# Topological data analysis of human vowels: Persistent homologies across representation spaces

Guillem Bonafos, Pierre Pudlo, Jean-Marc Freyermuth, Samuel Tronçon, Arnaud Rey

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## To cite this version:

Guillem Bonafos, Pierre Pudlo, Jean-Marc Freyermuth, Samuel Tronçon, Arnaud Rey. Topological data analysis of human vowels: Persistent homologies across representation spaces. Speech Communication, 2026, 178, pp.103363. 10.1016/j.specom.2026.103363. hal-04638297

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Submitted on 4 Mar 2026

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Topological data analysis of human vowels:
Persistent homologies across representation spaces

Guillem Bonafos<sup>a,b,c,d,*</sup>, Pierre Pudlo<sup>a,c</sup>, Jean-Marc Freyermuth<sup>a,c,1</sup>, Samuel Tronçon<sup>d</sup>, Arnaud Rey<sup>b,c</sup>

<sup>a</sup>Aix Marseille Univ, CNRS, I2M, Marseille, France

<sup>b</sup>Aix Marseille Univ, CNRS, LPC, Marseille, France

<sup>c</sup>Aix Marseille Univ, ILCB, Aix-en-Provence, France

<sup>d</sup>Résurgences R&D, Arles, France

**Abstract**

Topological Data Analysis (TDA) has been successfully used for various tasks in signal/image processing, from visualization to supervised/unsupervised classification. Here, we investigate the contribution of TDA to a human vowel classification task. We compare the contribution of the topological signature of three representation spaces for three classification tasks: i) vowel identification; ii) sex identification; iii) speaker identification. We show that topologically-augmented random forest improves the Out-of-Bag Error (OOB) over solely based Mel-Frequency Cepstral Coefficients (MFCC) for the vowels and speaker identification tasks. Our results also suggest that the topological information extracted from different signal representations is complementary to solve each classification tasks. These data therefore suggest that TDA provides new sources of information for improving performance in human voice processing.

**Keywords:** TDA, topologically-augmented machine learning, persistent homology, representation space, signal classification, human vowel

# 1. Introduction

Topological Data Analysis (TDA) is a fast-growing research area that relies on deep mathematical foundations [1, 2, 3]. It offers novel and potentially fruitful tools for digital audio signal processing and new angles of analysis. This innovative approach to data science is based on extracting information from the *shape of data*.

TDA has already been applied to various signal processing problems [4, 5, 6]. It starts from the assumption that the data have a shape [7] and it computes its persistent homologies, which provide a compact representation of its topological features. These topological representations are stable to perturbations of input data and independent of dimensions and coordinate systems. However, this shape strongly depends on the way a signal is represented (i.e., on the representation space). The purpose of this work is to study how the computation of persistent homologies depends on the chosen representation spaces, considering the specific task of vowels categorization in human language. We study the impact of the representation space on the extracted topological information and determine if accessing higher-dimensional persistent homologies allows getting more discriminant information. We also discuss what is the best way to summarize the information contained in a persistence diagram for our specific classification tasks.

This article is organized as follows. First, we introduce the problem and its rationale. Second, we describe in a nutshell the theory and the processing pipeline of TDA. Third, we present the strategy for investigating the problem, the nature of the acquired dataset and our classification aims. Fourth, we report the main results that are extensively discussed in the last section.

<footnote>*Corresponding author</footnote>

<footnote n="1">¹This publication is based upon work supported by King Abdullah University of Science and Technology Research Funding (KRF) under Award No. ORFS-2022-CRG11-5025.2</footnote>
## 2. The problem

### 2.1. Motivations for TDA

Topology is the branch of mathematics that deals with the qualitative geometric information of a space [1]. The tools provided by algebraic topology allow us to capture the shape of the data [8]. The topological approach frees itself from the question of metrics and coordinates by studying the properties of a space through its connectivity. It has an interesting explanatory power thanks to its great potential for visualization, and the topological features have a discriminant power which makes TDA a particularly interesting candidate for the classification of natural signals. In this section, we give an overview of application of TDA in signal processing. Additional details on the theoretical foundations of TDA are given in 3.

A central feature in TDA is the computation of persistent homologies [9]. Several pipelines have been proposed for the computation and use of such topological descriptors in data analysis. It typically consists in calculating persistent homologies on the input data (e.g., an audio signal) or a representation of this data (e.g., its spectrogram), vectorizing persistence diagrams, and using these characteristics in a model [3, 10]. Persistence diagrams and some of their representations have been shown to be stable against noise [11, 3, 12], meaning that small perturbations of the input data results in small changes in the persistent diagram. This stability makes the topological approach an excellent candidate for the description of noisy natural signals. Altogether, the interesting perspectives offered by TDA to face Big Data challenges combined with the fast development of tools for the efficient computation of topological descriptors has led to a proliferation of studies demonstrating the added value of TDA in a variety of contexts.

For example, [4, 5] demonstrate the usefulness of TDA for signal processing and for the study of signals on graphs. Topological tools are also useful in analyzing the shape of time series [13], for object detection in images [14] or in sound detection [15]. Besides detection tasks, topological descriptors are also useful for the classification of sound signals [16], musical signals [17] and speech signals [6]. It impacts multiple scientific domains alike biology, medicine, ecology [18], neurosciences, e.g., for fMRI [19] or EEG data [20, 21] for which TDA allows the construction of invariant signal descriptors. Topological features are complementary to more classical descriptors, and they allow capturing global and high-dimensional information useful for signal analysis. However, the impact of the representation space of the signal on the extracted topological information is still an unexplored question.

### 2.2. Representation space

Our study focuses on physical signals, in particular, sound signals. A raw data signal can be represented as different data objects, in different, here-called, representation spaces. First, we opted for the spectrogram because of its widespread use, notably in voice classification. Second, we have used a representation based on spectrogram's zeros based on the study by [22] which suggests that this representation has relevant features for signal classification tasks. Third, Takens representation is also a standard approach to signal processing from a topological perspective [18, 23, 15]. Unlike time-frequency representations, the Takens' embedding encompasses potentially different information since it provides a higher-dimensional ambient space, which allows us to assess the value of higher-dimensional homological features in voice classification tasks. Figure 1 shows these three different transformations of the same raw signal that corresponds to the waveform of a human vowel (i.e., the vowel phoneme /ä/). We now briefly characterize these three transformations.

First, among the plethora of existing representation methods of a speech signal, the spectrogram is certainly the most widely used (e.g., Flandrin [24]). Let us consider a sound signal $x \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$, its spectrogram $S_x^{(h)}(t, \omega)$ is defined as:

$$S_x^{(h)}(t, \omega) = |F_x^{(h)}(t, \omega)|^2, \quad (1)$$

where $(t, \omega) \in \mathbb{R}^2$ are time and frequency variables, $h(t) \in \mathbb{R}$ is a window function and $F_x^{(h)}(t, \omega)$ is the Short-Time Fourier Transform (STFT):

$$F_x^{(h)}(t, \omega) = \int_{-\infty}^{+\infty} x(s)h(s - t) \exp\{-i\omega(s - t/2)\}ds. \quad (2)$$

We can therefore represent the spectrogram of a one dimensional sound signal $x$, with time window function $h(\cdot)$, as a surface $S_x^{(h)}$ in a 3D ambient Euclidean space where the dimensions are time, frequency and amplitude, $S_x^{(h)} := \{(t, \omega, S_x^{(h)}(t, \omega)) | (t, \omega) \in \mathbb{R}^2\} \subset \mathbb{R}^3$ (see Figure 1b). This representation is particularly effective for revealing geometrical structures and enhancing classification performances [25].
<figure><img src="image_3.png" /><figcaption>Figure 1: Different representations of the vowel /a/: 1a) standard waveform representation; 1b) surface of its spectrogram; 1c zeros of its spectrogram in the time-frequency plane; 1d its Takens' embedding.</figcaption></figure>

We can therefore represent the spectrogram of a one dimensional sound signal $x$, with time window function $h(\cdot)$, as a surface $S_x^{(h)}$ in a 3D ambient Euclidean space where the dimensions are time, frequency and amplitude, $S_x^{(h)} := \{(t, \omega, S_x^{(h)}(t, \omega))|(t, \omega) \in \mathbb{R}^2\} \subset \mathbb{R}^3$ (see Figure 1b). This representation is particularly effective for revealing geometrical structures and enhancing classification performances [25].

A second possible representation is to consider the spectrogram's zeros as introduced in [26]. Choosing the window function to be Gaussian, i.e. $h(t) = \pi^{-1/4} \exp\{-t^2/4\}$, we readily show that the STFT can be rewritten as follows:

$$F_x^{(h)}(t, \omega) = \exp\{-|z|^2/4\} \mathcal{F}_x(z), \quad (3)$$

where, $z = \omega + it$ and $\mathcal{F}_x(z)$ is the Bargmann transform of $x$. It is easy to see that it is an entire function of order 2, which admits a Weierstrass-Hadamard form:

$$\mathcal{F}_x(z) \propto \prod_{n=1}^{\infty} \left(1 - \frac{z}{z_n}\right) \exp \left\{ \frac{z}{z_n} + \frac{1}{2} \left(\frac{z}{z_n}\right)^2 \right\}, \quad (4)$$

where $\mathcal{Z}_x := \{z_n = \omega_n + it_n\}_n$ is the set of zeros of $\mathcal{F}_x(z)$. The spectrogram is therefore completely characterized by its zeros, and it can be represented by a point cloud in the time-frequency plane. [24] pioneered the idea of using topological characteristics for describing a spectrogram by visualizing the distribution of the edges of its zero-based Delaunay-triangulation. Thereafter, we build on this idea for extracting topological characteristics using the tools described hereafter in Section (3).

A third representation of sound signals that is not related to time-frequency analysis is the Takens' embeddings. Considering a sound signal as a discrete-time digital audio recording (or time series) $\{x_1, \dots, x_T\}$ and assuming that it comes from a dynamic system, we can borrow the tools of dynamical system analysis to find out an informative representation. The Takens' theorem states that it is possible to obtain a representation of this time series that is topologically equivalent to the attractor of the system via a delay embedding, which contains useful information about the system [27]. We thus transform the digital audio recording into a point cloud in a higher dimensional space,
$\mathcal{P}_{D,x} = \{p_1, \dots, p_m\} \subset \mathbb{R}^D$, each element $p_i \in \mathcal{P}_{D,x}$ being a vector of dimension $D$, constructed by taking a time delay $\tau$:

$$p_i = (x_i, x_{i+\tau}, x_{i+2\tau}, \dots, x_{i+(D-1)\tau})' \quad (5)$$

We therefore embed a one dimensional digital audio recording into a higher dimensional space to obtain a point cloud. For this purpose, it is necessary to estimate two hyperparameters: $D$, the dimension of the space, and $\tau$, the time delay. $D$ is estimated using Cao's algorithm [28]. $\tau$ is selected using the Average Mutual Information (AMI). The coordinates of the phase-space embedding must be independent enough (to avoid aggregation around the diagonal in the embedding). $\tau$ is then chosen to be the smallest value such that $AMI(\tau) < \frac{1}{e}$. An example of this point cloud representation is shown in Figure 1d.

## 2.3. Same signal, different representations, different topological characteristics

We have described three ways of representing the same audio signal. It naturally raises some questions, such as, is there a representation that carries more topological (discriminant) information than the other? To illustrate this point, Figure 2 shows three persistence diagrams computed on the representations of the same signal illustrated on Figure 1 (see section 4.3 for more information about how this topological information is computed). Although the signal is the same, the extracted topological characteristics are very different because of the initial choice of the representation. Since the representation spaces can be of different dimensions, the homological features describing the topology of the object of these spaces will also be of different dimensions. This raises another, more specific question: does access to higher signal dimensions provide additional relevant information? Here we address this question in a quantitative way, through a case study, a classification task of human vowels.

# 3. Topological Data Analysis: An overview

This section presents the theory and data analysis pipeline of TDA in a nutshell. For more details, we refer the reader to two important textbooks that offer a fairly broad presentation of TDA and its theoretical foundations (i.e., [29, 30]) as well as an excellent introduction for data scientists [3].

## 3.1. TDA for voice classification

Let us consider the task of classifying vocal signals, which are represented as a set of wav files. Typically, we compute a representation of these signals, such as their spectrogram. The size of a spectrogram is determined by the duration of the recording, the recording device, and the parameters chosen for the STFT. Our goal is to analyze the shape of these points and extract meaningful information from it. We assume that the shape of the spectrogram corresponding to the sound of a phoneme differs from that of other existing phonemes.

Practically, we employ techniques from algebraic topology by first constructing a simplicial complex on the spectrograms to visualize their shape. To quantitatively and rigorously characterize this shape, we compute its homology, essentially counting the number of “holes” it contains. When building a simplicial complex from the data, we need to select a scaling parameter, prompting the question of how to choose this parameter appropriately. Through a process called filtration, instead of estimating a single parameter, we create a nested sequence of simplicial complexes using increasing values of the filtration parameter. We then compute the persistent homology of this sequence, summarizing the results in a persistence diagram. We use this diagram to construct a topological signature of the target signal, capturing essential information about the shape of its spectrogram.

## 3.2. Persistent homology

In a prosaic summary, TDA consists in the enumeration of holes of different dimensions in the shape of data. This intuitive notion of holes is formalized by the mathematical concept of a homology group. Homology groups allow us to treat the holes of a topological space mathematically, by studying the connectivity of the space.

Let us denote as $H_p(M)$ the $p^{th}$ homology group of the topological space $M$. For $p = 0$, $H_0(M)$ takes the connected elements of $M$, for $p = 1$, $H_1(M)$ takes the tunnels, for $p = 2$, the voids. The rank of $H_p(M)$ defines the $p^{th}$ Betti numbers, denotes as $\beta_p = rank(H_p)$ hereafter. Betti numbers basically count the number of different topological features. Homology groups and Betti numbers are topological invariants which characterizes the shape of $M$.
<figure><img src="image_4.png" /><figcaption>(a) Persistence diagram of the spectrogram's surface</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b) Persistence diagram of the spectrogram's zeros</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(c) Persistence diagram of the Takens' embedding</figcaption></figure>

Figure 2: Three persistent diagrams computed on the three different representations of the same signal reported in Figure 1. 2a is the persistence diagram of the spectrogram's surface using sublevel sets; 2b is the persistence diagram of the zeros of the spectrogram using an Alpha complex; 2c is the persistence diagram of the Takens' embedding using an Alpha complex. The points in the diagram represent the computed persistent homologies. The different colors represent the different dimensions: black for $p = 0$, red for $p = 1$ and blue for $p = 2$.
Instead of the homology groups of $\mathbb{M}$, we are interested in its persistent homology groups. We compute them through a filtration $F$ of $\mathbb{M}$. A filtration $F$ is a parametrized nested family of subspaces $F = (M_r)_{r \in T}$, where $T \subseteq \mathbb{R}$, such that for any $r, r' \in T$, if $r \le r'$, $M_r \subseteq M_{r'}$, and $\mathbb{M} = \cup_{r \in T} M_r$. The parameter $r \in T$ is the scale parameter. Let $f: \mathbb{M} \to \mathbb{R}$ be any function. Then the sublevel set of value $r \in \mathbb{R}$ is $M_r = f^{-1}(-\infty, r]$ and the sublevel set filtration is the family $\{M_r\}_{r \in T}$. A filtration allows multiscale topological description; for each value of $r$ we have the associated homology groups. The persistence of homologies makes it possible to track the lifetime of homologies, to determine their birth and their death according to the scale parameter $r$.

We denote as $\mathcal{X}$ our data objects that can be either $\mathcal{Z}_x$ or $\mathcal{P}_{x,D}$ (point clouds), or with the two-dimensional surface $S_x^{(h)}$. A point cloud does not usually contain any interesting topological information, but it is possible to extract some by building a simplicial filtration on the top of the cloud. We can define a simplicial filtration by constructing a simplicial complex on the data. The filtration is then a nested sequence of simplicial complexes, for each of which we compute the (simplicial) homologies. In this study, we use the Alpha complexes, a family of subcomplexes of the Delaunay complex, since it is equivalent but smaller than other complexes such as the Cech complexes. For any $x \in \mathcal{X}$, with $\mathcal{X} \subset \mathbb{R}^d$, and $r \in \mathbb{R}^+$, we define $B_x(r) = x + r\mathbb{B}^d$ the closed ball centered at $x$ and of radius $r$. The union of these balls is the set of points at a distance at most $r$ from at least one of the points of $\mathcal{X}$. We define the Alpha complex

$$ \text{Alpha}(r) = \{\sigma \subseteq \mathcal{X} | \cap_{x \in \sigma} R_x(r) \neq \emptyset\}, \qquad (6) $$

where $R_x(r)$ is the intersection of each Euclidean ball with its corresponding Voronoi cell. In the context of filtration, we construct a nested sequence of Alpha complexes on $\mathcal{X}$, taking an increasing value of $r$.

We recover the homological information of $\mathcal{X}$ computed for different scale values $r$. The birth and death of the topological features (the elements that make up each homology group) are recorded and summarized in the persistence diagram, for each value of $r$. Thus, each persistent topological feature resulting from the filtration is expressed by a pair $(b_i, d_j)$, the moment of its birth and its death, i.e., two values of $r$. The persistence is the difference $d_j - b_j$ [31]. We define a persistence diagram as a multiset of points in $\mathcal{D} := D \times \{1, ..., P\}$, where

$$ D := \{(b, d) \in \mathbb{R}^2 | d \ge b \ge 0\}. \qquad (7) $$

Each triplet $(b, d, p) \in \mathcal{D}$ represents a $p$-dimensional homological feature that appears when $r = b$ and disappears when $r = d$ [32].

In our study, we use filtration adapted to each representation spaces: the alpha-complex filtration for Takens' embeddings and spectrogram zeros and the sublevel set filtration for the spectrogram surface.

### 3.3. Exploitation of the information from the diagram space

Persistence diagrams provide multiscale homological descriptions of the data. However, the space of persistence diagrams has a very complicated geometry and topology which makes it difficult to exploit directly [33]. Many strategies were developed to extract the relevant information and enable machine learning applications. We refer the readers to [34, 35] that offer comprehensive review of existing methods.

Common approaches consist in computing functional representations alike persistent surfaces [36], persistence landscapes [37] or persistent silhouettes [12], and then, to discretize for use as vector inputs to machine learning algorithms. These representations have been shown to have stability properties [38], meaning that they are slightly modified by small variations in the input data. Their nice properties and ease to compute make them widely used in applications. For example, [16] uses persistence landscapes as input to a CNN for musical signal classification, and PLLay [39] is a topological layer based on persistence landscapes.

Instead of functional representations, persistence diagrams can be summarized by scalar descriptors, such as the $p$-norm of the persistence diagram [40] or the persistent entropy [41]. Both are proven to be stable to perturbations of the input data [42]. Aggregating those descriptors together appears to be an interesting strategy. [43] construct a stable vector of the persistence diagram from the distances between points and between each point and the diagonal, while [15] use a whole set of descriptors on the persistence diagrams. Hereafter, we will consider sets of descriptors as well as some standard functional summaries.
# 4. Experiments

In this section, we investigate how the choice of the representation space of a digital audio signal affects topological information. This information is quantified in terms of Out Of Bag (OOB) error in a supervised classification task inspired by the topical question of vowel classification [44, 45]. This task has applications in emotion classification [46], in evaluating developmental disorders [47] or to distinguish between healthy patients and patients with neurological disorders [48].

We collected our own dataset consisting of French-speaking adults pronouncing vowels indoors. For each of these recordings, we extract homological information from three possible representation spaces, as well as classical frequency descriptors from the associated spectrogram. Each recording corresponds to a specific vowel, pronounced by a speaker who is characterized by a certain sex. Thus, we can quantify (1) whether topological descriptors provide additional information to the more conventional frequency descriptors; (2) whether the contribution of the topological information depends on the representation space; (3) which is the best way to vectorize persistence diagrams. In order to visualize the extracted information in a more qualitative way, we project the extracted topological features on to a plane using UMAP [49].

In what follows, we describe the recorded data and three associated classification tasks on which the test is carried out. Then, we present the three representation spaces and associated filtration we use for each signal. Finally, we present different procedures for the vectorization of persistent diagrams and the methodology we use to compare them.

## 4.1. Presentation of the data

The data are digital audio recordings of French-speaking adults. These recordings were made in a controlled environment to keep low and stable signal-to-noise ratio and to avoid junk sounds. We used a Zoom H6 recording device with a stereophonic microphone XYH-6. The sampling rate of the recordings is 44100 Hz, 16 bits. The signal was downsampled to 16 kHz and converted to monophonic. We recorded 20 individuals, 15 women and 5 men. Each individual pronounced 8 vowels ( /a/, /a/, /ə/, /i/, /o/, /ɔ/, /u/, /y/ ) in 7 conditions: natural, low voice, high voice, short duration, long duration, with an ascending and descending tone. There were 10 utterances for each condition. Therefore, there was a total of 1400 recordings for each vowel and 560 recordings for each individual. In total, there were 8400 recordings from females and 2800 from males. The audio data set is freely accessible [50] and can be downloaded on https://zenodo.org/record/7961904.

## 4.2. Comparison on three supervised classification tasks

We compare the performance of our model on three classification tasks: prediction of the vowel, prediction of the individual who pronounced the vowel, prediction of the sex of the individual who pronounced the vowel. For each classification task and each representation space (i.e., spectrogram's surface, spectrogram's zeros, and Takens' embedding), we use a random forest as a classification model with the same number of 500 trees. We report the OOB error, which allows us to estimate the error by cross-validation at a lower cost, for models using either topological variables only, or frequency variables only, or both topological and frequency variables (i.e., topologically-augmented models).

## 4.3. Computation of the homological information

For each recording, we compute the spectrogram with a Gaussian window of 11.6 ms and an overlap of 90%. On the spectrogram's surface representation, we apply sublevel set filtration to compute the persistent homologies while on the spectrogram's zeros, we use an alpha-complex filtration (see Section 3).

For the representation using Takens' embeddings, we first estimate for each recording the two parameters required to build the embedding, $\tau$ according to the AMI and $D$ according to the Cao's algorithm. Then, we calculate the embeddings of each recording according to these two values. The recordings are then represented by point clouds in $\mathbb{D}^3$. On this space, we compute the persistent homologies using an alpha-complex filtration. The computational cost of calculating topological signatures can quickly become prohibitive, limiting their application to practical tasks. To address this, we propose a strategy that harmonizes the dimension of embedding spaces over recordings by reducing the dimension to 3 using UMAP. Selecting a dimension greater than 3 would have incurred a prohibitive computational
cost, as the alpha-complex exhibits poor scalability in ambient dimensions exceeding 3. Thus, each audio recording is associated with three persistent diagrams carrying potentially different topological information.

We also compute the Mel-Frequency Cepstral Coefficients (MFCC) [51, 52]. Those are more classical frequency descriptors of the signal, specifically designed for human speech analysis and generally used in machine learning for classification tasks involving human speech. They will serve as a baseline to compare our topological descriptors. Finally, in a topologically-augmented machine learning fashion, we will merge the MFCC with the topological descriptors.

### 4.4. Extraction of the information from the persistence diagram and comparison of the persistent variables

As discussed in Section 3.3, there are many ways of summarizing the information carried by persistence diagrams into variables that facilitate further statistical/machine learning usage. We follow here standard proposals from the literature to form a set of variables computed on persistence diagrams, here so-called persistent variables. In addition to persistent variables, we compute functional summaries. To ensure a fair comparison, we use the same classification model in each case.

#### 4.4.1. Persistent variables

Let $\mathcal{D}$ a persistence diagram, the multiset defined in Equation (7). We compute each variable for $p = \{0, 1\}$, and for the Takens' embeddings, we also compute for $p = 2$.

From [41], we compute the persistent entropy $E_p$. Let $L_p = \{\ell_i = d_i - b_i|1 \le i \le n\}$ the set containing the lifetime of each homological features of dimension $p$ of the diagram $\mathcal{D}$. We define the persistent entropy as

$$E_p(F) = - \sum_{i=1}^{n} p_i \log(p_i), \qquad (8)$$

where $p_i = \ell_i / S_L$ and $S_L = \sum_{i=1}^{n} \ell_i$. We compute it for each dimension.

We follow [15] to compute several descriptors of the persistence diagram. Using $L_p$, we compute the following statistics from the vector:

i) the mean $\mu_p$.

ii) the variance $\sigma_p^2$.

iii) the 5 top longest lifetimes $L_{p,i}$ for $i = 1, ..., 5$.

iv) two normalized longest lifetime,

$$\frac{L_{p,1}}{|L_p|}$$

v) and

$$\frac{L_{p,1}}{\mu_p}$$

vi) the number of $\alpha$-long lifetime $N_{p,\alpha}$. We choose $\alpha$ to distinguish between topological noise and information from the diagram. We take $\alpha = 0.05$.

$$N_{p,\alpha} = \#(L_{p,i} > \alpha).$$

vii) the ratio of means

$$\frac{\mu_{p_1}}{\mu_{p_2}}$$

$p_1$ and $p_2$ being two dimensions of homological features, with $p_1 < p_2$.

viii) the ratio of $\alpha$-long cycles

$$\frac{N_{p_1,\alpha}}{N_{p_2,\alpha}}$$
ix) the products of top longest lifetimes

for $i = 1, ..., 6$.

x) the products

for $i = 1, ..., 6$.

xi) the Periodicity Score

$$L_{p1,i} \cdot L_{p2,i},$$

$$L_{p1,i} \cdot (L_{p2,i} - L_{p2,i+1})$$

$$PS = 1 - \frac{L_{1,2}}{L_{1,1}}.$$

xii) the Quasi-Periodicity Score

$$QPS = L_{1,2} \cdot L_{2,1}.$$

xiii) the Frequency Shift Score

$$FSS = \frac{L_{2,1} \cdot L_{2,2}}{L_{1,1}}.$$

We compute four more variables from [18]:

* the persistent Betti number. For any pair of indices $0 \le k \le l \le n$ and any dimension $p$, the $p^{th}$ persistent Betti number is

$$\beta_p^{k,l} = \sum_{i \le k} \sum_{j > l} \kappa_p^{i,j}, \qquad (9)$$

where $\kappa_p^{i,j}$ is the number of $p$-dimensional homology that are born at $X_i$ and die at $X_j$, $X_i$ and $X_j$ being two subsets of the filtration. The persistent Betti number is the number of holes for each dimension [18]. We follow [29] and call it the persistent Betti number;

* the maximum hole lifetime in each dimension

$$\max_p = \max_{\ell_i \in L_p} (\ell_i); \qquad (10)$$

* the number of relevant holes

$$n\_rel_p = \sum_{\ell_i \in L_p} f(\ell_i, \max_p, \text{ratio}), \qquad (11)$$

where $f(\ell_i, \max_p, \text{ratio})$ equals 1 if $\ell_i \ge \max_p \times \text{ratio}$, 0 otherwise. It is the number of points in the persistent diagram that are relatively distant from the diagonal. We chose to count the holes with a lifetime at least greater than a quarter of the longest (i.e., ratio = 0.25);

* the sum of all lifetime

$$\text{sum}_p = \sum_{\ell_i \in L_p} (\ell_i). \qquad (12)$$

Following [40], we compute the $p$-norm of the persistence diagram,

$$\|D\|_p = \left[ \sum_{u \in D} \text{pers}(u)^p \right]^{\frac{1}{p}}, \qquad (13)$$

where $u$ is a point of the diagram and $\text{pers}(u)$ the absolute value of the difference between the coordinates. In practice, we compute it for $p = 2$.

The set of persistent variables is used alone in the different classification tasks, as well as in combination with the MFCCs. This data fusion method, which we expect to be the most efficient according to the results from the literature, can be found under the name “Topology augmented” in Table 1, which summarizes the results for the different tasks.
### 4.4.2. Functional summary of the persistence diagrams

In addition to the set of persistent variables, we consider functional summaries of the persistent diagrams. After being discretized, we can use it as input of a classification model. We present and test two of these methods.

First, we compute the silhouette of a persistence diagram [12]. It follows the persistence landscapes [37, 38]. The silhouette summarizes the persistence diagram in a single function. We choose this representation to illustrate the mapping of the persistence diagram into a Hilbert space. We define it as

$$ \phi^{(\gamma)}(t) = \frac{\sum_{j=1}^{m} |d_j - b_j|^{\gamma} \lambda_j(t)}{\sum_{j=1}^{m} |d_j - b_j|^{\gamma}}. \qquad (14) $$

The silhouette takes a parameter $\gamma$. This determines whether all points are treated equally ($\gamma$ small) or whether the most persistent pairs of points are given more weight ($\gamma$ large). We start by setting $\gamma = 1$, before exploring the impact of this choice in a second step. We also have to fix the number of sample on which we build the silhouette. We set $n = 2^9 = 512$.

Next, we compute the persistence images of the diagrams [36]. The idea is to map the persistence diagram to an integrable surface, so-called the persistence surface

$$ \rho_D(u, v) = \sum_{(x,y)=(b,d-b) \in D} w(x, y)g_{(x,y)}(u, v), $$

where $g_{(x,y)}(u, v) : \mathbb{R}^2 \to \mathbb{R}$ is bivariate Gaussian distribution centered at each point $(x, y) = (b, d - b) \in D$ and $w : \mathbb{R}^2 \to \mathbb{R}$ is a continuous and piecewise differentiable weighting function. $(u, v) \in \mathcal{D}$ is a compact domain, e.g., the domain of definition of the points $(x, y) = (b, d - b) \in D$. Then, we divide the domain in a collection of non-overlapping subdomains, the pixels $P_i$, with $\mathcal{D} = \bigcup P_i$. We integrate the persistence surface over the fixed grid to define the persistence image, taking the average of $\rho$ in each pixel,

$$ I_{P_i}(\rho_D) = \int \int_{P_i} \rho_D(u, v) dudv. \qquad (15) $$

This produces an image representing the persistence diagram, with a density distribution that is more or less important depending on the distribution of homologies on the persistence diagram. Again, we can use these vectors as input to a classification model. A sampling parameter must be set, which is fixed at 10.

### 4.5. Step-wise selection of the variables

As described in Section 3.3, the question of how to extract information from the persistence diagram for statistical purposes remains largely open and is not addressed in this study. We instead focus on the differences in persistent homologies between representation spaces of a given signal, and we bring complementary information to the field by comparing the most frequently occurring topological variables.

In order to carry out this comparison, we follow a step-wise strategy. For each classification task, we start with the complete set of persistent variables. We estimate the model with this set of variables, and then we remove the least important variable. We retrain the model with this new set of variables. This process continues until we remove all the variables in the training set. In the end, we retrain the model with the smallest OOB error. We report in Table 1 the results of the best model, and we list the variables present in the set of variables used to train this model. We count each time a variable is in the set of the best model, for each task and for each representation. This information can be found in Table 2.

We can then compare if there is a difference according to the input representation, if certain topological variables carry more interesting information than others, or at least if they are more often found in the best model for the classification tasks in question.

Note that our procedure includes MFCC, then we potentially remove some MFCC descriptors while keeping some topological variables in models producing the best results. If topological variables and MFCC contain the same cues at the same time, our procedure reduces these redundancies.

For each classification task and signal representation, we follow our step-wise strategy to find the best model, eliminating iteratively the less useful variable, with two conditions: with and without the MFCC. We save the 18 best
<table><caption>Table 1: Comparison of Out Of Bag error for different signal representations. Bold numbers emphasize the best signal representation for each task</caption><thead><tr><th rowspan="2">Signal Representation</th><th rowspan="2">Vectorization<sup>a</sup></th><th colspan="3">Error (%)</th></tr><tr><th>Vowel</th><th>Sex</th><th>Individual</th></tr></thead><tbody><tr><td>MFCC</td><td></td><td>8.71</td><td><strong>4.54</strong></td><td>11.54</td></tr><tr><td rowspan="10">Spectrogram's Surface</td><td>Silhouettes p = 0</td><td>72.97</td><td>20.37</td><td>73.72</td></tr><tr><td>Silhouettes p = 1</td><td>46.91</td><td>19.34</td><td>63.34</td></tr><tr><td>Silhouettes p = 0, 1</td><td>45.04</td><td>16.44</td><td>53.86</td></tr><tr><td>Persistent Image p = 0</td><td>79.31</td><td>28.46</td><td>88.27</td></tr><tr><td>Persistent Image p = 1</td><td>79.26</td><td>28.73</td><td>87.97</td></tr><tr><td>Persistent Image p = 0, 1</td><td>76.99</td><td>26.1</td><td>83.43</td></tr><tr><td>Persistent Variables</td><td>52.03</td><td>15.72</td><td>50.07</td></tr><tr><td>Topology augmented<sup>b</sup></td><td>8.43</td><td>4.71</td><td>10.42</td></tr><tr><td rowspan="10">Spectrogram's Zeros</td><td>Silhouettes p = 0</td><td>79.89</td><td>23.69</td><td>78.21</td></tr><tr><td>Silhouettes p = 1</td><td>73.33</td><td>17.61</td><td>75.69</td></tr><tr><td>Silhouettes p = 0, 1</td><td>70.11</td><td>16.14</td><td>67.5</td></tr><tr><td>Persistent Image p = 0</td><td>82.48</td><td>25.08</td><td>87.77</td></tr><tr><td>Persistent Image p = 1</td><td>70.5</td><td>16.79</td><td>71.15</td></tr><tr><td>Persistent Image p = 0, 1</td><td>70.7</td><td>17.19</td><td>70.68</td></tr><tr><td>Persistent Variables</td><td>69.85</td><td>15.19</td><td>62.15</td></tr><tr><td>Topology augmented<sup>b</sup></td><td>8.03</td><td>5.55</td><td><strong>9.24</strong></td></tr><tr><td rowspan="16">Takens' embeddings</td><td>Silhouettes p = 0</td><td>84.39</td><td>27.71</td><td>89.15</td></tr><tr><td>Silhouettes p = 1</td><td>81.51</td><td>25.32</td><td>84.44</td></tr><tr><td>Silhouettes p = 2</td><td>79.71</td><td>25.02</td><td>81.19</td></tr><tr><td>Silhouettes p = 0, 1</td><td>78.03</td><td>24.59</td><td>80.63</td></tr><tr><td>Silhouettes p = 0, 2</td><td>78.79</td><td>24.56</td><td>77.86</td></tr><tr><td>Silhouettes p = 1, 2</td><td>78.56</td><td>24.68</td><td>77.22</td></tr><tr><td>Silhouettes p = 0, 1, 2</td><td>76.41</td><td>23.98</td><td>75.53</td></tr><tr><td>Persistent Image p = 0</td><td>85.11</td><td>29.63</td><td>90.24</td></tr><tr><td>Persistent Image p = 1</td><td>81.89</td><td>27.06</td><td>84.87</td></tr><tr><td>Persistent Image p = 2</td><td>83.5</td><td>27.15</td><td>87.42</td></tr><tr><td>Persistent Image p = 0, 1</td><td>80.76</td><td>26.29</td><td>82.42</td></tr><tr><td>Persistent Image p = 0, 2</td><td>81.54</td><td>25.94</td><td>82.93</td></tr><tr><td>Persistent Image p = 1, 2</td><td>80.26</td><td>26.21</td><td>82.24</td></tr><tr><td>Persistent Image p = 0, 1, 2</td><td>79.53</td><td>25.76</td><td>79.86</td></tr><tr><td>Persistent Variables</td><td>69.85</td><td>20.27</td><td>60.89</td></tr><tr><td>Topology augmented<sup>b</sup></td><td>8.49</td><td>4.79</td><td>10.77</td></tr><tr><td rowspan="2">All together</td><td>Persistent Variables</td><td>41.56</td><td>10.89</td><td>32.13</td></tr><tr><td>Topology augmented<sup>b</sup></td><td><strong>7.98</strong></td><td>6.01</td><td>10.3</td></tr></tbody></table>

models, and we count the remaining persistent variables in each of them. Lastly, we follow the same procedure by including the topological variables of all representations in the same model, for the three tasks. Thus, a variable can be counted in a maximum of 24 models.

We put in bold in Table 2 the variables that appear at least in 50% of the best models. In the signal representation columns, each variable can appear at best 12 times. Here again, we put in bold those that appear at least in 50% of the best models. This way, it will be possible to compare whether certain variables appear more than others globally, whatever the representation and whatever the task. We can also check if there are differences depending on the representation space. For the sake of clarity, we resume the top 5 longest lifetimes $L_{p,i}$ and their product $L_{p_1,i} \cdot L_{p_2,i+1}$ by a row in Table 2 for each dimension. Also, the maximum count for these variables is not 24, but $24 \times 5 = 120$.

# 5. Results

## 5.1. Supervised task

The results for all supervised classification tasks can be found in Table 1. All models are random forests with the same number of trees. For models using as covariates either persistent variables or both persistent variables and MFCCs, the reported results are those of the best model (i.e., the model following the stepwise procedure, displaying the lowest error). Three main results emerge from these data.

<footnote><sup>a</sup> Strategy to extract information from the persistence diagram</footnote>

<footnote><sup>b</sup> Signal representation = MFCC + persistent variables. We follow the step-wise strategy presented in Section 4.5 on this set and present the result of the best model.</footnote>
### 5.1.1. TDA can be useful for vocal signal classification

Topological information improves the results of two over three classification tasks. For vowel classification, the MFCCs alone produce an error of 8.71%. The addition of topological information improves the results, whatever the chosen representation space. The error reduces to 8.43%, 8.03%, or 8.49% using the persistent variables obtained from spectrogram's surfaces, spectrogram's zeros and Takens' embeddings, respectively. The best improvement is obtained by taking all persistent variables from all representation spaces, resulting in an error of 7.98%. Including topological features reduces the error by 0.73 points, correcting 8.38% of errors.

The individuals' classification also benefits from topological information. The best results are obtained when the persistent variables are extracted from the spectrogram's zeros, lowering the error from 11.54% to 9.24%. Including topological features reduces the error by 2.3 points, correcting 19.93% of errors.

However, on the sex classification task, topology augmented approaches do not improve the results over the MFCCs alone, probably because sex classification can be done easily from the fundamental frequency of the signal, which is efficiently captured by MFCCs.

### 5.1.2. Topological information alone is not enough to discriminate vocal signal

Topological information alone, whatever the chosen representation space, is outperformed by classical frequency descriptors on each task. For example, for the vowel classification task, while the MFCCs alone achieve an error of 8.71%, the best result using all topological variables is 41.56%. Taken in isolation, each topological variable produces scores well below that obtained for MFCCs (i.e., 52.03%, 69.85%, and 69.85% for spectrogram's surface, spectrogram's zeros and Takens' embedding, respectively). For the sex classification task, MFCCs alone produce an error of 4.54% while topological information alone produce error ranging from 15.19% to 20.27%. Aggregating the persistent variables of each representation improves the results, with an error of 10.89%. For the individual classification task, MFCCs alone reach an error of 11.54% while the best score obtained by aggregating all persistence variables is 32.13%.

### 5.1.3. Complementarity of representations

Topology augmented approaches improve the results over the MFCCs for both vowel and individual classification. Comparing the topology augmented approaches for the three representations on the vowel and individual classification tasks, it is the variables extracted from the persistence diagrams of the spectrogram's zeros that achieve the minimum error.

## 5.2. Low-persistence features and topological signature

To evaluate the potential improvement of the method through better hyperparameter selection and gain insights into the underlying reasons for the improved results, we conduct an additional experiment.

We calculate the silhouette of each object's persistence diagram for 11 different values of $\gamma$, ranging from 0.001 to 4. Subsequently, we report the error for each classification task, using the silhouette as input. As a reminder, $\gamma$ serves as a trade-off parameter: lower values of $\gamma$ prioritize low-persistence pairs in the silhouette, while higher values emphasize high-persistence pairs. Analyzing the evolution of the error with varying $\gamma$ provides insights into the impact of low-persistence invariants and their significance. Figure 3 illustrates these results. Note that only the silhouette is used as input data in each case. The signal signature relies solely on the information encoded in the persistence diagram.

Irrespective of the value of $\gamma$ and the classification task, the silhouette computed on the persistence diagram of the Takens' embeddings consistently yields the poorest results. The error curves exhibit a clear positive trend, except for those associated with the silhouette of the spectrogram zeros for the sex classification task. This observation underscores the significance of the information encapsulated in the "topological noise" for addressing the three speech signal classification tasks. Notably, the most accurate predictions for identifying the sex from the topological signature derived from the spectrogram zeros, specifically for $\gamma = 1.34$. Conversely, for the remaining two tasks, the optimal results are achieved using persistence diagrams generated from the surface of the spectrogram, particularly for the minimum $\gamma$ value tested, which is 0.001.
<figure><img src="image_7.png" /><figcaption>Figure 3: Out Of Bag error, in percentage, for each classification task, according to the γ value used to compute the silhouette of the persistence diagram. The red, green and blue lines are the error of prediction of the model when its input is the silhouette computed on the persistence diagram of the spectrogram’s surface, the Takens’ embeddings and the spectrogram’s zeros, respectively.</figcaption></figure>
### 5.3. Best topological variables

For these classification tasks, persistent silhouettes or persistent images perform worse than the persistent variables in all scenarios. For this reason, we build the topologically augmented approach only with the persistent variables. They are therefore not taken into account in the stepwise procedure to identify the best topological variable (see Section 4.5). Results of this stepwise procedure are presented in Table 2.

In general, there are 8 persistent variables that we encounter at least in 50% of the best models. These are the persistent Betti number in dimension $p = 0$, the sum of $L_p$ in dimension $p = 1$, the 2-norm of the persistence diagram for $p = 1$, the average of $L_p$ for $p = 0$ and $p = 1$ and its variance for $p = 1$, the normalized longest lifetime for $p = 0$, with the normalization computed with the mean of $L_p$ of this dimension, the ratio of the mean of $L_0$ and $L_1$.

The number of variables retained in 50% of the best models varies significantly from one representation space to another. We identified 11, 5, 2 persistent variables retained for the spectrogram's surface, for the spectrogram's zeros and for the Takens' embedding, respectively.

Two persistent variables seem to stand out from the others: the persistent Betti number and the norm of the persistence diagram. The other persistent variables which appear regularly are linked and summarized in the vector $L_p$. We find statistics on this vector that summarize the information of the persistence diagrams (mean, variance, normalized maximum), whatever the initial representation of the signal.

### 5.4. Unsupervised analysis

In order to describe better the behavior of the homological features associated to the different representation spaces, we provide a visual and qualitative help using the Uniform Manifold Approximation and Projection for Dimension Reduction (UMAP) algorithm. Figure 4 represents the projection of the recordings onto the 2d space learned with the UMAP. It distinguishes MFCC, persistent variables accordingly to the representation space, and the three tasks considered therein.

Figure 4 reveals that the manifold learned on the MFCCs creates clear clusters for vowels. This is less the case for sex, where observations from both classes are quite close in this space. Same comment applies for individuals. Although some observations are centered, the distribution of individuals in this space can be quite spread out. It is even harder to see a pattern for the manifold learned on persistent variables, whatever the representation space. In fact, the manifolds learned from the persistent variables are much more connected, and we do not distinguish clear clusters in these spaces.

Interestingly, the overall shapes of manifolds learned on Takens' embeddings and on spectrogram's surface look quite similar, and rotated by 180°. While connectivity is still present, manifolds learned on topological variables extracted from zero's spectrogram look very different. This might be explained by the number of coordinates in the representation space within the persistent homology. Takens' embedding and the spectrogram's surface lie in a 3-dimensional space, whereas the spectrogram's zeros are in the plane.

## 6. Discussion

The main results that emerge from the present study are fourfold. First, we identify the top-performing topological descriptors that improve results of speaker identification and vowel classification tasks on the collected dataset. Second, the choice of the signal representation space has an impact on the topological information that is extracted. Third, the best way to use a persistence diagram for a classification task depends on the task. Finally, low-persistence features carry meaningful information for the classification of vocal signal.

### 6.1. Improving the prediction of labels

No matter how the information is extracted from the persistence diagrams, we found that topological information never outperformed MFCCs. However, adding topological variables to frequency variables improves the results in two of the three considered classification tasks. These results are therefore consistent with those reported in previous studies [43, 21, 3]. Topological descriptors of signals bring complementary information to MFCCs, which are specifically designed for human speech analysis.

These results contribute to fill an existing gap in the analysis of time-varying data, as noted in some recent reviews [35]. They provide encouraging results to continue investigating the performance of topologically-augmented
<table><caption>Table 2: Most frequently kept persistent variables in each best model</caption><thead><tr><th rowspan="2">Variable</th><th rowspan="2"></th><th rowspan="2">Frequency in the best model (%)<sup>a</sup></th><th colspan="3">Signal Representation</th></tr><tr><th>Spectrogram's Surface</th><th>Spectrogram's Zeros</th><th>Takens' embeddings</th></tr></thead><tbody><tr><td>β<sub>0</sub></td><td>(9)</td><td>62.5</td><td>33.33</td><td>41.67</td><td>50</td></tr><tr><td>β<sub>1</sub></td><td></td><td>33.33</td><td>50</td><td>0</td><td>16.67</td></tr><tr><td>β<sub>2</sub></td><td></td><td>25</td><td>0</td><td>0</td><td>50</td></tr><tr><td>max<sub>0</sub></td><td>(10)</td><td>29.17</td><td>58.33</td><td>0</td><td>0</td></tr><tr><td>max<sub>1</sub></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>max<sub>2</sub></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>n_rel<sub>0</sub></td><td>(11)</td><td>0</td><td>0</td><td>1</td><td>0</td></tr><tr><td>n_rel<sub>1</sub></td><td></td><td>4.17</td><td>0</td><td>0</td><td>0</td></tr><tr><td>n_rel<sub>2</sub></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>sum<sub>0</sub></td><td>(12)</td><td>41.67</td><td>66.67</td><td>0</td><td>2</td></tr><tr><td>sum<sub>1</sub></td><td></td><td>50</td><td>5</td><td>2</td><td>5</td></tr><tr><td>sum<sub>2</sub></td><td></td><td>8.33</td><td>0</td><td>0</td><td>16.67</td></tr><tr><td>E<sub>0</sub></td><td>(8)</td><td>29.17</td><td>50</td><td>0</td><td>8.33</td></tr><tr><td>E<sub>1</sub></td><td></td><td>33.33</td><td>50</td><td>0</td><td>16.67</td></tr><tr><td>E<sub>2</sub></td><td></td><td>4.17</td><td>0</td><td>0</td><td>8.33</td></tr><tr><td>||D<sub>0</sub>||<sub>ρ</sub></td><td>(13)</td><td>37.5</td><td>50</td><td>16.67</td><td>8.33</td></tr><tr><td>||D<sub>1</sub>||<sub>ρ</sub></td><td></td><td>50</td><td>25</td><td>50</td><td>25</td></tr><tr><td>||D<sub>2</sub>||<sub>ρ</sub></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>μ<sub>0</sub></td><td>(i)</td><td>87.5</td><td>83.33</td><td>75</td><td>16.67</td></tr><tr><td>μ<sub>1</sub></td><td></td><td>79.17</td><td>66.67</td><td>50</td><td>41.67</td></tr><tr><td>μ<sub>2</sub></td><td></td><td>4.17</td><td>0</td><td>0</td><td>8.33</td></tr><tr><td>σ<sub>0</sub><sup>2</sup></td><td>(ii)</td><td>45.83</td><td>33.33</td><td>41.67</td><td>16.67</td></tr><tr><td>σ<sub>1</sub><sup>2</sup></td><td></td><td>66.67</td><td>25</td><td>100</td><td>8.33</td></tr><tr><td>σ<sub>2</sub><sup>2</sup></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>L<sub>0,1</sub></td><td>(iv)</td><td>25</td><td>33.33</td><td>0</td><td>16.67</td></tr><tr><td>L<sub>1,1</sub></td><td></td><td>33.33</td><td>58.33</td><td>0</td><td>8.33</td></tr><tr><td>L<sub>2,1</sub></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>L<sub>0,1</sub></td><td>v)</td><td>54.17</td><td>41.67</td><td>41.67</td><td>25</td></tr><tr><td>L<sub>1,1</sub></td><td></td><td>37.5</td><td>50</td><td>8.33</td><td>16.67</td></tr><tr><td>L<sub>2,1</sub></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>μ<sub>2</sub></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>N<sub>0,α</sub></td><td>(vi)</td><td>29.17</td><td>0</td><td>58.33</td><td>0</td></tr><tr><td>N<sub>1,α</sub></td><td></td><td>25</td><td>0</td><td>8.33</td><td>41.67</td></tr><tr><td>N<sub>2,α</sub></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>μ<sub>0</sub></td><td>(vii)</td><td>87.5</td><td>100</td><td>33.33</td><td>41.67</td></tr><tr><td>μ<sub>1</sub></td><td></td><td>12.5</td><td>0</td><td>0</td><td>25</td></tr><tr><td>μ<sub>2</sub></td><td></td><td>4.17</td><td>0</td><td>0</td><td>8.33</td></tr><tr><td>μ<sub>2</sub></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>N<sub>0,α</sub></td><td>viii)</td><td>20.84</td><td>0</td><td>33.33</td><td>8.33</td></tr><tr><td>N<sub>1,α</sub></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>N<sub>2,α</sub></td><td></td><td>0</td><td>0</td><td>0</td><td>8.33</td></tr><tr><td>N<sub>1,α</sub></td><td></td><td>4.17</td><td>0</td><td>0</td><td>8.33</td></tr><tr><td>N<sub>2,α</sub></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>PS</td><td>(xi)</td><td>12.5</td><td>25</td><td>0</td><td>0</td></tr><tr><td>QPS</td><td>(xii)</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>FSS</td><td>(xiii)</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>L<sub>0,i</sub><sup>b</sup></td><td>(iii)</td><td>21.67</td><td>36.67</td><td>5</td><td>1.67</td></tr><tr><td>L<sub>1,i</sub><sup>b</sup></td><td></td><td>50.83</td><td>43.33</td><td>51.67</td><td>6.67</td></tr><tr><td>L<sub>2,i</sub><sup>b</sup></td><td></td><td>5</td><td>0</td><td>0</td><td>10</td></tr><tr><td>L<sub>0,i</sub> · L<sub>1,i</sub><sup>b</sup></td><td>(ix)</td><td>27.5</td><td>31.67</td><td>15</td><td>8.33</td></tr><tr><td>L<sub>0,i</sub> · L<sub>2,i</sub><sup>b</sup></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>L<sub>1,i</sub> · L<sub>2,i</sub><sup>b</sup></td><td></td><td>2.5</td><td>0</td><td>0</td><td>5</td></tr><tr><td>L<sub>0,i</sub> · (L<sub>1,i</sub> - L<sub>0,i+1</sub>)<sup>b</sup></td><td>(x)</td><td>5</td><td>5</td><td>3.33</td><td>1.67</td></tr><tr><td>L<sub>0,i</sub> · (L<sub>2,i</sub> - L<sub>0,i+1</sub>)<sup>b</sup></td><td></td><td>0.3</td><td>0</td><td>0</td><td>1.67</td></tr><tr><td>L<sub>1,i</sub> · (L<sub>2,i</sub> - L<sub>1,i+1</sub>)<sup>b</sup></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td></tr></tbody></table>

<footnote><sup>a</sup> We follow the step-wise strategy to choose the best model for each classification task. We count in how many models each variable appears.</footnote>

<footnote><sup>b</sup> We count if one of them is kept, $\forall i \in \{1, ..., 6\}$</footnote>
<figure><img src="image_8.png" /><figcaption>(a) MFCC Vowel</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(b) MFCC Sex</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(c) MFCC subject</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(d) Surface Vowel</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(e) Surface Sex</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(f) Surface Subject</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(g) Zeros Vowel</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(h) Zeros Sex</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(i) Zeros Subject</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(j) Takens Vowel</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(k) Takens Sex</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(l) Takens Subject</figcaption></figure>

<caption>Figure 4: Projection of the records on the 2d space learned with UMAP, depending on the input of the algorithm: MFCC, topological variables from the spectrogram's surface, the spectrogram's zeros or the Takens' embedding.</caption>

machine learning approaches even for natural signals suffering from much lower signal-to-noise ratio, as might be considered for robust filtration along the lines of the Distance-To-Measure introduced in [53].
One can also note that our results show no improvement for the sex classification task. This is the simplest task as it is a binary classification, while the other have 8 and 20 classes, for vowel and individual predictions, respectively. Another possible reason for this lack of improvement is the pitch difference between men and women, which provides a main difference between these classes. MFCCs certainly capture most of the variance here and topological characteristics do not probably provide any additional information.

When the task becomes more complicated, topological information proved to be useful. The most notable improvement is for the prediction of the speaker. Persistent homologies seem to carry here additional information needed to improve the classification performance.

## 6.2. Different objects, different topologies

### 6.2.1. The difference between representation spaces

As already noticed, it is difficult to identify one representation of the signal as being better than the others. Nevertheless, there are two points to bear in mind from the classification results according to the initial representation space:

* For the two tasks where topological information improves the results (i.e., individual and vowel classification), the persistent homologies extracted from the time-frequency plane of the spectrogram zeros improve the error the most. Takens' embeddings are the representation with the least improvement for both problems. Thus, access to higher dimensional homologies does not seem to provide discriminative information about the signal.

* The aggregation of topological information is also advantageous. It allows the best improvement in the prediction of vowels and, when it does not allow the best improvement in the prediction of individuals, the improvement is better than for the other two representations. Moreover, for the models trained only on the persistent variables, without the MFCCs, the best results are always obtained when all the representations are combined. The homological information of the representation spaces therefore seems complementary.

This shows that a particular representation space reveals different salient topological features. Frequently, Takens' embeddings are considered in topological signal analysis. In this study, we introduced two ways to derive topological information from spectrogram that have the advantage of being more interpretable.

### 6.2.2. A different perspective

The topological signatures we computed on the different signal representation spaces are complementary to more classical descriptors such as MFCCs. This has also been found in several supervised classification tasks [43, 21, 3].

The projections we learn from the persistent variables, presented in Figure 4, are strongly connected. The topological approach is metric-free and based on connectivity [1]. But in our case, a topological latent space does not clearly cluster the data.

Similar observation have been reported with topological autoencoders [54]. A latent space which takes into account the topology of the data makes entangled structures appear that are impossible in a clear spatial separation. This could be useful to illustrate a hierarchy in the data, specifically when a parent-child structure is present.

The value of the topological approach depends on the task [2] and is particularly suited to tasks requiring analysis of nested categories. If one expects such a structure in the data, it is certainly worth examining its topology in order to look at the problem from a blind spot of more conventional analysis tools.

## 6.3. Construction of the topological signature

Regarding the vectorization of persistence diagrams, it seems difficult to characterize topological signatures, at least among those tested. We confirm a result already present in other reviews studying this issue [34], which find that the most interesting topological signatures seem to depend on the studied data set. Here we found that mapping the persistence diagram onto a functional space, such as [12, 36], was less efficient than taking a set of scalars that summarize the information contained in the diagram. Nevertheless, these approaches have the advantage of having well established stability properties and have been used elsewhere [38, 16, 39]. This does not disqualify them for other tasks, and they might even be effective on this task, if handled more carefully (e.g., by tuning finely the different parameters they depend on, or by computing well-chosen summary data).
We did not test all the existing approaches to this question. In fact, we adopted a strategy consisting of extracting information from the persistence diagrams, either by mapping persistence diagrams onto function space (persistent landscape lives in Banach space) or by calculating vectors to describe them. While we did not conduct a comprehensive search for all hyperparameters due to computational constraints, our experiments revealed that refining the parameters utilized for constructing topological signatures could enhance the results.

## 6.4. Importance of topological noise

Finally, we raise the problem of topological noise, which generally designates homological features with very short lifespans. It can be seen on persistent diagrams as the set of points along the main diagonal. But should we really consider topological noise as noise? Indeed, it has already been noticed that topological noise can be useful to characterize a signal [55]. The persistence diagrams obtained after alpha-complex filtration on the zeros of the spectrograms are visually the diagrams with the most persistent homologies, many of them being close to the diagonal, as can be seen in Figure 2. Yet, topological information extracted from them appeared to be particularly complementary to the MFCCs and provided the best results out of all tested models for individual prediction. Therefore, these homologies are likely not just being noise but to carry some information. The experiment depicted in Figure 3 shows that assigning greater importance to low-persistence features enables capturing more information about the signal. Low-persistence features are particularly sensitive to the curvature of the manifold from which the data is sampled [55]. Persistence diagrams of the spectrogram's zeros, which contain a greater number of these points, are assumed to capture these curvatures more effectively compared to other representations. As a result, they offer complementary information to MFCCs, which are adept at clearly partitioning the space into distinct clusters. This combination results in a more comprehensive and detailed understanding of the data than would be achievable with either representation alone.

## 6.5. Limitations and potential challenges

The first limitation lies in the considerable computational time necessary for computing the topological signatures. The process involves calculating persistent homology for multiple representations, which is computationally expensive. In this work, we adopted a strategy to ensure that embedding dimensions do not exceed three. However, this does not fully resolve the issue of the computational cost of TDA, and it prevents us from assessing the potential relevance of topological information encoded in homologies of dimension higher than two. Further efforts are required to enhance the efficiency of algorithms, particularly to mitigate costs when dealing with larger datasets, commonly encountered in real-world applications. By addressing this limitation, future research can unlock the full potential of topological data analysis in various domains.

This challenge is exacerbated by the fact that several hyperparameters could potentially be learned. In our current approach, we have treated each of the model's hyperparameters as fixed. Implementing a procedure to dynamically select hyperparameters from the data, e.g. through cross-validation, could enhance classification performance but necessitates recalculating the topological signatures multiple times. While an iterative hyperparameter search would undoubtedly increase the computational burden, our experiment with the silhouette parameter $\gamma$ demonstrates that there is indeed room for improvement in performance through more careful tuning.

The third limitation concerns the choice of representation spaces. Others could still have been chosen, like other time-frequency representation (e.g., MPS [56]), but also embeddings learned by various models [57, 58, 59]. These embeddings, which can achieve better results than MFCCs on classification tasks [60], are difficult to interpret and rather made to serve directly as features: we do representation learning on the data to a particular task. In this paper, we have preferred to adopt a feature extraction strategy, in order to have a clear comparison between topological features according to the space on which they have been computed. It would be very interesting in a future work to look at the shape of these embeddings and see if it would be possible to extract topological invariants summarizing the contained information.

The fourth limitation pertains to the vectorization of persistence diagrams, for which a general-purpose pipeline is lacking. Addressing this challenge would involve developing a standardized pipeline for automatically vectorizing persistence diagrams. We did not develop a new vectorization method in this study, but rather chose to experimentally evaluate whether an existing technique from the literature would emerge as most effective for our dataset. Alternatively, another approach would be to explore methods that bypass the need for persistence diagrams altogether. One promising avenue is leveraging representation learning to incorporate data topology directly [54, 61]. By integrating
representation learning techniques, we can potentially overcome the first three limitations as well, thereby establishing
an end-to-end approach that seamlessly incorporates topological considerations into the classification process.

7. Conclusion

We discussed the potential added value of the topological approach to sound signal processing by studying classifi-
cation performances according to the representation space of the signal. We tested this approach on three classification
tasks, predicting the sex of the speaker, the pronounced vowel and the identity of the speaker. For two of these tasks,
vowel and identity prediction, the topological features improved the results compared to the baseline. Although it
is difficult to distinguish one representation space as being more informative than another, it seems that the topo-
logical descriptors computed on each of them are complementary. Our results also suggest the use of less common
representations than Takens' embeddings, such as spectrogram's surfaces or spectrogram's zeros. For individual clas-
sification, the zeros give the best results with the topologically-augmented approach. Moreover, parameter selection
and interpretation of a spectrogram is simpler than Takens' embeddings. We analyzed different ways of vectorizing
information from persistence diagrams. The best results were obtained using a set of persistent variables regardless
of the representation space. It shows that the topological approach offers a complementary and interesting angle of
analysis. It is however not sufficient on its own to discriminate the signal, but it does provide additional information
about its hierarchical structure. It would be interesting to analyze more theoretically in the future the complementarity
of the representation spaces of the signal, and to see the possibility of working directly in the space of persistence
diagrams. In a next study, it would be valuable to compare modern deep learning approaches by analyzing persistent
homology across different latent spaces, conditioned on the training set, architecture, and learning task. It would also
be relevant to carry out a similar study on other components of speech, such as consonants, to assess the broader
applicability of our findings.

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