# Magnitude-aware Probabilistic Speaker Embeddings

###### Abstract

Recently, hyperspherical embeddings have established themselves as a dominant
technique for face and voice recognition. Specifically, Euclidean space vector
embeddings are learned to encode person-specific information in their
direction while ignoring the magnitude. However, recent studies have shown
that the magnitudes of the embeddings extracted by deep neural networks may
indicate the quality of the corresponding inputs. This paper explores the
properties of the magnitudes of the embeddings related to quality assessment
and out-of-distribution detection. We propose a new probabilistic speaker
embedding extractor using the information encoded in the embedding magnitude
and leverage it in the speaker verification pipeline. We also propose several
quality-aware diarization methods and incorporate the magnitudes in those. Our
results indicate significant improvements over magnitude-agnostic baselines
both in speaker verification and diarization tasks.

Keywords: speaker verification, magnitude-aware embeddings, probabilistic
embeddings, uncertainty propagation, speaker diarization

##  1 Introduction

In recent years, deep learning-based methods have shown impressive performance
in speaker verification and diarization. However, recognition in the wild is
still difficult due to the large variability in speech signals acquired in
unconstrained conditions. This variability is caused by different channels or
recording environments involving background noise and reverberation.

Therefore, the audio segments provided as an input to a speaker recognition
system are not always appropriate for recognition. A typical speaker
recognition system operates under the assumption that the input audio contains
recognizable information, and this assumption usually holds during training
because the training datasets are specifically collected and filtered in this
way. However, this is not always the case after deployment, as the input
signal may be of low quality or may not even contain human speech, which could
lead to incorrect predictions because the embedding extractor has never seen
such inputs before.

One solution to this issue is to train an extra model to assess the quality of
the input recordings. Thus, ground-truth labels must be provided for a
training dataset, including recordings of varying quality. However, the
quality labels provided by humans or those derived from the signal-to-noise
ratio (SNR) or signal duration [1, 2, 3] may not be the most relevant
characteristics in terms of recognition performance.

Figure 1: Distributions of embeddings magnitudes for three publicly available
pre-trained embedding extractors.
Left111https://github.com/clovaai/voxceleb_trainer: ResNetSE34 by CLOVA [4].
Center222https://huggingface.co/speechbrain/spkrec-ecapa-voxceleb: ECAPA-TDNN
[5] from the SpeechBrain package [6].
Right333https://github.com/ranchlai/speaker-verification: ResNetSE34 by [7].
Only in the last case the magnitudes serve as weak descriptors of speech
quality. Best viewed in color.

Another group of related studies revealed how to obtain the information about
the input quality directly from the extracted embeddings. In [8] it was
observed that face embeddings learned using the softmax loss tend to have a
smaller magnitude 444“Magnitude” hereafter stands for $L_{2}$-norm. for the
inputs of lower quality (_e.g._ blurry images). Further, [9, 10] proposed new
classification-based losses to explicitly encourage magnitudes to represent
feature confidence in the embedding space. An appealing property of magnitudes
is providing useful information for the predictions confidence estimation
without any extra cost.

There also exists a concept of probabilistic embeddings, which naturally
provides the quality estimates as variance-related probability distribution
parameters. Such embeddings are well studied in both face recognition [11, 12,
13] and speaker recognition research [14, 15].

The main contributions of this paper are as follows.

  * •

First, inspired by [9], we train speaker embedding extractor with
interpretable embedding magnitudes and examine properties of the resulting
embeddings pertaining to the speech recording quality.

  * •

We introduce a specific form of probabilistic embeddings with Gaussian
precision interpretation for magnitudes and evaluate it for the speaker
verification task.

  * •

We also explore the ability of embeddings magnitudes to filter out non-speech
or low-quality speech segments. Based on this, we propose a two-step speaker
diarization method that relies on a subset of reliable segments to robustly
identify initial clusters.

  * •

Last, we modify a popular diarization method known as VBx [16] by enabling
uncertainty propagation in this model.

##  2 Background

###  2.1 Magnitude-aware embeddings

Earlier works in face recognition identified an interesting property of
embeddings trained with the softmax loss function [8, 17]. It was found that
embedding’s magnitude can be a predictor for image quality, where the quality
refers to various factors affecting the ability to identify a person such, as
a blur, low resolution, extreme pose, or bad lighting. Similar observations
were later made in [10]. As discussed in [10], the explanation could be that
since the magnitude affects the peakedness of the class posterior
distribution, hard and noisy samples will be forced to have smaller magnitudes
during training. Moreover, these authors also found that this effect exists
for angular losses, such as ArcFace [18], despite their magnitude-agnostic
nature.

Based on these findings, we studied magnitudes of a few modern top-performing
publicly available speaker embedding extractors, all of which were trained
with the softmax-based losses. For this estimation we employed the VoxCeleb1
test set [19], the evaluation set from the VOiCES from a Distance Challenge
2019 [20] and the MUSAN noise collection [21]. These datasets represent clean
speech, noisy speech, and non-speech recordings, respectively. Figure 3 shows
the distributions of magnitudes for these datasets. Surprisingly, only one of
the embedding extractors allows interpretation of magnitudes as quality
estimates. This leads to the conclusion that such interpretation is not
provided “by default”.

It is worth mentioning that the magnitudes of speaker embeddings were already
successfully applied for the voice activity detection task [22]. However, this
work lacks any qualitative analysis of embedding magnitudes properties. To
fill this gap, we present such analysis in section 6.

Based on these observations, we decided to train the embedding extractor using
the recently proposed MagFace loss [9] designed to explicitly learn quality-
aware representations. As a reference, we choose the ArcFace (also referred to
as AAM-Softmax [23]) as a loss without an explicit mechanism to affect the
embedding magnitude.

ArcFace [18], is among the most widely adopted loss functions in speaker
recognition. This loss function can be obtained from the softmax loss by
normalizing the classifier weights and input embeddings and by introducing the
_constant_ angular margin into the target logits. Although this loss function
is magnitude-agnostic, it still features the confidence propagation
behavior111But not necessary, _e.g._ , the pre-trained ECAPA-TDNN [5] from
SpeechBrain [6] lacks this property, despite having AAM-Softmax as a training
objective..

MagFace [9] loss function can be seen as a generalization of ArcFace with an
_adaptive_ angular margin depending on the magnitude of the input embedding
$\bm{\mathrm{x}}$. The loss can be formulated as follows:

$\mathcal{L}_{Mag}=-\frac{1}{N}\sum_{i=1}^{N}\log\frac{e^{\psi(\theta_{y_{i}}^{i})}}{e^{\psi(\theta_{y_{i}}^{i})}+\sum\limits_{j\neq
y_{i}}e^{s\cos\theta_{j}^{i}}}+\lambda_{g}g(||\bm{\mathrm{x}}_{i}||),$ (1)

where $i$ runs over the training batch of size $N$, and
$\theta_{j}^{i}\in\mathbb{R}$ represents the angle between the $j$-th
classifier weight and $d$-dimensional embedding vector
${\mathbf{x}}_{i}\in\mathbb{R}^{d}$ with the corresponding label $y_{i}$. In
addition, $\psi$ denotes the angle function depending on the base loss, which
is defined as $\psi(\theta_{y_{i}}^{i})=s\cos(\theta_{y_{i}}^{i}+m)$ for
ArcFace, where $m$ is a margin and $s$ is a fixed scale factor. In MagFace the
margin $m\equiv m(\left\lVert{\mathbf{x}}_{i}\right\rVert)$ is a strictly
increasing convex function of the embedding’s magnitude. Additionally, the
MagFace loss has a second term, weighted by $\lambda_{g}$, where $g$ is a
strictly decreasing convex function.

###  2.2 Probabilistic embeddings

In the representation learning, the extracted vector representations are
conventionally called _embeddings_. These embeddings usually are point-wise,
or _deterministic_ , estimates of some latent variable in the representation
space. Alternatively, one can consider _probabilistic_ representations,
usually in a form of probability distributions in the latent space [24]. Such
representations are of sufficient interest, as they can capture more
information than simple point-wise estimation, _e.g._ quantify the data
uncertainty for input sample, and provide more flexibility to define
similarity measures.

Recently, probabilistic embeddings gained popularity for face recognition in
unconstrained conditions where the recognition systems may heavily suffer from
low-quality inputs [11, 12, 13, 25]. For instance, the authors of [11]
proposed to represent the input images by Gaussian distributions. These
distributions are parameterized by diagonal covariance matrices, estimated by
an additional branch to the embedding extractor. In this architecture, the
covariance matrix represents the dimension-wise uncertainty of face embeddings
treated as hidden variables.

In this work, we consider a particular instance of probabilistic embeddings
known as _meta-embeddings_ [14] that provide the likelihood distribution for
embeddings treated as hidden variables. While the choice of particular
functional form of meta-embeddings is flexible, [14] proposed _Gaussian meta-
embeddings_ (GME) as a tractable solution convenient for practical
application. Later, similar to [11], the authors of [15] proposed an
architecture to extract meta-embeddings by introducing an additional branch
whose outputs are interpreted as precisions for the Gaussian distribution.
This model yielded a noticeable performance gains for the speaker diarization
task.

##  3 Magnitude-aware Gaussian meta-embeddings

In this work, we propose a probabilistic embedding extractor that takes an
advantage of the information encoded in the magnitudes of deterministic
embeddings. Our approach relies on the assumption that embeddings magnitudes
can serve as indicators of the input quality. Specifically, we use the
magnitude to define a precision parameter in the Gaussian meta-embeddings.

###  3.1 From magnitude to precision

Our approach can be seen as a specific recipe to design the probabilistic
embeddings in the form described in [14] or [15]. In this section, we
illustrate the proposed idea for the Gaussian meta-embeddings introduced in
[14] while it is also applicable to the model [15] which uses probabilistic
linear discriminant analysis (PLDA) as the embedding prior.

_Meta-embedding_ $f({\mathbf{z}})$ is a likelihood function for the latent
speaker identity variable ${\mathbf{z}}\in\mathbb{R}^{d}$, _i.e._ :

$f({\mathbf{z}})\propto p(x|{\mathbf{z}}),$

where $x$ denotes the input audio recording represented by waveform, Mel-
spectrogram, or vector embedding. Given the embedding prior $p({\mathbf{z}})$,
one can compare possible inputs partitions by computing the corresponding
likelihood ratios expressed in terms of meta-embeddings. In practice, however,
one needs to restrict the functional form of meta-embeddings for the sake of
tractability. One convenient option is Gaussian meta-embeddings (GME) where
$f$ has the form of the Gaussian likelihood function.

Our proposal can be summarized as follows: we resort to the isotropic
precision matrix and use the embedding magnitude as its scale parameter. This
leads to the following formulation of GME:

$f({\mathbf{z}};x)=\exp\left[-\frac{||\bm{\mu}||}{2}{\mathbf{z}}^{\mathsf{T}}{\mathbf{z}}+{\mathbf{z}}^{\mathsf{T}}\bm{\mu}\right],$

where $\bm{\mu}$ denotes the $d$-dimensional speaker embedding extracted from
the speech recording $x$ by a deep neural network:
$\bm{\mathrm{\mu}}=\mathrm{net}(x)$. By using the decomposition
$\bm{\mu}=\overline{\bm{\mu}}\cdot||\bm{\mu}||$ where $\overline{\bm{\mu}}$
denotes the length-normalized emebdding, one can see that $||\bm{\mu}||$ can
be interpreted as a scalar precision parameter in the isotropic Gaussian.

The potential advantage of this approach is that it does _not_ require
training a separate model, allowing to obtain the precision parameter without
extra cost. This makes it different from similar models [15, 26], which
augment the embedding extractor with an additional branch that provides an
additional output.

Given a pair of inputs $\\{x_{1},x_{2}\\}$ representing the verification trial
and assuming the standard Gaussian prior
$p({\mathbf{z}})=\mathcal{N}(\bm{\mathrm{0}},\bm{\mathrm{I}})$, as in [14],
this results in the following expression for the log-likelihood ratio (LLR)
score:

$\text{LLR}(x_{1},x_{2})=\frac{1}{2}\frac{(\bm{\mathrm{\mu}}_{1}+\bm{\mathrm{\mu}}_{2})^{\mathsf{T}}(\bm{\mathrm{\mu}}_{1}+\bm{\mathrm{\mu}}_{2})}{r_{1}+r_{2}+1}-\frac{1}{2}\frac{\bm{\mathrm{\mu}}_{1}^{\mathsf{T}}\bm{\mathrm{\mu}}_{1}}{r_{1}+1}\\\
-\frac{1}{2}\frac{\bm{\mathrm{\mu}}_{2}^{\mathsf{T}}\bm{\mathrm{\mu}}_{2}}{r_{2}+1}+\frac{d}{2}\log\frac{(r_{1}+1)(r_{2}+1)}{(r_{1}+r_{2}+1)},$
(2)

where $r=||{\bm{\mu}}||$ denotes the $L_{2}$-norm. One may notice that this
LLR is a shifted and scaled cosine similarity with both shift and scale
parameters depending only on the embeddings magnitudes.

To ensure that the embedding magnitudes are indeed positively correlated with
speech quality, _e.g._ signal-to-noise ratio, we employ the MagFace loss [9]
for training the extractor of deterministic embeddings. However, as we
observed in the experiments, magnitudes may depend on the duration of speech
recording. That is why we introduced a learnable parametric transform
conditioned on the input duration. See also section 6 for related discussion.

###  3.2 Compensation for utterance duration variability

We start with several assumptions and observations:

  * •

The embedding magnitude should be a good descriptor of recording quality
allowing for Gaussian precision interpretation. We assume that there exists a
_monotonic mapping_ from magnitudes to precisions.

  * •

The qualitative analysis (see section 6) indicates the _descent_ of embedding
magnitudes with the increase of recording duration. This effect contradicts
the confidence-based interpretation of magnitudes. We compensate for this by
introducing the duration-based additive adjustment for the magnitude.

  * •

Finally, the adjusted magnitude should be transformed to the appropriate range
for serving as the precision parameter.

Based on the above, we have chosen the following transformation rule:

$r_{i}=s\cdot(||\bm{\mu}_{i}||+\gamma\cdot\min\\{20,\text{len}(x_{i})\\}),$
(3)

where $s>0$ indicates the global scaling parameter, which provides the
appropriate range, $\text{len}(\cdot)$ stands for the recording duration (in
seconds; we empirically constrain the maximum allowed duration to 20 seconds),
and $\gamma>0$ determines the impact of duration adjustment. It is important
to note that described transformation is only needed for LLR scoring, while
the raw magnitudes can be used for quality assessment purposes. The parameters
in (3) can be estimated from a development set of recordings.

##  4 Speaker verification

We start by evaluating the proposed models on the speaker verification task.

ResNet-ArcFace | ResNet-MagFace | Large ResNet-MagFace
---|---|---
13.84M / 13.82 GFLOPs | 13.84M / 13.82 GFLOPs | 50.45M / 34.66 GFLOPs
c=7x7, f=128, s=1, p=3
|  c=3x3, f=128, s=1, p=0
---
c=3x3, f=128, s=1, p=0
$\times 3$ |  |  |  c=3x3, f=128, s=1, p=0
---
c=3x3, f=128, s=1, p=0
$\times 3$
|  c=3x3, f=128, s=2, p=0
---
c=3x3, f=128, s=2, p=0
$\times 4$ |  |  |  c=3x3, f=128, s=2, p=0
---
c=3x3, f=128, s=2, p=0
$\times 8$
|  c=3x3, f=256, s=2, p=0
---
c=3x3, f=256, s=2, p=0
$\times 6$ |  |  |  c=3x3, f=256, s=2, p=0
---
c=3x3, f=256, s=2, p=0
$\times 36$
|  c=3x3, f=256, s=2, p=0
---
c=3x3, f=256, s=2, p=0
$\times 3$ |  |  |  c=3x3, f=256, s=2, p=0
---
c=3x3, f=256, s=2, p=0
$\times 3$
Statistic Pooling [27]
FC (d=256)
ArcFace Loss | MagFace Loss

Table 1: The description of architectures, where c denotes a size of each
convolutional filter, f corresponds to a number of convolutional filters, s
denotes a stride size, p denotes a padding size, and d indicates the output
embedding dimension. The Second line in the table shows models’ efficiency
characteristics. The first number in each pair denotes a number of parameters
(in millions), the second one indicates amount of GFLOPs to process one second
of audio.

###  4.1 Implementation details

####  4.1.1 Architecture

We implemented three different embedding extractors for comparison:

  * •

ResNet-ArcFace: ResNet trained with the ArcFace loss,

  * •

ResNet-MagFace: ResNet trained with the MagFace loss,

  * •

Large ResNet-MagFace: ResNet with more layers trained with the MagFace loss.

Architectures of these networks are shown in Table 1.

As features, we used 80-dimensional log Mel-filterbank energies extracted with
a window length of 25 ms and a frame-shift of 10 ms.

####  4.1.2 Training

All the speaker embedding extractors were trained on the development part of
the VoxCeleb2 dataset [28] using segments of 3 seconds randomly cropped from
the original audios. The parameters of the embedding extractors were updated
via the Ranger optimizer [29] with a cosine annealing learning rate scheduler.
The minimum learning rate was set to $10^{-5}$ with a scheduler’s period equal
to 100K iterations and the initial learning rate was equal to $10^{-3}$. The
batch size was set to 64.

We applied an online augmentation strategy based on noises and room impulse
responses from the MUSAN corpus [21], following the recipe from [30]. All the
embedding extractors were trained for 600K iterations, and the checkpoint with
the best performance on the test part of the VoxCeleb2 dataset was used for
the evaluation.

Following [9], we used the same functional form for the functions $m$ and $g$
in MagFace loss, Eq. (1):

$g(a)=\frac{a}{n_{u}^{2}}+\frac{1}{a}\text{ and
}m(a)=\frac{m_{u}-m_{l}}{n_{u}-n_{l}}(a-n_{l})+m_{l},$

where $n_{u}=110,n_{l}=10,m_{u}=1,m_{l}=0.1$ are the maximum and minimum
values of the magnitudes and margins, respectively, used for clamping during
training.

The parameters of the precision transformation (3) were estimated on the
VoxCeleb2 test set by minimizing equal error rate on a custom verification
protocol. Precisely, we firstly estimated the relative impact of duration
based on desired correlation of duration-adjusted confidence with duration,
and then selected the global scale to obtain the desired range of confidence
values.

Embedder | Loss | Score | VC1-test (cleaned) | VC1-test (cleaned, 2s-pool) | VB-pool | VOiCES Eval
---|---|---|---|---|---|---
EER | minDCF | EER | minDCF | EER | minDCF | EER | minDCF
ResNet | ArcFace | cosine | 1.13 | 0.123 | 3.47 | 0.317 | 6.42 | 0.472 | 4.50 | 0.295
GME-LLR | 0.98 | 0.112 | 3.02 | 0.284 | 5.76 | 0.468 | 3.97 | 0.288
MagFace | cosine | 1.07 | 0.109 | 3.27 | 0.292 | 6.34 | 0.462 | 4.18 | 0.263
GME-LLR | 0.92 | 0.105 | 2.85 | 0.277 | 5.40 | 0.452 | 3.61 | 0.256
Large ResNet | cosine | 0.66 | 0.064 | 2.59 | 0.250 | 6.09 | 0.433 | 4.14 | 0.246
GME-LLR | 0.57 | 0.064 | 2.30 | 0.237 | 5.93 | 0.437 | 3.55 | 0.232

Table 2: Results of speaker verification with three different embedding
extractors. Performance is measured in terms of EER and minDCF with
$P_{\mathrm{target}}=0.01$.

###  4.2 Verification performance

####  4.2.1 Datasets

We used two commonly adopted evaluation protocols: the original VoxCeleb1 test
(cleaned) [19] and the evaluation set from the VOiCES from a Distance
Challenge 2019 [20]. These tests represent the “clean” and “noisy” conditions,
respectively. For the VoxCeleb1 test we created two additional copies with (a)
both recordings cropped to 2 seconds from the beginning and (b) only the
enrollment recording cropped to 2 seconds. All the three copies (original, (a)
and (b)) were further concatenated to create a new protocol called 2s-pool
with the purpose to demonstrate robustness to duration variability.

We also created a custom evaluation protocol using a parallel database of
clean and noisy speech recordings called VoiceBank [31]. For each present
identity, we took 20 random audios and created a Cartesian product on all
taken audios, then converted this list into clean-vs-clean, clean-vs-noisy,
and noisy-vs-noisy versions. We then concatenated them and eliminated
duplicates. This pipeline produced $\sim 5.9$M pairs with $1.1\%$ positives.
This protocol is referred to as VB-pool and includes “mixed” conditions,
intended to represent real-world applications with uncontrolled conditions.

####  4.2.2 Metrics

We evaluated the speaker verification performance using two metrics: the
_equal error rate_ (EER) and the _minimum normalized detection cost function_
(minDCF) with $P_{\mathrm{target}}=0.01$ [32].

####  4.2.3 Results

Table 2 provides the speaker verification results. One can see that the GME
back-end achieves lower EER on all the test sets. In addition, the MagFace
objective provides slight improvements over ArcFace, on both clean and noisy
data. Effects of moving from deterministic to probabilistic embeddings are
precisely noticeable for ResNet-ArcFace and ResNet-MagFace networks, although,
for Large ResNet-MagFace model the improvement is relatively smaller,
especially in terms of minDCF, which even degrades for VB-pool.

##  5 Speaker diarization

We now turn our attention to the speaker diarization task. This section
presents how to improve the speaker diarization performance by using the
_segment-specific_ quality scores. In particular, we used embedding magnitudes
to provide such scores. The following two sections detalize our proposals.

###  5.1 Quality-aware two-step pipeline

First, we propose the following _two-step_ diarization method which may serve
as a general template independent of a clustering back-end. The first step
filters out the less reliable segments and finds initial clusters within the
selected segments. The second step uses all (or a subset of) available
segments and refines their labels based on the clusters estimated in the first
step.

The proposed two-step method sorts embeddings magnitudes and divides
embeddings into two subsets based on the percentile tuned on the development
set. Then, the first step applies a general clustering algorithm with an
unknown number of clusters to the set formed from embeddings with larger
magnitudes. Intuitively, this should provide a better estimate for the number
of clusters and their locations than one-step algorithms. The resulting labels
were used as input to the second step of the algorithm, which has three
possible implementations.

Algorithm 2.1. We computed the average embeddings (centroids) within each
cluster found in the first step and assigned the remaining embeddings to the
closest centroid using the similarity function.

Algorithm 2.2. We applied a clustering algorithm with a fixed number of
clusters (estimated in the first step) to all embeddings.

Algorithm 2.3. We applied a clustering algorithm with a fixed number of
clusters to the remaining embeddings.

To solve the label ambiguity problem in (2.3) algorithm, we computed the
centroids of the clusters from the first step and assigned the remaining
embeddings to the closest centroid by cosine similarity. Finally, we found the
dominating labels within each cluster found in the second step and assigned
this label to all the embeddings within this cluster.

###  5.2 Bayesian HMM with uncertainty propagation

Second, we implement uncertainty propagation for the Bayesian hidden Markov
model (HMM) back-end [16] in a similar way as described in [33, 34].

We assume that the extracted utterance embeddings were generated by the PLDA
model with the full-rank speaker subspace in the form of a two-covariance
model [35]. The model is specified by the following probability distributions:

$\displaystyle p(\bm{\mathrm{y}})$
$\displaystyle=\mathcal{N}(\bm{\mathrm{y}}|\bm{\mathrm{0}},\bm{\mathrm{B}})$
(4) $\displaystyle p(\bm{\mathrm{x}}|\bm{\mathrm{y}})$
$\displaystyle=\mathcal{N}(\bm{\mathrm{x}}|\bm{\mathrm{y}},\bm{\mathrm{W}})$

where the two matrices $\bm{\mathrm{B}}$ and $\bm{\mathrm{W}}$ model between-
and within-speaker covariances. For each speaker, represented by latent
variable $\bm{\mathrm{y}}$, the distribution of embedding vectors
$\bm{\mathrm{x}}$ is modeled by a single Gaussian. These Gaussians are
parameterized by speaker-dependent means $\bm{\mathrm{y}}$ and a shared
covariance matrix $\bm{\mathrm{W}}$.

For clustering purposes, we construct a HMM where states correspond to
speakers and the speaker-specific distributions are derived from the PLDA
model (4). Assuming that $K$ speakers are present in the collection of
embeddings,
$\bm{\mathrm{X}}=\\{\bm{\mathrm{x}}_{1},\dots,\bm{\mathrm{x}}_{N}\\}$, the
model has $K$ latent speaker mean vectors
$\bm{\mathrm{Y}}=\\{\bm{\mathrm{y}}_{1},\dots,\bm{\mathrm{y}}_{K}\\}$, one for
each speaker-specific Gaussian. This model can be seen as a particular case of
the model in [16] that also supports the PLDA model with low-rank speaker
subspace.

The uncertainty propagation mechanism is implemented by introducing an _input-
dependent_ covariance matrix $\bm{\mathrm{\Sigma}}$:

$\displaystyle p(\bm{\mathrm{x}}_{i}|\bm{\mathrm{y}})$
$\displaystyle=\mathcal{N}(\bm{\mathrm{x}}_{i}|\bm{\mathrm{y}},\bm{\mathrm{W}}+\bm{\mathrm{\Sigma}}_{i})$
(5)

By setting $\bm{\mathrm{\Sigma}}$ to zero the conventional PLDA model is
recovered.

In practice, we use a PLDA model with _spherical covariances_ , _i.e._ with
$\bm{\mathrm{B}}=\sigma_{\text{B}}^{2}\bm{\mathrm{I}}$,
$\bm{\mathrm{W}}=\sigma_{\text{W}}^{2}\bm{\mathrm{I}}$, and
$\bm{\mathrm{\Sigma}}_{i}=\sigma_{i}^{2}\bm{\mathrm{I}}$, where
$\bm{\mathrm{I}}$ denotes an identity matrix. Here, $\sigma_{i}^{2}$ is the
inverse of precision computed from the embedding magnitude by (3).

We modified the publicly available implementation of [16] to support
uncertainty propagation and used the original implementation as a
baseline111https://github.com/BUTSpeechFIT/VBx, further referred to as VBx.

###  5.3 Experiment Setup

We conducted the experiments using the popular AMI [36] and VoxConverse [37]
corpora. We followed the development/evaluation split for the AMI corpus from
the official website 222https://groups.inf.ed.ac.uk/ami/corpus/datasets.shtml
and for the VoxConverse we used dev v0.0.1/eval v0.0.2 sets from the official
GitHub repository333https://github.com/joonson/voxconverse, where the
development set is used only for tuning the parameters of the clustering
algorithm. That is, we used the same embedding extractors as described in the
previous section without fine-tuning on the AMI or VoxConverse data.

We used a simple speaker diarization pipeline including the following steps:
voice activity detection (VAD), overlapped speech detection (OSD; both from
[38, 39]), fixed-length segmentation, clustering, and post-processing. The
post-processing includes merging the adjacent sub-segments from the same
speaker and distributing the overlapped segments equally among the adjacent
segments with different speakers. We extracted embeddings from segments of
length $1.5$ sec with $0.75$ overlap within the boundaries computed by VAD.
These embeddings were further clustered using one of the following algorithms:

AHC: agglomerative hierarchical clustering (AHC) algorithm described in [16].
The stopping threshold was tuned on the development set.

VBx: clustering model based on Bayesian HMM from [16] with an initial
assignment by the AHC output. Similar to AHC, the VBx parameters were tuned on
development set.

VBx-UP: the proposed modification of VBx with uncertainty propagation that
uses reciprocals of magnitudes as segment-specific variances as shown in (5).

Finally, we reassign labels for segments with overlaps detected by the OSD
module. We used a simple heuristic: we assign two closest speakers by time to
every segment marked by OSD.

###  5.4 Evaluation results

We report the results using the _diarization error rate_ (DER) [40] and
_Jaccard error rate_ (JER) [41] metrics. In this experiment, we did not use
the forgiveness collar. In addition, the speaker overlap regions were included
during scoring at the evaluation step. We followed this evaluation setup in
all the reported experiments except the one shown in Table 4.

First, each of the three variants of the proposed two-step method outperformed
the baseline in the DER and JER metrics on the AMI Headset-Mix channel. In the
evaluation results provided in Table 3, we found out that the Algorithm 2.1
provides better results than the other two. Therefore, we report the results
only for the Algorithm 2.1 in Tables 4, 5 and 6.

Here and further adding a postfix to the clustering algorithm name indicates
that the two-step clustering procedure was used. The specific variant is shown
in brackets with the Algorithm 2.1 being a default option.

Clustering | DER, % | JER, %
---|---|---
AHC | 22.75 | 28.29
AHC-2step | 13.13 | 23.62
AHC-2step (2.2) | 19.07 | 25.75
AHC-2step (2.3) | 17.04 | 26.68
|  |
Table 3: Comparison of two-step clustering algorithms based on ResNet-MagFace embedding extractor and AHC clustering algorithm. The evaluation was performed on the AMI Headset-Mix channel without any forgiveness collar and with overlaps included. Dataset | Clustering | OSD | Setup | DER, % | JER, %
---|---|---|---|---|---
|  |  | collar | overlap |  |
AMI | AHC | No | 0.25 | No | 6.89 | 32.59
0.25 | Yes | 16.58
0 | Yes | 27.27
AHC-2step | No | 0.25 | No | 2.76 | 28.12
0.25 | Yes | 11.74
0 | Yes | 20.66
AHC | Yes | 0.25 | No | 6.89 | 28.29
0.25 | Yes | 13.91
0 | Yes | 22.75
AHC-2step | Yes | 0.25 | No | 2.76 | 23.62
0.25 | Yes | 7.07
0 | Yes | 13.13
VoxConverse | AHC | No | 0.25 | No | 5.96 | 28.01
0.25 | Yes | 7.61
0 | Yes | 11.83
AHC-2step | No | 0.25 | No | 4.24 | 28.53
0.25 | Yes | 5.84
0 | Yes | 9.41
AHC | Yes | 0.25 | No | 5.96 | 25.07
0.25 | Yes | 6.74
0 | Yes | 11.39
AHC-2step | Yes | 0.25 | No | 4.24 | 24.79
0.25 | Yes | 5.23
0 | Yes | 9.21
|  |  |  |  |  |
Table 4: Comparison of speaker diarization methods based on different clustering algorithms with and without OSD module. The evaluation was performed on the AMI test and VoxConverse test v0.0.2. Dataset | Model | Clustering | DER, % | JER, %
---|---|---|---|---
AMI | ResNet-ArcFace | AHC | 21.68 | 25.28
AHC-2step | 13.06 | 24.11
ResNet-MagFace | AHC | 22.75 | 28.29
AHC-2step | 13.13 | 23.62
Large ResNet-MagFace | AHC | 20.15 | 25.95
AHC-2step | 15.87 | 24.34
VoxConverse | ResNet-ArcFace | AHC | 10.71 | 24.40
AHC-2step | 7.95 | 25.28
ResNet-MagFace | AHC | 11.39 | 25.07
AHC-2step | 9.21 | 24.79
Large ResNet-MagFace | AHC | 8.87 | 20.47
AHC-2step | 7.68 | 22.65
|  |  |  |
Table 5: Comparison of speaker diarization methods based on different embedder models with Oracle VAD and OSD modules. The evaluation was performed on the AMI Headset-Mix channel and the VoxConverse test v0.0.2. Dataset | Clustering | DER, % | JER, %
---|---|---|---
AMI | AHC | 22.75 | 28.29
AHC-2step | 13.13 | 23.62
VBx | 16.82 | 23.33
VBx-2step | 11.15 | 22.38
VBx-UP | 17.60 | 25.50
VBx-UP-2step | 10.87 | 22.25
VoxConverse | AHC | 11.39 | 25.07
AHC-2step | 9.21 | 24.79
VBx | 10.73 | 24.33
VBx-2step | 9.76 | 23.93
VBx-UP | 8.28 | 26.93
VBx-UP-2step | 6.99 | 23.14
|  |  |
Table 6: Comparison of speaker diarization methods based on different
clustering algorithms with Oracle VAD and OSD modules. The evaluation was
performed on the AMI Headset-Mix channel and VoxConverse datasets. The
embedding extractor is the ResNet-MagFace.

Table 4 provides the results of AHC and AHC-2step methods with three
evaluation setups with different settings for the forgiveness collar and
overlap parameters. The proposed two-step approach allowed to achieve a
$60\%$, $49\%$ and $42\%$ relative reduction in DER with different evaluation
setups (from the most forgiving to the most strict, respectively) on the AMI
corpus. The JER metric was improved by $17\%$ and $13\%$ in cases with and
without the OSD module, correspondingly. On the VoxConverse data the DER
metric has similar behavior ($17\%$ relative reduction), however,
surprisingly, the JER metric remained stationary. Additionally, the two-step
approach with the overlapped speech detection module achieved higher relative
improvement ($49\%$) than the setup without the overlapped speech detection
($24\%$).

Next, Table 5 shows a comparison of different embedding extractors. The
ResNet-Arcface model performed slightly better than the ResNet-MagFace, but
the relative improvement in both DER and JER of the AHC-2step algorithm is
higher for the ResNet-MagFace on the AMI data ($40\%$ vs $42\%$ and $4\%$ vs
$17\%$ in DER and JER, respectively). Also, the Large ResNet-MagFace performed
the best among all setups with the one-step AHC algorithm, but it achieved the
least relative improvement in the DER metric. A similar observation can be
made for the VoxConverse data.

Finally, Table 6 demonstrates experimental results for different clustering
algorithms. One can observe that the two-step procedure achieved lower DER and
JER for all the reported clustering algorithms. Also, uncertainty propagation
with the two-step clustering further helped to improve DER and JER metrics
compared to the original VBx algorithm especially in combination with the two-
step clustering.

##  6 Analysis

In this section, we examine the quality propagation abilities of the learned
magnitudes for the ResNet-MagFace model.

First, we augmented the recordings from the VoxCeleb1 test with additive white
Gaussian noise with random SNR selected uniformly in $[-10;30]$. Figure 2
(Left) demonstrates a scatter plot of SNRs versus magnitudes for the
embeddings extracted from the noisy recordings. This result confirms the
interpretation of the embedding magnitudes as quality scores.

Figure 2: Left: SNR versus magnitude (VC1-test, additive white noise). Center:
duration versus magnitude (VC2-test). Right: distributions of the embedding
magnitudes for three different datasets. All the results are reported for the
ResNet-MagFace model. Best viewed in color.

Further, we examined the relation of the utterance duration to the embedding
magnitude. To reduce the effect of other possible sources of uncertainty, we
used the relatively clean VoxCeleb data (VC2-test). We picked a set of 2000
recordings and cropped them at a random moment selected uniformly in
$[1\text{s};15\text{s}]$. Figure 2 (Center) depicts the results. Surprisingly,
the magnitudes become larger for shorter segments, which somewhat contradicts
the interpretation of magnitudes as confidence scores. However, this effect
becomes more apparent only for small durations and can be easily compensated,
_e.g._ , by introducing the duration-dependent adjustment, which we describe
in section 3.1.

Finally, we estimated the distribution of magnitudes for three different sets
of recordings: the noise part of MUSAN corpus [21], the VoxCeleb1 test and the
VOiCES Eval set. From Figure 2 (Right) one can see that the learned magnitudes
allow for more accurate separation between the clean speech (VoxCeleb) and
non-speech (MUSAN) than those from the extractors shown on Figure 3.
Therefore, magnitudes can also be utilized for the out-of-distribution
detection in the embedding space. This property can possibly be employed as an
additional layer of defense to compensate for failures of a voice activity
detector, as suggested by [22].

We also conducted a quantitative evaluation of the ability of magnitudes to
serve as quality measures. We considered the speaker verification task, where
a recognizer can reject a trial if there is a high risk of making a wrong
prediction. In detail, given a trial list, we computed the confidence scores
as a sum of magnitudes of embeddings forming a trial pair. Then, we discarded
a subset of trials with the lowest confidence scores and computed the
performance metrics using the remaining trials. For this scenario we adopt the
VOiCES Eval set, which provides the recordings with varying quality. Fig. 3
illustrates how the equal error rate (EER) depends on the fraction of rejected
trials.

Figure 3: Results of speaker verification with a reject option for the VOiCES
Eval set. The plot presents EER vs. the fraction of rejected trials. Trials
with the lowest sum of embeddings magnitudes are discarded.

The magnitudes emerge useful for filtering out trials with incorrect
predictions. Even with the rejection fraction of 0.2, the EER on the VOiCES
Eval test is reduced by half, revealing the high capability of magnitudes for
quality assessment.

##  7 Conclusion

We presented a speaker embedding extractor providing interpretable magnitudes
suitable for out-of-the-box quality assessment or propagating the quality
information into the back-end model. We proposed a new probabilistic embedding
representation that can be obtained from a pre-trained extractor network with
almost no additional effort. We evaluated the proposed model for the speaker
verification task and achieved up to 20% relative improvements over the
magnitude-agnostic baselines, for both clean and noisy test sets. In addition,
we proposed a two-step clustering algorithm for speaker diarization, which
improved the DER and JER metrics relatively up to $60\%$ and $17\%$,
respectively, on two popular public benchmarks. Moreover, the proposed
modification of the VBx model with uncertainty propagation achieved better
performance compared to the original model. It is worth noting that the two-
step approach is faster than the one-step as it takes only a subset of
embeddings to perform computationally expensive clustering. We also provided a
comprehensive analysis of magnitudes properties, which highlights the
potential application for speech quality assessment tasks. We plan to make our
models and evaluation protocols available online.

In the future work, we aim at training a magnitude-aware embedding extractor
from scratch to get rid of the ad hoc duration variability compensation
transform. Another direction includes integrating the magnitude-based quality
assessment into the two-step pipelines based on the target-speaker VAD such as
[42].

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Geometrical properties of the space $I_{f}(X)$
of idempotent probability measures

Zaitov A. A.1, Ishmetov A. Ya.2
1 Tashkent institute of architecture and civil engineering

<EMAIL_ADDRESS>
2 Tashkent institute of architecture and civil engineering

<EMAIL_ADDRESS>
###### Abstract

In this paper we prove that for a compact space $X$ inclusion $I_{f}(X)\in
ANR$ holds if and only if $X\in ANR$. Further, it is shown that the functor
$I_{f}$ preserves property of a compact to be $Q$-manifold or a Hilbert cube,
properties of maps fibres to be $ANR$-compact, $Q$-manifold, Hilbert cube (the
finite of Hilbert cube).

Keywords: idempotent measure, compact Hausdorff space (compact), retract,
$AR$-space, $ANR$-space.

The theory of idempotent measures belongs to idempotent mathematics, i. e. the
field of the mathematics based on replacement of usual arithmetic operations
with idempotent (as, for example, $x\oplus y=\max\\{x,y\\}$). The idempotent
mathematics intensively develops at this time (see, for example, [1], survey
article [2] and the bibliography in it). Its communication with traditional
mathematics is described by the informal principle [2] according to which
there is a heuristic compliance between important, interesting and useful
designs the last and similar results of idempotent mathematics.

In the present article we investigate a subfunctor $I_{f}$ of a functor of
idempotent probability measures in category compact Hausdorff spaces. In
traditional mathematics to it there corresponds the functor $P_{f}$ of
probability measures. The concept of an idempotent measure (Maslov’s measure)
finds numerous applications in various field of mathematics, mathematical
physics and economy. In particular, such measures arise in problems of dynamic
optimization [3]; the analogy between Maslov’s integration and optimization is
noted also in [4]. In [5] it is claimed that use of measures of Maslov for
modeling of uncertainty in mathematical economy can be so relevant as far as
also use of classical probability theory.

Unlike a case of probability measures to which consideration extensive
literature is devoted (see [6]) geometrical and topological properties of the
spaces of idempotent measures were practically not investigated.

Consider set $\mathbb{R}_{+}=\mathbb{R}\bigcup\left\\{-\infty\right\\}$ with
two algebraic operations: addition $\oplus$ and multiplication $\odot$
determined as follows: $u\oplus v=\max\\{u,v\\}$ and $u\odot v=u+v$,
$u,v\in\mathbb{R}_{+}$, where $\mathbb{R}$ is the set of real numbers.

Let $X$ be a compact Hausdorff space ($\equiv$ a compact), $C(X)$ be the
algebra of continuous functions on $X$ with usual algebraic operations. On
$C(X)$ operations $\oplus$ and $\odot$ we will determine by rules
$\varphi\oplus\psi=\max\\{\varphi,\psi\\}$ and $\varphi\odot\psi=\varphi+\psi$
where $\varphi$, $\psi\in C(X)$.

Remind a functional $\mu:C(X)\to\mathbb{R}$ is called $\left[7\right]$ to be
an idempotent probability measure on $X$ if it satisfies the following
properties:

(1) $\mu({{\lambda}_{X}})=\lambda$ for all $\lambda\in\mathbb{R}$, where
${{\lambda}_{X}}$ – constant function;

(2) $\mu(\lambda\odot\varphi)=\lambda\odot\mu(\varphi)$ for all
$\lambda\in\mathbb{R}$ $\varphi\in C(X)$;

(3) $\mu(\varphi\oplus\psi)=\mu(\varphi)\oplus\mu(\psi)$ for all $\varphi$,
$\psi\in C(X)$.

For a compact $X$ we denote by $I(X)$ the set of all idempotent probability
measures on $X$. We consider $I(X)$ as subspace of ${{\mathbb{R}}^{C(X)}}$.

For given compacts $X$, $Y$ and continuous map $f:X\to Y$ it is possible to
check that determined by a formula $I(f)(\mu)(\psi)=\mu(\psi\circ f)$
naturally way arised map $I(f):I(X)\to I(Y)$ continuous. The construction $I$
is a normal functor. Therefore for each idempotent probability measure $\mu\in
I(X)$ one may determine its support:

$\text{supp}\mu=\bigcap\left\\{A\subset X:\overline{A}=A,\,\,\mu\in
I(A)\right\\}.$

For a positive integer $n$ and compact $X$ define the following set

${{I}_{n}}(X)=\left\\{\mu\in I(X):\,\,\left|\operatorname{supp}\mu\right|\leq
n\right\\}.$

Put

${{I}_{\omega}}(X)=\bigcup\limits_{n=1}^{\infty}{{{I}_{n}}(X)}.$

Set ${{I}_{\omega}}(X)$ everywhere dense $\left[7\right]$ in $I(X)$.
Idempotent probability measure $\mu\in{{I}_{\omega}}(X)$ is called an
idempotent probability measure with finite support.

We determine a subfunctor $I_{f}$ of the functor $I$ of idempotent probability
measures. For a given compact $X$ a set $I_{f}(X)$ consists of measures with
finite support such that if support of a measure $\mu$ consists from $n$
points $x_{1},x_{2},...,x_{n}$, that only a unique of these points
$max$-$plus$-barycenter mass is equal to zero, all other masses are not big
$-ln(n+1)$. By definition

$I_{f}(X)=\\{\mu=\bigoplus_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}\in
I_{\omega}(X):\mbox{ equality }\lambda_{i_{0}}=0\mbox{ holds for only unique
}$ $\mbox{ index }i_{0}\mbox{ and }\lambda_{i}\leq-ln(n+1)\mbox{ for all
}i\in\\{1,...,n\\}\setminus\\{i_{0}\\}\\}$

This functor is interesting that it is a functor with the finite support, and
has not finite degree. Functor $I_{f}:Comp\rightarrow Comp$ is a normal
subfunctor of a functor $I$ idempotent probability measures.

From definition of elements of the space $I_{f}(X)$ follows that a set
$\delta(X)$ Dirac’s measures lies in $I_{f}(X)$.

Let $X$ $Y$ be two compacts lying in spaces $M$ and $N$ respectively, where
$M,N\in AR$. A sequence of maps $f_{k}:M\rightarrow N$, $k=1,2,...,$ is called
to be fundamental sequence from $X$ into $Y$, if for each neighbourhood $V$ of
the compact $Y$ (in $N$) there is such neighbourhood $U$ of the compact $X$
(in $M$), that $f_{k}|_{U}=f_{k+1}|_{U}$ at $V$ almost for all $k$. It means
that there is such homotopy $f_{k}:U\times[0,1]\rightarrow V$,
$f_{k}(x,0)=f_{k}(x)$ $f_{k+1}(x,1)=f_{k+1}(x)$ for all $x\in U$. We will
denote this fundamental sequence through $\left\\{f_{k},X,Y\right\\}$ or
shortly through $f$, also we will write $f:X\rightarrow Y$.

A fundamental sequence $f=\left\\{f_{k},X,Y\right\\}$ is generated by map
$f:X\rightarrow Y$ if $f_{k}(x)=f(x)$ for all $x\in X$ and for all
$k=1,2,...$.

Spaces $X$ and $Y$ are fundamentally equivalent if there are such two
fundamental sequences $f:X\rightarrow Y$ and $g:Y\rightarrow X$ that
$gf=id_{X}$ and $fg=id_{Y}$.

The relation of fundamental equivalence is the equivalence relation therefore
the class of all spaces decomposes to in pairwise disjoint classes of spaces
which are called shape. So two spaces belong to the same shape if and only if
when they are fundamentally equivalent. A shape containing a space $X$ is
called [10] a shape of the space $X$ and denote by $Sh(X)$. It is known that
for two neighbourhood retract of $A$ and $B$ fairly $Sh(A)=Sh(B)$ if and only
if when they are homotopically equivalent.

Let’s remind that maps $f:X\rightarrow Y$ is called cellular and similar
(briefly – $CE$) [11], if any compact $A\subset Y$ prototype $f^{-1}(A)$ is a
compact and for each point $y\in Y$ prototype $f^{-1}(y)$ has sheyp points
(i.e. a prototype $f^{-1}(y)$ homotopically it is equivalent to a point).

If $r:X\rightarrow F$ is a retraction and there also exists such a homotopy
$h:X\times[0,1]\rightarrow F$ that $h(x,0)=x$, $h(x,1)=r(x)$ for all $x\in X$
then $r$ is deformation retraction, and $F$ is deformation retract of the
space $X$. Deformation retraction $r:X\rightarrow F$ is strongly deformation
retraction if for a homotopy $h:X\times[0,1]\rightarrow F$ we have $h(x,t)=x$
for all $x\in F$ and all $t\in[0,1]$ [12].

Take any measure
$\mu=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}\in I_{f}(X)$.
Let $\lambda_{i_{0}}=0>-ln(n+1)$. Measure $\mu$ puts in correspondence a point
$\delta_{x_{i_{0}}}$ of a compact $\delta(X)$. The obtained correspondence
$I_{f}(X)\rightarrow\delta(X)$ denote by $r_{\delta(X)}^{I(X)}$.

1. For any compact $X$ a map $r_{\delta(X)}^{I(X)}:I_{f}(X)\rightarrow\delta(X)$ is continuous, open, cellular-similar (all fibres are collapsible) retraction.

Proof. By construction the map
$r_{\delta(X)}^{I(X)}:I_{f}(X)\rightarrow\delta(X)$ is defined correctly. It
is clear that $r_{\delta(X)}^{I(X)}(\delta_{x})=\delta_{x}$ for each $x\in X$,
i. e. every point of the space $\delta(X)$ is fixed-point according to the map
$r_{\delta(X)}^{I(X)}$. So, it is established that $r_{\delta(X)}^{I(X)}$ is
retraction.

It is clear, that for each point $x\in X$ the fibre
$\left(r_{\delta(X)}^{I(X)}\right)^{-1}(\delta_{x})$ is compact. It is clear,
that for each point $\mu\in\left(r_{\delta(X)}^{I(X)}\right)^{-1}(\delta_{x})$
an interval
$[\mu,\delta_{x}]=\\{\lambda_{1}\odot\delta_{x}\oplus\lambda_{2}\odot\mu:\lambda_{1}\oplus\lambda_{2}=\mathbf{1};\,\,\lambda_{1}\geq\mathbf{0},\lambda_{2}\geq\mathbf{0}\\}$
lies in the fibre $\left(r_{\delta(X)}^{I(X)}\right)^{-1}(\delta_{x})$.

Fix a fibre $\left(r_{\delta(X)}^{I(X)}\right)^{-1}(\delta_{x})$ and define a
map
$h:\left(r_{\delta(X)}^{I(X)}\right)^{-1}(\delta_{x})\times[0,1]\rightarrow\left(r_{\delta(X)}^{I(X)}\right)^{-1}(\delta_{x})$
by the rule

$h(\mu,t)=\left(ln(1-t)-lnt\oplus
ln(1-t)\right)\odot\delta_{x}\oplus\left(lnt-lnt\oplus
ln(1-t)\right)\odot\mu,$

where
$\mu=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}\in\left(r_{\delta(X)}^{I(X)}\right)^{-1}(\delta_{x})$
and $t\in[0,\ 1]$.

It is easy to see that $h$ is a homotopy, connecting identity mapping,
$h_{1}=id_{\left(r_{\delta(X)}^{I(X)}\right)^{-1}(\delta_{x})}$ and stationary
map
$h_{0}=\left(r_{\delta(X)}^{I(X)}\right)^{-1}(\delta_{x})\rightarrow\\{\delta_{x}\\}$.

So each pre-image has shape of a point, i. e. a retraction
$r_{\delta(X)}^{I(X)}$ is cellularity-similar. Moreover, these pre-images
subtend to a point.

By virtue of $I_{f}(X)=\bigcup\limits_{x\in
X}\left(r_{\delta(X)}^{I(X)}\right)^{-1}(\delta_{x})$ and
$\left(r_{\delta(X)}^{I(X)}\right)^{-1}(y)\cap\left(r_{\delta(X)}^{I(X)}\right)^{-1}(z)=\varnothing$
on $y\neq z$, we obtain that each fibre
$\left(r_{\delta(X)}^{I(X)}\right)^{-1}(x)$, $x\in X$, is open and closed in
$I_{f}(X)$. Therefore map $r_{\delta(X)}^{I(X)}:I_{f}(X)\rightarrow\delta(X)$
is continuous.

It is remain to show the map $r_{\delta(X)}^{I(X)}$ is open. Let $\mu\in
I_{\omega}(X)$,
$\mu=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}$. Define a set

$\left\langle\mu;U_{1},...,U_{n};\varepsilon\right\rangle=\left\\{\nu=\bigoplus\limits_{j=1}^{k}\gamma_{i}\odot\delta_{y_{i}}\in
I_{\omega}(X):supp\nu\cap U_{i}\neq\varnothing,i=1,2,...,n,\right.$
$\left.supp\nu\subset\bigcup\limits_{i=1}^{n}U_{i},\ \
\bigoplus\limits_{y_{j}\in
U_{i}}|\lambda_{i}-\gamma_{j}|<\varepsilon\right\\},$

where $U_{i}$ are open neighbourhoods of points $x_{i}$, $i=1,...,n$,
respectively. It is clear, that set
$\left\langle\mu;U_{1},...,U_{n};\varepsilon\right\rangle$ is open in
$I_{\omega}(X)$. Let’s show that sets of a type
$\left\langle\mu;U_{1},...,U_{n};\varepsilon\right\rangle$ form base of weak
topology in $I_{\omega}(X)$. Let
$\left\langle\mu;\varphi;\varepsilon\right\rangle$ be an element of the
prebase where $\varphi\in C(X)$, $\varepsilon>0$ and
$\mu=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}\in
I_{\omega}(X)$. As function $\varphi$ is continuous, for each point $x_{i}$
there is its open neighbourhood $U_{i}$ such that for any point $y\in U_{i}$
the inequality $|\varphi(x_{i})-\varphi(y)|<\frac{\varepsilon}{2}$ holds.
Further, for every
$\nu=\bigoplus\limits_{j=1}^{k}\gamma_{j}\odot\delta_{y_{j}}\in\left\langle\mu;U_{1},...,U_{n};\frac{\varepsilon}{2}\right\rangle$
we have $\bigoplus\limits_{y_{i}\in
U_{i}}\left|\lambda_{i}-\gamma_{j}\right|<\frac{\varepsilon}{2}$. Let’s
estimate an absolute value
$|\mu(\varphi)-\nu(\varphi)|=\left|\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\varphi(x_{i})-\bigoplus\limits_{j=1}^{k}\gamma_{j}\odot\varphi(y_{j})\right|=a$.
Two cases are possible:

1)
$\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\varphi(x_{i})\geq\bigoplus\limits_{j=1}^{k}\gamma_{j}\odot\varphi(y_{j})$.
Let
$\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\varphi(x_{i})=\lambda_{i^{\prime}}\odot\varphi(x_{i^{\prime}})$.
Then

$a=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\varphi(x_{i})-\bigoplus\limits_{j=1}^{k}\gamma_{j}\odot\varphi(y_{j})=\lambda_{i^{\prime}}\odot\varphi(x_{i^{\prime}})-\bigoplus\limits_{j=1}^{k}\gamma_{j}\odot\varphi(y_{j})\leq(\mbox{for
each }y_{j}\in U_{i^{\prime}})$.
$\leq\lambda_{i^{\prime}}\odot\varphi(x_{i^{\prime}})-\gamma_{j}\odot\varphi(y_{j})=|\lambda_{i^{\prime}}\odot\varphi(x_{i^{\prime}})-\gamma_{j}\odot\varphi(y_{j})|\leq|\lambda_{i^{\prime}}-\gamma_{j}|+|\varphi(x_{i^{\prime}})-\varphi(y_{j})|<\varepsilon$.

2)
$\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\varphi(x_{i})\leq\bigoplus\limits_{j=1}^{k}\gamma_{j}\odot\varphi(y_{j})$.
Let
$\bigoplus\limits_{j=1}^{n}\gamma_{j}\odot\varphi(y_{j})=\gamma_{j^{\prime}}\odot\varphi(y_{j^{\prime}})$.
Then

$a=\bigoplus\limits_{j=1}^{n}\gamma_{j}\odot\varphi(y_{j})-\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\varphi(x_{i})=\gamma_{j^{\prime}}\odot\varphi(y_{i^{\prime}})-\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\varphi(x_{i})\leq(\mbox{for
all }{i},\mbox{ for which }y_{j^{\prime}}\in
U_{i})\leq\gamma_{i}\odot\varphi(y_{j^{\prime}})-\lambda_{i}\odot\varphi(x_{i})=|\gamma_{j^{\prime}}\odot\varphi(y_{j^{\prime}})-\lambda_{i}\odot\varphi(x_{i})|\leq|\lambda_{i}-\gamma_{j^{\prime}}|+|\varphi(x_{i})-\varphi(y_{j^{\prime}})|<\varepsilon$.

So, $a<\varepsilon$ for two cases, i. e.
$|\mu(\varphi)-\nu(\varphi)|<\varepsilon$. From here
$\nu\in\left\langle\mu;\varphi;\varepsilon\right\rangle$, in other words,

$\left\langle\mu;U_{1},...,U_{n};\frac{\varepsilon}{2}\right\rangle\subset\left\langle\mu;\varphi;\varepsilon\right\rangle.$

Let now $\mu=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}\in
I_{f}(X)$, $r_{\delta(X)}^{I(X)}(\mu)=\delta_{x_{i_{0}}}$, and $U_{i}$ be
neighbourhoods of points $x_{i}$ in $X$, respectively,
$\left\langle\mu;U_{1},...,U_{n};\varepsilon\right\rangle$ be a neighbourhood
of an idempotent probability measure $\mu$ in $I_{f}(X)$. Denote
$V=U_{i_{0}}$. It is easy to check that at $0<\varepsilon<ln3$ relations
$V\cap U_{i}=\varnothing$, $i=1,...,n$, $i\neq i_{0}$ take place. Therefore
$r_{\delta(X)}^{I(X)}(\nu)\in\delta(V)$ for any
$\nu\in\left\langle\mu;U_{1},...,U_{n};\varepsilon\right\rangle$, where
$\delta(V)=\\{\delta_{x}:x\in V\\}$ is open in $\delta(X)$ set. From here
$r_{\delta(X)}^{I(X)}(\left\langle\mu;U_{1},...,U_{n};\varepsilon\right\rangle)\subset\delta(V)$.
Now for each point $y\in V$ we construct an idempotent probability measure

$\mu_{y}=0\odot\delta_{y}\oplus\bigoplus\limits_{\begin{subarray}{c}i=1\\\
i\neq i_{0}\end{subarray}}^{n}\lambda_{i}\odot\delta_{x_{i}}.$

Then, as it is easy to see,
$\mu_{y}\in\left\langle\mu;U_{1},...,U_{n};\varepsilon\right\rangle$ and
$r_{\delta(X)}^{I(X)}(\mu_{y})=\delta_{y}$. Thus, for each point $y\in V$ we
have
$\left(r_{\delta(X)}^{I(X)}\right)^{-1}(\delta_{y})\in\left\langle\mu;U_{1},...,U_{n};\varepsilon\right\rangle$,
i. e.
$r_{\delta(X)}^{I(X)}(\left\langle\mu;U_{1},...,U_{n};\varepsilon\right\rangle)=\delta(V)$.
Openness of the map $r_{\delta(X)}^{I(X)}$ is established. Theorem 1 is
proved.

Proposition 1. For any compact $X$ the space $\delta(X)$ is a strong
deformation retract of the compact $I_{f}(X)$.

Proof. Consider a map $h:I_{f}(X)\times[0,1]\rightarrow I_{f}(X)$, determined
by a formula

$h(\mu,t)=h_{t}(\mu)=\left(ln(1-t)-lnt\oplus
ln(1-t)\right)\odot\mu\oplus\left(lnt-lnt\oplus ln(1-t)\right)\odot
r_{f}^{x}(\mu),$ $(\mu,t)\in I_{f}(X)\times[0,1].$

It is easy to check that the map $h$ is defined correctly. Moreover,
$h_{0}=id_{I_{f}(X)}$ and $h_{1}=r_{\delta(X)}^{I(X)}$, i. e. $h$ is the
homotopy connecting maps $id_{I_{f}(X)}$ and $r_{\delta(X)}^{I(X)}$. Further,
we have

$h(\delta_{x},t)=\left(ln(1-t)-lnt\oplus
ln(1-t)\right)\odot\delta_{x}\oplus\left(lnt-lnt\oplus ln(1-t)\right)\odot
r_{f}^{x}(\delta_{x})=\delta_{x},$

i. e. $h_{t}(\delta_{x})=\delta_{x}$ for all $\delta_{x}\in\delta(X)$ and
$t\in[0,1]$. Thus, $\delta(X)$ is a strong deformation retract of a compact
$I_{f}(X)$. Proposition 1 is proved.

Proposition 2. For any finite compact $X$ the set $I_{f}(X)$ is an
neighbourhood retract of the compact $I(X)$.

Proof. For each measure of Dirac $\delta_{x}$, $x\in X$, we build an open set

$O\delta_{x}=\left\\{\mu=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}\in
I(X):\lambda_{i}=0\mbox{ at }x_{i}=x\mbox{ and }\lambda_{i}<-ln2\mbox{ at
}x_{i}\neq x\right\\}.$

It is clear, that $O\delta_{x}\cap O\delta_{y}=\varnothing$ at $x\neq y$.

Consider open set $\bigcup\limits_{x\in X}O\delta_{x}$ in $I(X)$. We have
$I_{f}(X)\subset\bigcup\limits_{x\in X}O\delta_{x}$. Now we will show that
$I_{f}(X)$ is retract of the open set $\bigcup\limits_{x\in X}O\delta_{x}$.
Take $\nu\in\bigcup\limits_{x\in X}O\delta_{x}$, where
$\nu=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}$,
$\bigoplus\limits_{i=1}^{n}\lambda_{i}=\mathbf{1}$,
$\lambda_{i}\geq\mathbf{0}$, $i=1,...,n$. Then $\lambda_{i_{0}}=0$ for some
(only unique) $i_{0}$ and, therefore
$r_{\delta(X)}^{I(X)}(\nu)=\delta_{x_{i_{0}}}$. Define a map
$r:\bigcup\limits_{x\in X}O\delta_{x}\rightarrow I_{f}(X)$ by the rule
$r(\nu)=$

$=\begin{cases}0\odot\delta_{x_{i_{0}}}\oplus\bigoplus\limits_{\begin{subarray}{c}i=1,\\\
i\neq i_{0}\end{subarray}}^{n}(-ln(n+1))\odot\delta_{x_{i}},&\text{if
$\lambda_{i}\geq-ln(n+1)$ for any $i\in\\{1,...,n\\}\setminus\\{i_{0}\\}$},\\\
\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}},&\text{if
$\lambda_{i}\leq-ln(n+1)$ for all
$i\in\\{1,...,n\\}\setminus\\{i_{0}\\}$.}\end{cases}$

The map $r$ is defined correctly. It is continuous. Besides $r(\mu)=\mu$ for
any measure $\mu\in I_{f}(X)$. It means the map $r$ is a retraction.
Proposition 2 is proved.

Remind [13] that a set $A\subset X$ is called to be collapsible by space $X$
to a set $B\subset X$ if embedding map $i_{A}:A\rightarrow X$ is homotopic to
some map $f:A\rightarrow X$ such that $f(A)\subset B$. If $B$ consists of one
point, they say that $A$ is collapsible by $X$.

It is clear, if there is a homotype $h:A\times I\rightarrow A$, such that
$h(y,0)=i_{A}$, and $h(y,1)=\\{\text{a point}\\}$ then $A$ collapsible by $X$.

A space $X$ is called to be locally collapsible to a point $x_{0}\in X$ if any
neighbourhood $U$ of the point $x_{0}$ contains a neighbourhood $U_{0}$ such,
that collapsible to by $U$ to a point. A space $X$ is called to be locally
collapsible if it is locally collapsible to each point.

Theorem 2. The functor $I_{f}$ preserves a collapsibility of compacts i. e. if
$X$ is collapsible compact, then $I_{f}(X)$ is also collapsible compact.

Proof. We will show more: the functor $I_{f}$ preserves a homotopy of maps.
Let $h_{0}$, $h_{1}:X\rightarrow Y$ be homotopic maps,
$h:X\times[0,1]\rightarrow Y$ is the homotopy connecting the maps $h_{0},\ \
h_{1}$, i. e. $h(x,0)=h_{0}(x)$, $h(x,1)=h_{1}(x)$. The identity map
$i_{t_{0}}:X\times\\{t_{0}\\}\rightarrow X\times I$, determined by equality
$i_{t_{0}}(x,t_{0})=(x,t_{0})$, $x\in X$, defines an identity map
$I_{f}(i_{t_{0}}):I_{f}(X\times\\{t_{0}\\})\rightarrow I_{f}(X\times I)$. But,
for every $t_{0}\in[0,1]$ the space $I_{f}(X\times\\{t_{0}\\})$ is naturally
homeomorphic to $I_{f}(X)\times\\{t_{0}\\}$. This homeomorphism can be carried
out, as it is easy to see, by means of correspondence
$\mu_{t_{0}}\leftrightarrow(\mu,t_{0})$, where
$\mu_{t_{0}}=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{(x_{i},t_{0})}\in
I_{f}(X\times\\{t_{0}\\})$ and
$\mu=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}\in I_{f}(X)$.

Define now a map $I_{f}(h):I_{f}(X)\times[0,1]\rightarrow I_{f}(Y)$ by
equality
$I_{f}(h)\left(\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}},t\right)=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{h(x_{i},t)}$.
We have

$I_{f}(h)\left(\bigoplus_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}},0\right)=\bigoplus_{i=1}^{n}\lambda_{i}\odot\delta_{h(x_{i},0)}=\bigoplus_{i=1}^{n}\lambda_{i}\odot\delta_{h_{0}(x_{i})}=I_{f}(h_{0})\left(\bigoplus_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}\right),$
$I_{f}(h)\left(\bigoplus_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}},1\right)=\bigoplus_{i=1}^{n}\lambda_{i}\odot\delta_{h(x_{i},1)}=\bigoplus_{i=1}^{n}\lambda_{i}\odot\delta_{h_{1}(x_{i})}=I_{f}(h_{1})\left(\bigoplus_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}\right),$

i. e. $I_{f}(h)(\mu,0)=I_{f}(h_{0})(\mu)$ and
$I_{f}(h)(\mu,1)=I_{f}(h_{1})(\mu)$ for any $\mu\in I_{f}(X)$. In other words,
$I_{f}(h)$ is the homotopy connecting the maps $I_{f}(h_{0})$ and
$I_{f}(h_{1})$. Thus, functor $I_{f}$ preserves a homotopyness of maps.
Theorem 2 is proved.

Lemma 1. For a compact $X$ the set $I_{f}(X)$ is the neighbourhood retract of
the space $I_{\omega}(X)$.

Proof. Consider the following set

$O_{\delta_{x}}=\left\\{\mu=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}\in
I_{\omega}(X):\lambda_{i}=0\mbox{ at }x_{i}=x\mbox{ and
}\lambda_{i}<-ln2\mbox{ at }x_{i}\neq x\right\\}.$

We will show that $O_{\delta_{x}}$ is open in $I_{\omega}(X)$. Let
$\nu=\bigoplus\limits_{i=1}^{k}\lambda_{i}\odot\delta_{x_{i}}\in
O_{\delta_{x}}$. Then $\lambda_{i}=0$ at $x_{i}=x$ and $\lambda_{i}<-ln2$ at
$x_{i}\neq x$. For points $x_{1},\ ...,\ x_{k}$ we choose disjoint open
neighbourhoods $U_{i}$, $i=1,\ ...,\ k$. Put
$\varepsilon=\min\left\\{-\lambda_{i}-ln2:\ i\in\\{1,\ ...,\
k\\}\setminus\\{i_{0}\\}\right\\}>0$. Let
$\mu=\bigoplus\limits_{j=1}^{s}\gamma_{j}\odot\delta_{y_{j}}\in\left\langle\nu;U_{1},...,U_{n};\varepsilon\right\rangle$.
Then $supp\mu\cap U_{i}\neq\varnothing$, and
$supp\mu\subset\bigcup\limits_{i=1}^{k}U_{i}$ and $\bigoplus\limits_{y_{j}\in
U_{i}}|\lambda_{i}-\gamma_{j}|<\varepsilon$. On construction for every $i$ and
for all $j$, $y_{j}\in U_{i}$, we have

$|\lambda_{i}-\gamma_{j}|\leq\bigoplus\limits_{y_{j}\in
U_{i}}|\lambda_{i}-\gamma_{j}|<\varepsilon=\min\left\\{-\lambda_{i}-ln2:\
i=1,\ ...,\ k,i\neq i_{0}\right\\}\leq-\lambda_{i}-ln2,$

i. e. $|\lambda_{i}-\gamma_{j}|<-\lambda_{i}-ln2.$ From where we have
$\gamma_{j}<-ln2$. Hence,
$\left\langle\nu;U_{1},...,U_{n};\varepsilon\right\rangle\subset
O_{\delta_{x}}$. Thus, the set $O_{\delta_{x}}$ is open.

It is obvious that $I_{f}(X)\subset\bigcup\limits_{x\in X}O_{\delta_{x}}$. If
$\nu\in\bigcup\limits_{x\in X}O_{\delta_{x}}$, that exists only $x\in X$, that
$\nu\in O_{\delta_{x}}$.Let’s construct map $r:\bigcup\limits_{x\in
X}O_{\delta_{x}}\rightarrow I_{f}(X)$ as follows

$r(\nu)=$

$=\begin{cases}0\odot\delta_{x_{i_{0}}}\oplus\bigoplus\limits_{\begin{subarray}{c}i=1,\\\
i\neq i_{0}\end{subarray}}^{n}(-ln(n+1))\odot\delta_{x_{i}},&\text{if
$\lambda_{i}\geq-ln(n+1)$ for some
$i\in\\{1,...,n\\}\setminus\\{i_{0}\\}$},\\\
\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}},&\text{if
$\lambda_{i}\leq-ln(n+1)$ for all
$i\in\\{1,...,n\\}\setminus\\{i_{0}\\}$,}\end{cases}$

$\nu=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{x_{i}}\in
O_{\delta_{x}}$, $\lambda_{i_{0}}=0$ for unique $i_{0}$ and $\lambda_{i}<-ln2$
at all $i\in\left\\{1,...,n\right\\}\backslash\\{i_{0}\\}$.

It is easy to see, the map $r$ is a retraction. Lemma 1 is proved.

Theorem 3. Let $X$ be an $A(N)R$-compact. Then $I_{f}(X)$ is also
$ANR$-compact.

Proof. Let $X$ be neighbourhood retract some compact $Y$, $U$ be open set in
$Y$ such that $U\supset X$ and there is a retraction $r:U\rightarrow X$.

Consider open set $\bigcup\limits_{x\in U}O\delta_{x}$ in $I_{f}(Y)$. As we
above note $I_{f}(X)\subset\bigcup\limits_{x\in U}O\delta_{x}$, and if
$\nu\in\bigcup\limits_{x\in X}O_{\delta_{x}}$ then there exists unique $x\in
X$ such that $\nu\in O_{\delta_{x}}$. For $\nu\in\bigcup\limits_{x\in
U}O\delta_{x}\subset I_{f}(Y)$,
$\nu=\bigoplus\limits_{i=1}^{n}\lambda_{i}\odot\delta_{y_{i}}$,
$\bigoplus\limits_{i=1}^{n}\lambda_{i}=\mathbf{1}$,
$\lambda_{i}\geq\mathbf{0}$, $i=1,2,...,n$, $\lambda_{i_{0}}=0$ put

$r_{U}^{Y}(\nu)=\left(\lambda_{i_{0}}\oplus\bigoplus\limits_{y_{i}\in
Y\backslash
U}\lambda_{i}\right)\odot\delta_{y_{i_{0}}}\oplus\bigoplus\limits_{y_{i}\in
U}\lambda_{i}\odot\delta_{y_{i}}.$

It is obvious that $r_{U}^{Y}(\nu)\in\bigcup\limits_{x\in U}O\delta_{x}$.
Besides, $r_{U}^{Y}(\nu)=\nu$ for any measure $\nu\in I_{f}(Y)$ such that
$supp\nu\subset U$. As the operation of capture of a maximum is continuous,
the constructed map $r_{U}^{Y}:\bigcup\limits_{x\in
U}O\delta_{x}\rightarrow\bigcup\limits_{x\in U}O\delta_{x}$ is continuous.
Further, we will put

$R\left(r_{U}^{Y}(\nu)\right)=\left(\lambda_{i_{0}}\oplus\bigoplus\limits_{y_{i}\in
Y\backslash
U}\lambda_{i}\right)\odot\delta_{r\left(y_{i_{0}}\right)}\oplus\bigoplus\limits_{y_{i}\in
U}\lambda_{i}\odot\delta_{r\left(y_{i}\right)}.$

On construction we have $R\left(r_{U}^{Y}(\nu)\right)\in I_{f}(X)$. The map
$R:\bigcup\limits_{x\in U}O\delta_{x}\rightarrow I_{f}(X)$ is defined
correctly. As retraction $r:U\rightarrow X$ is continuous, the map $R$ is also
continuous. It is easy to check that $R\left(r_{U}^{Y}(\nu)\right)=\nu$ for
any measure $\nu\in I_{f}(X)$. Thus, $R\circ r_{U}^{Y}:\bigcup\limits_{x\in
U}O\delta_{x}\rightarrow I_{f}(X)$ is required retraction. So, the set
$I_{f}(X)$ is neighbourhood retract of the compact $I_{f}(Y)$.

Now application of a Lemma 1 and theorem (3.1) from work [13] finishes the
proof of Theorem 3.

In funktorial mean Theorem 3 looks as so:

Corollary 1. Functor $I_{f}$ preserves $ANR$-compacts.

Theorem 3 and Proposition 1 imply the following important result.

Corollary 2. Let $X$ be a compact. $I_{f}(X)\in ANR$ if and only if $X\in
ANR$.

Further, Theorem 3 implies the following statements.

Corollary 3. Functor $I_{f}$ preserves property of a compacts to be
$Q$-manifold or Hilbert cube.

Corollary 4. Functor $I_{f}$ preserves property of fibres of maps to be
$ANR$-compact, compact $Q$-manifold and Hilbert cube (finite sum of Hilbert
cube).

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