prevalent on Wednesday and Friday. A negative mood
was not prevalent in this community on any days in that week. In terms of
stress data, high stress was prevalent on Wednesday, medium stress was
prevalent on Monday, Tuesday, Thursday, and Friday, and low stress was
prevalent on Thursday. Of course, these observations are true for that
specific week, and might not necessarily hold true for other weeks. We saw the
lowest positive comments on Friday, which may mean this community is exhausted
after a long busy week, and thus did not think much or see many positive
aspects of this new type of interaction.

Figure 2: Written Prompt Used in the Study & Responses in Post-its.

###  3.5 Analysis

The lead author of this paper conducted a thematic analysis of the study data
containing prompt responses of the study participants as shown in Fig. 2. A
specific theme was assigned based on the nature of prompt response. For
instance, comments like “Sharing my mental health data makes me feel more
aware of my stress.” has been assigned to the theme of “self-awareness”. And
comments like “I am a bit scared about sharing my data but hope it will go
towards a good cause.” has been assigned to the theme of “privacy” as it shows
concerns about how participants’ data will be used and possible risks to their
privacy.
The themes are mostly positive supporting our hypothesis that sharing mental
health data with a social robot can support mental healthcare. The only
negative themes involved privacy concerns about data sharing and feedback for
improvement on robot design. We can mitigate these negative themes by
incorporating participants’ feedback on the robot’s design and ensuring
participants about the privacy of their shared data in our secured data
storage. One observation we made was that this community is part of an
academically competitive department, so these results seem to align with this
competitive mentality. For instance, from overall community counts, it seems
this community most commonly has a good or positive mood and medium stress.
Good mood could be a result of the satisfaction and achievement that comes
with being part of a prestigious academic department, and being able to pursue
the profession they are very passionate about in a competitive place. The
medium stress might result from the busy nature of academic or engineering
work– deadlines of homework, projects, papers, presentations, demos, etc.,
that all are constantly working on.

We discuss specific themes below:
Theme 1: Emotional Sharing
Emotions revealed by users’ post-interaction comments ranged between positive
(e.g., “happy”, “good, could be myself”), unchanged (e.g., “Neutral. No change
in how I feel”, “didn’t feel anything”), and validated (e.g., “Accepted”,
“welcomed, nice to check-in with someone”, “Heard, seen, like someone cares
:)”).
Implications: This theme reinforces that sharing humans’ mental health data
with a social robot can indeed have various effects in their emotions. These
findings can inform future designs of data sharing interaction with social
robots targeted to impact the users’ mental health for the better.
Theme 2: Robot Interaction
Some users expressed how they felt interacting with a robot specifically while
sharing their mental health data. One type of comments expressed the robot’s
figure positively impacting mental health: “I feel relaxed by the face”, “It
made my day to share with the robot”.
Another type of comments expressed doubts in the robot’s capability: “As it’s
a robot, I know that it will lack human level empathy. The set of responses
would be fixed. I knew it won’t be capable of doing much like releasing my
stress.”, “confused, not sure what happened or what’s happening here, does it
have voice and movements”.
Implications: These findings indicate the need to design or use social robots
that can transparently and positively support mental health data sharing
interaction. In general, the term “creepy technology” is increasingly
appearing in literature along with concerns for ethical boundaries,
cybersecurity, privacy, and mistaken identity [13]. Creepiness has been noted
to have three core factors– implied malice, undesirability, and
unpredictability– in the first work that devoted detailed attention to
creepiness in the context of interactive technologies [22]. Therefore, when
designing these interactions, ensuring transparency, desirability,
predictability, and privacy can help avoid the robot being perceived as
creepy.
Theme 3: Inward-Looking
This theme is comprised of comments that show users contemplating themselves
during the interaction. Some keywords users conveyed: “reflective”,
“Transparent”, “introspective”.
Implications: These findings suggest this interaction’s ability to compel
humans for self-reflection, which can inform future design of such
interactions to allow space for reflection.
Theme 4: Impact on Stress Level
This theme reveals how this interaction impacts users’ stress levels. Some
comments demonstrated alleviation in stress: “Less Stressed”, “like some
weight has been lifted off of me”, whereas one comment shows no change in
stress level: “same level of stress”.
Implications: These findings show potential for this type of interaction as a
means of alleviating stress. Future research needs to be conducted to solidify
this possibility. Diverse communities, targeted interaction experiments, and a
larger number of participants can be used to conduct further research in
different settings.
Theme 5: Community Benefit
This theme suggests how this interaction could be beneficial to the community.
One comment towards this: “helpful to others”.
Implications: These findings point out the need to conduct further research to
evaluate how this interaction can be beneficial for people or communities.
Depending on in what ways these can be beneficial, the design of such
interactions or used social robots could be modified for maximum benefit in
specific contexts.
Theme 6: Feedback
This theme contains comments by users as suggestions for improvement. Some
comments on interactions: “weird…as I’m not sure if my mental health can be
quantified by data alone”, “unusual”, and one comment on robot design: “It
didn’t feel like a robot. Maybe one integrated system might make it feel like
a robot. A frame around the belly might help hide the belly ipad, and make it
look like an integrated system. But, you can think more about it like arms
coming out, or to make it more humanoid; maybe remove distance between face
and belly. If the question appears in the face and I answer in the belly, it
might be more like an integrated or full system interaction.”
Implications: These findings indicate some concrete areas for improvement in
this interaction and robot design. In general, mental health is a bit abstract
and it can be hard to design convincing short (e.g., 20-seconds) interactions
to collect accurate mental health data. Incorporating these feedback and
conducting further research might lead to new developments in this regard.
Theme 7: Self-Awareness
This theme shows user comments with regards to increased awareness in various
ways, such as: “more aware of my stress”, “more conscious about my overall
health, since mental health is even more important than physical health in my
opinion :)”, “I feel more self-aware”.
Implications: These findings are promising in the context of using this type
of interactions in different communities as a means of raising awareness of
mental health care or fighting associated stigma.
Theme 8: Privacy
This theme indicates users’ feelings when sharing their “private” mental
health data in this interaction. Some comments didn’t express any concerns: “I
typically don’t like sharing my data, but I liked this one. I liked that there
were options given that I could choose from, so I didn’t have to think a lot
about my mood, etc.”, “I am not concerned at all about sharing my data this
way, as it seems low granularity. Not invasive.” Some other comments showed
users’ concerns: “I am a bit scared about sharing my data but hope it will go
towards a good cause.”, “I’m sort of uncomfortable w/ it but it’s probably
better if I do share.”
Implications: These findings demonstrate the need for further work in making
users feel comfortable and gaining their trust with their private data shared
with a robot. Establishing a good relationship with users may increase
effectiveness of such interactions.

##  4 Limitations

As shown in Fig. 1, the robot used in this study had a belly and a face
covered with external outfits which made it look a bit like a humanoid robot
[14] perhaps making participants feel they are sharing their mental health
data with a human-like companion or friend. Prior work showed humans prefer
interacting with human-looking robots [2]. So, the results of this study might
be biased towards this type of robot compared to that of a non-humanoid robot
design like a Knightscope robot [21]. Therefore, based on robot designs this
study results might vary, and thus we cannot claim this study’s inferences are
true for all social robots. Another potential limitation is data bias. As we
collected data through the robot placed in an open space of a community, this
might have caused data bias. For instance, if the user is aware that the study
coordinators or other community members can observe them when they are
entering their data, they might not be truthful (especially if it’s negative
or sensitive mental health data, they might feel hesitation or shame to share
that in public). This could be the reason why we didn’t get very high counts
of negative data (high stress, negative mood, or negative comments about
mental health). One way to avoid this bias might be to experiment with data
collection in a closed space. For example, placing the robot in a closed room,
where only the user can enter to share their data, or placing a cover around
the belly of the robot that will not allow the user’s interaction (or at least
what data they are entering) to be seen.
Lastly, the medium of data collection can pose limitations. Instead of using
an emoji Likert scale, if we had used a 10-pt slider scale on a 0-9 range, we
might have more nuanced data of mood and stress levels indicated by specific
values (e.g., 7) as well as provided participants with more options to
accurately express their mental health data.

##  5 Conclusion & Future Directions

To conclude, our study findings revealed mostly positive feelings about this
type of human-robot mental health data sharing interaction among participants,
as well as different arousal this interaction can have on people such as self-
awareness, emotional sharing, etc. as discussed in the “Analysis” section
above. The majority of positive results indicate such interactions with social
robots need to be studied further in different settings to evaluate their
potential in mental healthcare holistically. Future studies can be conducted
with some changed variables in current study settings such as a different
location (e.g., hospital, high schools, public library, online), population
(e.g., teens, seniors, wheelchair users), and robot (e.g., NAO, Moxi). It
would also be interesting to gather information about participant demographics
(e.g., age, ethnicity, citizenship, gender), and experience or attitude
towards robotic technologies. Furthermore, conducting studies in a closed
space to give participants a bit more comfort in sharing personal data, and
informing them about their data security in our secure storage might produce
more positive or interesting results. Another consideration in future studies
is to have the robot respond based on participants’ shared data as a two-way
interaction, since valuable responses tying to their mental health can be
impactful [3].

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# A new bound in the Littlewood–Offord problem

Friedrich Götze  and  Andrei Yu. Zaitsev<EMAIL_ADDRESS>Fakultät für Mathematik,
Universität Bielefeld, Postfach 100131,
D-33501 Bielefeld, Germany
<EMAIL_ADDRESS>St. Petersburg Department of Steklov Mathematical
Institute
Fontanka 27, St. Petersburg 191023, Russia
and St.Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg,
199034 Russia

###### Abstract.

The paper deals with studying a connection of the Littlewood–Offord problem
with estimating the concentration functions of some symmetric infinitely
divisible distributions. It is shown that the concentration function of a
weighted sum of independent identically distributed random variables is
estimated in terms of the concentration function of a symmetric infinitely
divisible distribution whose spectral measure is concentrated on the set of
plus-minus weights.

###### Key words and phrases:

concentration functions; inequalities; the Littlewood–Offord problem; sums of
independent random variables

###### 1991 Mathematics Subject Classification:

Primary 60F05; secondary 60E15, 60G50

The authors were supported by SFB 1283/2 2021 –- 317210226 and by the RFBR-DFG
grant 20-51-12004. The second author was supported by grant RFBR 19-01-00356.

The aim of the present work is to provide a supplement to the paper of
Eliseeva and Zaitsev [1]. We study a connection of the Littlewood–Offord
problem with estimating the concentration functions of some symmetric
infinitely divisible distributions. In the proof, we repeat the arguments of
[1] adding at the last step an application of Jensen’s inequality.

Let $X,X_{1},\ldots,X_{n}$ be independent identically distributed (i.i.d.)
random variables. The concentration function of a $\mathbf{R}^{d}$-dimensional
random vector $Y$ with distribution $F=\mathcal{L}(Y)$ is defined by the
equality

$Q(F,\lambda)=\sup_{x\in\mathbf{R}^{d}}\mathbf{P}(Y\in x+\lambda B),\quad
0\leq\lambda\leq\infty,$

where $B=\\{x\in\mathbf{R}^{d}:\|x\|\leq 1/2\\}$. Of course, $Q(F,\infty)=1$.
Let $a=(a_{1},\ldots,a_{n})$, where
$a_{k}=(a_{k1},\ldots,a_{kd})\in\mathbf{R}^{d}$, $k=1,\ldots,n$. In this paper
we study the behavior of the concentration functions of the weighted sums
$S_{a}=\sum\limits_{k=1}^{n}X_{k}a_{k}$ with respect to the properties of
vectors $a_{k}$. Interest in this subject has increased considerably in
connection with the study of eigenvalues of random matrices (see, for
instance, Friedland and Sodin [2], Rudelson and Vershynin [3], [4], Tao and Vu
[5], [6], Nguyen and Vu [7], Vershynin [8], Tikhomirov [9], Livshyts,
Tikhomirov and Vershynin [10], Campos et al. [11]). For a detailed history of
the problem we refer to a review of Nguyen and Vu [12]. The authors of the
above articles (see also Halász [13]) called this question the
Littlewood–Offord problem, since, for the first time, this problem was
considered in 1943 by Littlewood and Offord [14] in connection with the study
of random polynomials. They considered a special case, where the coefficients
$a_{k}\in\mathbf{R}$ are one-dimensional, and $X$ takes values $\pm 1$ with
probabilities $1/2$.

The recent achivements in estimating the probabilities of singularity of
random matrices [9], [10], [11] were based on the Rudelson and Vershynin [3],
[4], [8] method of least common denominator. Note that the results of [2],
[4], [8] (concerning the Littlewood–Offord problem) were improved and refined
in [15], [16], [17].

Now we introduce some notation. In the sequel, let $F_{a}$ denote the
distribution of the sum $S_{a}$, let $E_{y}$ be the probability measure
concentrated at a point $y$, and let $G$ be the distribution of symmetrized
random variable $\widetilde{X}=X_{1}-X_{2}$. For $\delta\geq 0$ we denote

$p(\delta)=G\big{\\{}\\{z:|z|>\delta\\}\big{\\}}.$ (1)

The symbol $c$ will be used for absolute positive constants which may be
different even in the same formulas.

Writing $A\ll B$ means that $|A|\leq cB$. Also we will write $A\asymp B$, if
$A\ll B$ and $B\ll A$. We will write $A\ll_{d}B$, if $|A|\leq c(d)B$, where
$c(d)>0$ depends on $d$ only. Similarly, $A\asymp_{d}B$, if $A\ll_{d}B$ and
$B\ll_{d}A$. The scalar product in $\mathbf{R}^{d}$ will be denoted
$\left\langle\,\cdot\,,\,\cdot\,\right\rangle$. Later $\lfloor x\rfloor$ is
the largest integer $k$ such that $k<x$. For
${x=(x_{1},\dots,x_{n})\in\mathbf{R}^{n}}$ we will use the norms
$\|x\|^{2}=x_{1}^{2}+\dots+x_{n}^{2}$ and $|x|=\max_{j}|x_{j}|$. We denote by
$\widehat{F}(t)$, $t\in\mathbf{R}^{d}$, the characteristic function of
$d$-dimensional distributions $F$.

Products and powers of measures will be understood in the convolution sense.
For infinitely divisible distribution $F$ and $\lambda\geq 0$, we denote by
$F^{\lambda}$ the infinitely divisible distribution with characteristic
function $\widehat{F}^{\lambda}(t)$.

The elementary properties of concentration functions are well studied (see,
for instance, [18], [19], [20]). It is known that

$Q(F,\mu)\ll_{d}(1+\lfloor\mu/\lambda\rfloor)^{d}\,Q(F,\lambda)$ (2)

for any $\mu,\lambda>0$. Hence,

$Q(F,c\lambda)\asymp_{d}\,Q(F,\lambda).$ (3)

Let us formulate a generalization of the classical Esséen inequality [21] to
the multivariate case ([22], see also [19]):

######  Lemma 1.

Let $\tau>0$ and let $F$ be a $d$-dimensional probability distribution. Then

$Q(F,\tau)\ll_{d}\tau^{d}\int_{|t|\leq 1/\tau}|\widehat{F}(t)|\,dt.$ (4)

In the general case $Q(F,\tau)$ cannot be estimated from below by the right
hand side of inequality (4). However, if we assume additionally that the
distribution $F$ is symmetric and its characterictic function is non-negative
for all $t\in\mathbf{R}$, then we have the lower bound:

$Q(F,\tau)\gg_{d}\tau^{d}\int\limits_{|t|\leq 1/\tau}{\widehat{F}(t)\,dt},$
(5)

and, therefore,

$Q(F,\tau)\asymp_{d}\tau^{d}\int\limits_{|t|\leq 1/\tau}{\widehat{F}(t)\,dt},$
(6)

(see [23] or [18], Lemma 1.5 of Chapter II for $d=1$). In the multivariate
case relations (5) and (6) may be found in Zaitsev [24]. Exactly the use of
relation (6) allows us to simplify the arguments of Friedland and Sodin [2],
Rudelson and Vershynin [4] and Vershynin [8] which were applied to
Littlewood–Offord problem (see [15], [16], [17]).

The main result of this paper is a general inequality which reduces the
estimation of concentration functions in the Littlewood–Offord problem to the
estimation of concentration functions of some infinitely divisible
distributions. This result is formulated in Theorem 1.

For $z\in\mathbf{R}$, introduce the distribution $H_{z}$ with the
characteristic function

$\widehat{H}_{z}(t)=\exp\Big{(}-\frac{\,1\,}{2}\;\sum_{k=1}^{n}\big{(}1-\cos(\left\langle\,t,a_{k}\right\rangle
z)\big{)}\Big{)}.$ (7)

It depends on the vector $a$. It is clear that $H_{z}$ is a symmetric
infinitely divisible distribution. Therefore, its characteristic function is
positive for all $t\in\mathbf{R}^{d}$.

Recall that $G=\mathcal{L}(X_{1}-X_{2})$ and $F_{a}=\mathcal{L}(S_{a})$, where
$S_{a}=\sum\limits_{k=1}^{n}X_{k}a_{k}$.

######  Theorem 1.

Let $V$ be an arbitrary one-dimensional Borel measure such that
$\lambda=V\\{\mathbf{R}\\}>0$, and $V\leq G$, that is, $V\\{B\\}\leq
G\\{B\\}$, for any Borel set $B$. Then, for any $\tau>0$, we have

$Q(F_{a},\tau)\ll_{d}\int_{z\in\mathbf{R}}Q(H_{1}^{\lambda},\tau|z|^{-1})\,W\\{dz\\},$
(8)

where $W=\lambda^{-1}V$.

######  Corollary 1.

For any $\varepsilon,\tau>0$, we have

$Q(F_{a},\tau)\ll_{d}Q(H_{1}^{p(\tau/\varepsilon)},\varepsilon),$ (9)

where $p(\,\cdot\,)$ is defined in (1).

In order to verify Corollary 1 we note that the distribution
$G=\mathcal{L}(\widetilde{X})$ may be represented as the mixture

$G=p_{0}G_{0}+p_{1}G_{1},\quad\text{where}\quad
p_{j}={\mathbf{P}}\bigl{\\{}\widetilde{X}\in A_{j}\bigr{\\}},\quad j=0,1,$

$A_{0}=\\{x\colon|x|\leq\tau/\varepsilon\\}$,
$A_{1}=\\{x\colon|x|>\tau/\varepsilon\\}$, $G_{j}$ are probability measures
defined for $p_{j}>0$ by the formula $G_{j}\\{B\\}=G\\{B\cap A_{j}\\}/{p_{j}}$
, for any Borel set $B$. In fact, $G_{j}$ is the conditional distribution of
$\widetilde{X}$ given that $\widetilde{X}\in A_{j}$. If $p_{j}=0$, then we can
take as $G_{j}$ an arbitrary measure.

The conditions of Theorem 1 are satisfied for $V=p_{1}G_{1}$.
$\lambda=p_{1}=p(\tau/\varepsilon)$, $W=G_{1}$.

Inequalities (2) and (6) imply that

$\displaystyle Q(F_{a},\tau)$ $\displaystyle\ll_{d}$ $\displaystyle\int_{z\in
A_{1}}Q(H_{1}^{\lambda},\tau|z|^{-1})\,W\\{dz\\}$ (10) $\displaystyle\leq$
$\displaystyle\sup_{z\geq\tau/\varepsilon}\;Q\big{(}H_{1}^{p(\tau/\varepsilon)},\tau/z\big{)}=Q\big{(}H_{1}^{p(\tau/\varepsilon)},\varepsilon\big{)},$

proving (9).

Applying Theorem 1 with $V$ of the form

$V\\{dz\\}=\big{(}1+\lfloor\tau(\varepsilon|z|)^{-1}\rfloor\big{)}^{-d}\,G\\{dz\\},$
(11)

and using inequality (2), we obtain

######  Corollary 2.

For any $\varepsilon,\tau>0$, we have

$Q(F_{a},\tau)\ll_{d}\lambda^{-1}\,Q(H_{1}^{\lambda},\varepsilon),$ (12)

where

$\lambda=\lambda(G,\tau/\varepsilon)=V\\{\mathbf{R}\\}=\int\limits_{z\in\mathbf{R}}\big{(}1+\lfloor\tau(\varepsilon|z|)^{-1}\rfloor\big{)}^{-d}\,G\\{dz\\}.$
(13)

It is clear that $\lfloor\tau(\varepsilon|z|)^{-1}\rfloor=0$ if
$|z|>\tau/\varepsilon$. Therefore. $\lambda=\lambda(G,\tau/\varepsilon)\geq
p(\tau/\varepsilon)$ and hence $Q(H_{1}^{\lambda},\varepsilon)\leq
Q(H_{1}^{p(\tau/\varepsilon)},\varepsilon)$. Thus, if $\lambda\gg_{d}1$, then
inequality (12) of Corollary 2 is stronger than inequality (9) of Corollary 1.

The proof of Theorem 1 is based on elementary properties of concentration
functions. We repeat the arguments of [1] adding at the last step an
application of Jensen’s inequality. In [1], inequality (2) was used instead.
The main result of [1] does not imply Corollary 2. Note that $H_{1}^{\lambda}$
is an infinitely divisible distribution with the Lévy spectral measure
$M_{\lambda}=\frac{\,\lambda\,}{4}\;M^{*}$, where
$M^{*}=\sum\limits_{k=1}^{n}\big{(}E_{a_{k}}+E_{-a_{k}}\big{)}$. It is clear
that the assertions of Theorem 1 and Corollaries 1 and 2 may be treated as
statements about the measure $M^{*}$.

Corollary 1 was already proved earlier in [1] and [25], see also [26] for the
case $\tau=0$. It was used essentially in [25] and [27] to show that Arak’s
inequalities for concentration functions may be used for investigations of the
Littlewood–Offord problem. Arak has shown that if the concentration function
of infinitely divisible distribution is relatively large, then the spectral
measure of this distribution is concentrated in a neighborhood of a set with
simple arithmetical structure. Together with Corollary 1, this means that
large value of $Q(F_{a},\tau)$ implies a simple arithmetical structure of the
set $\\{\pm a_{k}\\}_{k=1}^{n}$. This statement is similar to the so-called
”inverse principle” in the Littlewood–Offord problem (see [5], [7], [12]).

Note that using the results of Arak [23], [28] (see also [18]) one could
derive from Corollary 1 inequalities similar to boumds for concentration
functions in the Littlewood–Offord problem, which were obtained in a paper of
Nguyen and Vu [7] (see also [12]). A detailed discussion of this fact is
presented in [25] and [27]. We noticed that Corollary 2 may be stronger than
Corollary 1. Therefore, the results of [25] and [27] could be improved (in the
sense of dependence of constants on the distribution of $X_{1}$) replacing an
application of Corollary 1 by an application of Corollary 2. The authors are
going to devote a separate publication to this topic.

Proof of Theorem 1. Let us show that, for arbitrary probability distribution
$W$ and $\lambda,T>0$,

$\log\int_{|t|\leq
T}\exp\Big{(}-\frac{\,1\,}{2}\;\sum_{k=1}^{n}\int_{z\in\mathbf{R}}\big{(}1-\cos(\left\langle\,t,a_{k}\right\rangle
z)\big{)}\,\lambda\,W\\{dz\\}\Big{)}\,dt\\\
\leq\int_{z\in\mathbf{R}}\bigg{(}\log\int_{|t|\leq
T}\exp\Big{(}-\frac{\,\lambda\,}{2}\;\sum_{k=1}^{n}\big{(}1-\cos(\left\langle\,t,a_{k}\right\rangle
z)\big{)}\Big{)}\,dt\bigg{)}\,W\\{dz\\}\\\
=\int_{z\in\mathbf{R}}\bigg{(}\log\int_{|t|\leq
T}\widehat{H}_{z}^{\lambda}(t)\,dt\bigg{)}\,W\\{dz\\}.$ (14)

It suffices to prove (14) for discrete distributions
$W=\sum_{j=1}^{\infty}p_{j}E_{z_{j}}$, where $0\leq p_{j}\leq 1$,
$z_{j}\in\mathbf{R}$, $\sum_{j=1}^{\infty}p_{j}=1$. Applying in this case the
generalized Hölder inequality, we have

$\int\limits_{|t|\leq
T}\exp\Big{(}-\frac{\,1\,}{2}\;\sum\limits_{k=1}^{n}\int\limits_{z\in\mathbf{R}}\big{(}1-\cos(\left\langle\,t,a_{k}\right\rangle
z)\big{)}\,\lambda\,W\\{dz\\}\Big{)}\,dt\\\ =\int\limits_{|t|\leq
T}\exp\Big{(}-\frac{\,\lambda\,}{2}\;\sum\limits_{j=1}^{\infty}p_{j}\sum\limits_{k=1}^{n}\big{(}1-\cos(\left\langle\,t,a_{k}\right\rangle
z_{j})\big{)}\Big{)}\,dt\\\
\leq\prod\limits_{j=1}^{\infty}\bigg{(}\int\limits_{|t|\leq
T}\exp\Big{(}-\frac{\,\lambda\,}{2}\;\sum\limits_{k=1}^{n}\big{(}1-\cos(\left\langle\,t,a_{k}\right\rangle
z_{j})\big{)}\Big{)}\,dt\bigg{)}^{p_{j}}.$ (15)

Taking logarithms of the left and right-hand sides of (15), we get (14). In
general case we can approximate $W$ by discrete distributions in the sense of
weak convergence and pass to the limit. Note also that the integrals
$\int_{|t|\leq T}dt$ may be replaced in (14) by the integrals
$\int\mu\\{dt\\}$ with an arbitrary Borel measure $\mu$.

Since for characteristic function $\widehat{U}(t)$ of a random vector $Y$, we
have

$|\widehat{U}(t)|^{2}=\mathbf{E}\exp(i\langle\,t,\widetilde{Y}\rangle)=\mathbf{E}\cos(\langle\,t,\widetilde{Y}\rangle),$

where $\widetilde{Y}$ is the corresponding symmetrized random vector, then

$|\widehat{U}(t)|\leq\exp\Big{(}-\cfrac{\,1\,}{2}\,\big{(}1-|\widehat{U}(t)|^{2}\big{)}\Big{)}=\exp\Big{(}-\cfrac{\,1\,}{2}\,\mathbf{E}\,\big{(}1-\cos(\langle\,t,\widetilde{Y}\rangle)\big{)}\Big{)}.$
(16)

According to Lemma 1 and relations $V=\lambda\,W\leq G$, (14) and (16),
applying Jensen’s inequality of the form
$\exp(\mathbf{E}\,f(\xi))\leq\mathbf{E}\,\exp(f(\xi))$ for any measurable
function $f$ and any random varialble $\xi$, we have

$\displaystyle Q(F_{a},\tau)$ $\displaystyle\ll_{d}$
$\displaystyle\tau^{d}\int_{\tau|t|\leq 1}|\widehat{F_{a}}(t)|\,dt$ (17)
$\displaystyle\ll_{d}$ $\displaystyle\tau^{d}\int_{\tau|t|\leq
1}\exp\Big{(}-\frac{\,1\,}{2}\,\sum_{k=1}^{n}\mathbf{E}\,\big{(}1-\cos(\left\langle\,t,a_{k}\right\rangle\widetilde{X})\big{)}\Big{)}\,dt$
$\displaystyle=$ $\displaystyle\tau^{d}\int_{\tau|t|\leq
1}\exp\Big{(}-\frac{\,1\,}{2}\;\sum_{k=1}^{n}\int_{z\in\mathbf{R}}\big{(}1-\cos(\left\langle\,t,a_{k}\right\rangle
z)\big{)}\,G\\{dz\\}\Big{)}\,dt$ $\displaystyle\leq$
$\displaystyle\tau^{d}\int_{\tau|t|\leq
1}\exp\Big{(}-\frac{\,1\,}{2}\;\sum_{k=1}^{n}\int_{z\in\mathbf{R}}\big{(}1-\cos(\left\langle\,t,a_{k}\right\rangle
z)\big{)}\,\lambda\,W\\{dz\\}\Big{)}\,dt$ $\displaystyle\leq$
$\displaystyle\exp\bigg{(}\int_{z\in\mathbf{R}}\log\bigg{(}\tau^{d}\int_{\tau|t|\leq
1}\widehat{H}_{z}^{\lambda}(t)\,dt\bigg{)}\,W\\{dz\\}\bigg{)}$
$\displaystyle\leq$
$\displaystyle\int_{z\in\mathbf{R}}\bigg{(}\tau^{d}\int_{\tau|t|\leq
1}\widehat{H}_{z}^{\lambda}(t)\,dt\bigg{)}\,W\\{dz\\}.$

Using (6), we have

$\displaystyle\tau^{d}\int_{\tau|t|\leq 1}\widehat{H}_{z}^{\lambda}(t)\,dt$
$\displaystyle\asymp_{d}$ $\displaystyle
Q(H_{z}^{\lambda},\tau)=Q\big{(}H_{1}^{\lambda},\tau{|z|}^{-1}\big{)}.$ (18)

Substituting this formula into (17), we obtain (8). In (18), we have used that
$H_{z}^{\lambda}=\mathcal{L}(z\eta)$, where $\eta$ is a random vector with
$\mathcal{L}(\eta)=H_{1}^{\lambda}$. $\square$

Acknowledgment. We are grateful to the reviewers for valuable remarks.

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