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Explainable AI in Speaker Recognition -- Making Latent Representations Understandable
Yanze Xu, Wenwu Wang, Mark D. Plumbley
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Summary
This paper investigates the organizational patterns of latent representations in speaker recognition neural networks through the lens of Explainable AI (XAI). While previous studies focused on 'flat clustering' using algorithms like K-means, this work demonstrates the existence of 'inner hierarchical clustering' using Single-Linkage Clustering (SLINK) and HDBSCAN. To semantically interpret these hierarchical structures, the authors propose the Hierarchical Cluster-Class Matching (HCCM) algorithm, which matches hierarchical clusters to predefined semantic classes (e.g., 'male', 'UK', or conjunctions like 'male and UK'). Additionally, the paper introduces the 'Liebig score' (L-score), a new metric designed to provide better diagnostic interpretability than the traditional F-score by identifying the specific limiting factor (precision or recall) in the matching process.
Entities (8)
Relation Signals (5)
SLINK → isatypeof → Hierarchical Clustering
confidence 100% · this work applies two algorithms -- Single-Linkage Clustering (SLINK) and Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN)
HDBSCAN → isatypeof → Hierarchical Clustering
confidence 100% · this work applies two algorithms -- Single-Linkage Clustering (SLINK) and Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN)
HCCM → performsmatchingfor → Hierarchical Clusters
confidence 100% · HCCM is designed to perform one-to-one matching between predefined semantic classes and hierarchical representation clusters
Liebig's score → quantifies → HCCM performance
confidence 100% · A new metric, Liebig's score, is proposed to quantify the performance of each matching behaviour
Speaker Recognition → uses → Neural Networks
confidence 100% · the speaker recognition network that recognises the speaker identity of utterances.
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Abstract
Abstract:Neural networks can be trained to learn task-relevant representations from data. Understanding how these networks make decisions falls within the Explainable AI (XAI) domain. This paper proposes to study an XAI topic: uncovering unknown organisational patterns in network representations, particularly those representations learned by the speaker recognition network that recognises the speaker identity of utterances. Past studies employed algorithms (e.g. t-distributed Stochastic Neighbour Embedding and K-means) to analyse and visualise how network representations form independent clusters, indicating the presence of flat clustering phenomena within the space defined by these representations. In contrast, this work applies two algorithms -- Single-Linkage Clustering (SLINK) and Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) -- to analyse how representations form clusters with hierarchical relationships rather than being independent, thereby demonstrating the existence of hierarchical clustering phenomena within the network representation space. To semantically understand the above hierarchical clustering phenomena, a new algorithm, termed Hierarchical Cluster-Class Matching (HCCM), is designed to perform one-to-one matching between predefined semantic classes and hierarchical representation clusters (i.e. those produced by SLINK or HDBSCAN). Some hierarchical clusters are successfully matched to individual semantic classes (e.g. male, UK), while others to conjunctions of semantic classes (e.g. male and UK, female and Ireland). A new metric, Liebig's score, is proposed to quantify the performance of each matching behaviour, allowing us to diagnose the factor that most strongly limits matching performance.
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- Source: https://arxiv.org/abs/2604.23354v1
- Canonical: https://arxiv.org/abs/2604.23354v1
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Explainable AI in Speaker Recognition - Making Latent Representations Understandable Yanze Xu, Wenwu Wang, and Mark D. Plumbley Yanze Xu, Wenwu Wang are in the Centre for Vision, Speech and Signal Processing, University of Surrey, Guildford, GU2 7XH, UK. Mark D. Plumbley is in Department of Informatics, King’s College London, London, WC2R 2LS, UK. Corresponding author: Yanze Xu [yanze.xu@outlook.com] Abstract Neural networks can be trained to learn task-relevant representations from data. Understanding how these networks make decisions falls within the Explainable AI (XAI) domain. This paper proposes to study an XAI topic: uncovering unknown organisational patterns in network representations, particularly those representations learned by the speaker recognition network that recognises the speaker identity of utterances. Past studies employed algorithms (e.g. t-distributed Stochastic Neighbour Embedding and K-means) to analyse and visualise how network representations form independent clusters, indicating the presence of flat clustering phenomena within the space defined by these representations. In contrast, this work applies two algorithms—Single-Linkage Clustering (SLINK) and Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN)—to analyse how representations form clusters with hierarchical relationships rather than being independent, thereby demonstrating the existence of hierarchical clustering phenomena within the network representation space. To semantically understand the above hierarchical clustering phenomena, a new algorithm, termed Hierarchical Cluster–Class Matching (HCCM), is designed to perform one-to-one matching between predefined semantic classes and hierarchical representation clusters (i.e. those produced by SLINK or HDBSCAN). Some hierarchical clusters are successfully matched to individual semantic classes (e.g. male, UK), while others to conjunctions of semantic classes (e.g. male and UK, female and Ireland). A new metric, Liebig’s score, is proposed to quantify the performance of each matching behaviour, allowing us to diagnose the factor that most strongly limits matching performance. Index Terms: Explainable AI, Deep Learning, Speaker Recognition, Hierarchical Clustering, Cluster-class Matching I Introduction Explainable AI (XAI) aims to make the decision-making processes of AI systems, particularly those implemented using neural networks [16, 21], transparent and understandable to humans [15, 56, 23]. In the deep learning field [16, 21], neural networks are typically trained in a supervised manner to learn a nonlinear mapping from inputs (e.g. images or signals) to pre-labelled outputs (e.g. classes or values), with the intermediate outputs of this mapping process serving as task-relevant representations of the inputs [35]. We are interested in two high-level XAI questions: i) Inspired by the fact that humans can organise knowledge or information [27], we assume that the neural network has a similar ability to organise representations of different inputs, and we question what the network’s organisation of representations is. i) Inspired by the fact that human attention mechanism is the cognitive ability to selectively process relevant stimuli [19], we assume that the network has the computational ability to selectively process task-relevant information during decision-making, and we question what information is selectively processed. This paper focuses exclusively on the first question. To address this, we propose to analyse, visualise, and understand the representation space of a well-trained neural network111The representation space of the neural network is a multi-dimensional space formed by representations extracted from the same network layer after feeding diverse model inputs into the network.. Our experiments are conducted on a speaker recognition task, in which most neural networks are trained in a supervised manner to map an utterance signal or its spectrogram to a pre-labelled speaker identity [1, 34, 6]. Figure 1: An approximate 2-dimensional visualisation for the representation space of a well-trained speaker recognition network, originated from Li et al.’s paper [22]. Previous work employs flat clustering algorithms [29], such as K-means [28], to analyse flat clustering phenomena within the network representation space [37, 3, 4]. Other studies [39, 51, 22] use dimensional reduction algorithms [54, 31, 39] to visualise these flat clustering phenomena. In other words, both the analytical and visualisation results of these works indicate that the network representations are well organised into independent clusters. More specifically, as shown in Fig. 1, Li et al. [22] visualise a two-dimensional space that approximates the representation space of their speaker recognition network, in which representations are arranged into independent, flat clusters. To understand the observed flat clustering phenomenon, it is generally interpreted that each distinct cluster in Fig. 1 corresponds to a particular speaker identity. Peiffer et al. [37] refer to the flat clustering phenomenon within the network representation space as inner or internal clustering. The research problem is that it remains largely underexplored whether there are underlying relationships among these flat representation clusters, rather than their being entirely independent. To address this, this work applies hierarchical clustering algorithms [41] to analyse whether hierarchical clustering phenomena are preserved within the representation space of a speaker recognition network. Specifically, two well-established hierarchical clustering algorithms—Single Linkage (SLINK) [14, 46] and Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) [2]—are applied separately to analyse representations extracted from a well-trained speaker recognition network [6]. The results produced by both algorithms indicate that the representations of our network are really organised into clusters exhibiting hierarchical relationships (i.e. hierarchical representation clusters), rather than forming entirely independent clusters. We refer to this hierarchical clustering phenomenon observed within the examined speaker recognition network as inner hierarchical clustering. The performance of SLINK and HDBSCAN in analysing these representations is evaluated on the basis of the quality of the hierarchical representation clusters they produce, using the method proposed by Rosenberg et al. [43]. With respect to the topic of analysing, visualising and understanding the network representation space, we further aim to visualise and semantically interpret the inner hierarchical clustering phenomena, rather than merely analysing it. Specifically, given the optimal hierarchical representation clusters produced by a particular algorithm (i.e. SLINK or HDBSCAN) that achieves the highest evaluation score, we present these clusters in the form of a tree-like structure, namely a dendrogram [49]. This tree structure illustrates how smaller, low-level clusters are hierarchically combined as larger, high-level clusters. Furthermore, we design an algorithm, termed Hierarchical Cluster–Class Matching (HCCM), to automatically interpret the hierarchical representation clusters visualised in the dendrogram. Specifically, HCCM identifies, in a one-to-one manner, which representation clusters can be most suitably matched to the predefined representation divisions of different classes222The predefined representation division of a class, refers to the set of all representations in the dataset whose corresponding model inputs belong to a certain predefined semantic class. Fig. 2 illustrates the interpretations offered by the HCCM method in general. It shows seven exemplar representation clusters with hierarchical relationships. As an example, ‘cluster 5’ is interpreted as the ‘UK’ semantic class considering representations inside ‘cluster 5’ match the best with a predefined representation division belonging to the ‘UK’ class. ‘cluster 6’ is interpreted as the semantic class of ‘UK&male’ or ‘UK and male’ because representations in this cluster match the best with the predefined representation division belonging to both the ‘UK’ semantic class and the ‘male’ semantic class simultaneously. We refer to the semantic class of an individual modifier (e.g. ‘UK’) as an individual semantic class, and the semantic class of an intersective modifier [32]—that is, a logical conjunction such as ‘UK&male’—as an and-logic-based conjunctive semantic class. Figure 2: An illustration of the Hierarchical Cluster-Class Matching (HCCM)’s interpretation offered to hierarchical representation clusters Lastly, inspired by Liebig’s law of the minimum [8]—which states that the overall performance of a system is limited by its most restrictive factor—we propose a new metric, termed the Liebig score (L-score), to quantify the matching degree in HCCM. Fig. 3 illustrates the difference between the traditional F-score and the L-score when matching ‘cluster 5’ with the UK class. In the left panel, the F-score is calculated as the harmonic mean of 0.81 precision and 0.73 recall, which conceals the fact that recall is the limiting factor. By contrast, the L-score in the right panel directly reflects the 0.73 recall. This allows the red annotation to be interpreted: the matching performance is constrained by the 27% of UK samples not retrieved by cluster 5. The ability to provide a diagnostic interpretation of matching performance is the reason for designing the L-score metric. Figure 3: An illustration for interpreting the matching degree quantified by F-score and L-score I Related works This section reviews the work most relevant to our study. We first examine the differences in research focus of explainable artificial intelligence (XAI) between recognition and generative tasks, and then discuss prior studies on clustering analysis of neural network representations. I-A XAI in Recognition and generation Compared with XAI research for generative tasks, XAI research for recognition or classification tasks tends to have a different focus. In our opnion, research for recognition tasks is generally more theoretical in nature, concentrating predominantly on analysing and visualising network attention mechanisms [26, 42, 44, 60, 38], or clustering phenomena within network representation space [22, 24, 37]. However, few studies have yet to elucidate what practical, application-level benefits these insights into neural networks can offer [3, 4]. In contrast, research for generative tasks is more practically oriented, with most studies focusing on analysing vectors that capture the principal variance across multiple representations extracted from generative models [45, 58, 13]. The benefit of doing so is that, by modifying representations along such vectors, the generated outputs can be progressively controlled in specific semantic directions, which can be translated into real-world engineering practice. I-B Flat clustering and dimensionality reduction for network representations Flat clustering methods [29] and dimensionality reduction methods [9] can be used separately to analyse or visualise the inner clustering phenomena of neural networks [37, 9]. Specifically, flat clustering algorithms operate directly on the original high-dimensional representations to analyse whether they form flat clusters, whereas dimensionality reduction methods transform these high-dimensional representations into lower-dimensional spaces that can be visualised. While low-dimensional representations may appear as flat clusters, it remains uncertain how faithfully they preserve the flat clustering phenomena present in the original representation space, which requires further verification. I-C Hierarchical clustering for network representations Hierarchical clustering algorithms [33] can be used to analyse whether network representations are organised into clusters with hierarchical relationships, or more broadly, whether hierarchical clustering phenomena occur within the network representation space (i.e. inner hierarchical clustering). Specifically, Naumov et al. [36] surveyed the application of various hierarchical clustering algorithms to representations extracted from networks trained for image classification and Natural Language Processing (NLP) tasks. In the speaker recognition task, some studies apply hierarchical clustering algorithms to network representations to subsequently build speaker diarisation systems addressing the ‘who speaks when’ problem [11, 47, 25]. In contrast, this work not only offers a new XAI perspective for applying hierarchical clustering algorithms to representations (i.e. analysing inner hierarchical clustering phenomena), but also extends the analysis by providing a visualisation of the clusters using dendrogram and a semantic interpretation via the Hierarchical Cluster–Class Matching (HCCM) method. I-D Cluster-class matching (CCM) and HCCM The HCCM method we designed is inspired by a similar approach proposed by Rosenberg et al. [43]. Specifically, Rosenberg et al.’s method, also termed the Cluster–Class Matching (CCM) method, is originally used to measure a global matching degree, representing how well independent data clusters produced by a flat clustering algorithm match the predefined data groups (i.e. divisions) of different classes. The better flat clustering algorithm is expected to produce data clusters that align more closely with predefined data divisions. In contrast, HCCM focuses on quantifying one-to-one matches rather than measuring overall matching. In particular, it identifies cluster-class pairs, each of which shows which individual hierarchical data cluster best aligns with the predefined data divisions of a given class. The more data groups that are predefined, the more cluster-class pairs can be identified, leading to increasingly detailed interpretations of the hierarchical representation clusters. I Preliminary knowledge Two hierarchical clustering algorithms [41] are used to analyse representations from the speaker recognition neural network: Single Linkage (SLINK) [14, 46] and Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) [2]. Section I-A presents the background on the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) [10] algorithm, which forms the foundation for HDBSCAN and is related to SLINK. Furthermore, we design Hierarchical Cluster–Class Matching (HCCM) to semantically interpret the hierarchical representation cluster produced by SLINK and HDBSCAN. Section I-B introduces the CCM method proposed by Rosenberg et al. [43], as it inspired the design of HCCM. I-A DBSCAN Figure 4: Pesudocode of the DBSCAN from RJGB Campello’s paper [2]. DBSCAN [10] is one of the most well-known flat clustering algorithms [55] that can analyse whether flattened clustering phenomena remain in the data, whether data form flat clusters. It is commonly referred to as a density-based flat clustering method, since each data point assigned to a cluster must satisfy a density constraint, requiring that it has a sufficient number of neighbouring points within a specified search radius. We now give a short discussion of the pseudocode of DBSCAN presented in its original paper [2], as shown in Fig. 4. Generally, this pseudocode includes two stages. In the first stage (i.e. lines 2∼92 9), the algorithm identifies a initial point, a start point p that has at least minPtsminPts neighbours within the search radius ε . Once the density requirement of having minPtsminPts neighbours is satisfied, p is treated as the core point [10]. Neighbours within the search radius ε of p are stored in the set S. Notably, RangeQuery() is the function returning neighbours within a certain range of a query point; the first parameter DBDB represents the set of all data points; the second parameter distdist specifies the distance metric (i.e. usually Euclidean distance space) used to measure distances between data points; the third parameter is the query point; and the fourth parameter ε denotes the search radius for finding neighbours near the query point. In the second stage (i.e. lines 10∼ 16), this algorithm iteratively searches for new core points within the set S and expands S by incorporating the reachable neighbour points of each newly identified core point. All core points discovered in this process are labelled as members of a certain cluster using the function label()label(). Eventually, by re-running the first and second stages from different unlabeled starting points, the members of different independent, flat clusters can be identified. In the following Section IV-A, the relationship between this DBSCAN and the hierarchical clustering algorithms HDBSCAN and SLINK will be discussed. I-B Cluster-class matching for evaluation Diverse flat clustering algorithms have been developed to analyse flat clustering phenomena within data, necessitating the evaluation of which flat clustering algorithms provide better analyses. To do so, Rosenberg et al. [43] use the the Cluster-class matching (CCM) method to evaluate how well flat data clusters analysed by each algorithm align with the predefined grouping information of the data. In more detail, the CCM requires that data are pre-labelled into groups of different semantic classes, which we refer to as the predefined data divisions of classes. CCM then quantify an overall matching degree between data clusters with predefined data divisions. A better flat clustering algorithm should produce clusters that align more closely with the predefined grouping information, resulting in a higher matching degree. Subsequent studies [18, 59] generalised the CCM method to evaluate hierarchical clustering algorithms, employing the F-score [5] or V-score [43] as quantitative measures of the overall matching degree We now present the mathematical formulation of the CCM method in detail. Specifically, let C=c1,c2,…,cnclassC=\c_1,c_2,…,c_n_class\ represents the collection of predefined data divisions of nclassn_class number of semantic classes, where each cic_i contains the indices of all data points in the dataset whose ground-truth labels belong to the semantic class i. Notably, the CCM method does not specify whether the semantic classes used to label the dataset must be independent, nor whether the semantic classes function as individual modifiers or as compound modifiers. Besides, let K=k1,k2,…,knclusterK=\k_1,k_2,…,k_n_cluster\ denote the collection of nclassn_class data clusters analysed by a certain clustering algorithm, where each kjk_j contains the indices of all data points assigned to cluster j. The overall matching degree between data clusters K and predefined data divisions C is defined as follows: Fscore(C,K)=∑ci∈C|ci|maxkj∈Kfscore(ci,kj)F_score(C,K)= _c_i∈ C |c_i|N _k_j∈ K\f_score(c_i,k_j)\ (1) Here, |ci||c_i| represents the number of data points that are pre-labelled as semantic class i, and N is the total number of data points in the dataset; thus, |ci|/|c_i|/N specifies the proportion of data points belonging to that semantic class. Given the predefined data partition corresponding to a semantic class (i.e. cic_i), the expression maxkj∈Kfscore(ci,kj) _k_j∈ Kf_score(c_i,k_j) iterates over all clusters in K to identify the cluster that best matches cic_i, where the matching degree is quantified by the F-score metric fscoref_score. The overall matching degree is computed as the weighted average of the individual matching degrees across all best-matched pairs between data clusters (i.e., K) and predefined data partitions (i.e., C). A higher overall matching degree indicates that the data clusters (e.g., K) exhibit stronger alignment with the predefined class partitions (e.g., C), reflecting superior clustering performance. The matching degree for each cluster–class pair, quantified using the F-score metric fscoref_score in Eq. (1), is defined as follows: fscore(c,k)=2(|c∩k||c|)−1+(|c∩k||k|)−1=2|c∩k||c|+|k| f_score(c,k)= 2 ( |c∩ k||c| )^-1+ ( |c∩ k||k| )^-1= 2|c∩ k||c|+|k| Here, |c∩k||c| |c∩ k||c| calculates the recall value for matching the predefined data division of a certain class with a certain data cluster. The recall in this context is defined as the proportion of all data points belonging to the semantic class that are successfully retrieved by the matched cluster [43]. In addition, |c∩k||k| |c∩ k||k| is the precision, which is defined as the proportion of data points in the cluster that truly belong to, or are predicted as, the matched semantic class [43]. On this basis, the F-score value is a harmonic mean of the values of precision and recall. However, Christen et al. [5] criticised the fact that, although both precision and recall are proportions with clear representational meaning, their harmonic mean (i.e. the F-score) lacks such interpretability. In the context of document retrieval task, a recall of 0.30 can be directly interpreted as ‘30% of the relevant documents are retrieved by the system’, while a precision of 0.30 indicates that ‘30% of the retrieved documents are relevant’. By contrast, an F-score of 0.30 does not admit a similarly direct interpretation: it reflects a balance between precision and recall, but cannot be construed as representing any specific proportion of documents. Algorithm 1 Find a flat cluster by DBSCAN (asdfasdfasdfasdfasdfasdf) 1:N←RangeQuery(DB,dist,p,ε)N← RangeQuery(DB,dist,p, ) 2:if |N|>minPts|N|>minPts then 3: Seed set S = p\p\ 4:end if 5:for each q in S do 6: if label(q)≠undefinedlabel(q)≠ undefined then 7: end if 8: N←RangeQuery(DB,dist,q,ε)N← RangeQuery(DB,dist,q, ) 9: if |N|>minPts|N|>minPts then 10: 11: 12: 13: label(q)←clabel(q)← c 14: S←S∪NS← S∪ N 15: end if 16:end for Algorithm 2 Find a flat cluster by constructing MST in mutual reachability distance space) 1:N←RangeQuery(DB,distmr,p,ε)N← RangeQuery(DB,dist_mr,p, ) 2:Seed set S←pS←\p\ 3:Edge set E←∅E← 4:while 1 do 5: N←∅N← 6: for each s∈Ss∈ S do 7: N←N∪RangeQuery(DB,distmr,s,ε)N← N∪ RangeQuery(DB,dist_mr,s, ) 8: end for 9: if N=N= empty then 10: break 11: end if 12: (q,d)←NearestLink(S,N,distmr)(q,d)← NearestLink(S,N,dist_mr) 13: label(q)←clabel(q)← c 14: S←S∪qS← S∪\q\ 15: E←E∪dE← E∪\d\ 16:end while IV Methodology This section discusses the two hierarchical clustering algorithms used to analyse whether the representations form hierarchical clusters, and introduces the Hierarchical Cluster–Class Matching (HCCM) method designed to interpret hierarchical representation clusters at the semantic level. In particular, Section IV-A introduces SLINK [14, 46] and HDBSCAN [2] by examining their relationships. Section IV-B introduces the HCCM method in detail based on the F-score quantification metric. Section IV-C presents the new quantification metric we propose, Liebig’s score (L-score), and demonstrates how to interpret the L-score value in the context of the HCCM method. IV-A Relationships of SLINK and HDBSCAN SLINK, proposed by Sibson [46], is an efficient implementation of classical single-linkage hierarchical clustering [14]. One equivalent implementation is based on the Minimum Spanning Tree (MST) [40, 20]: a spanning tree covering all samples with minimal total edge weight is constructed (i.e. edges represent the shortest connections between samples), and the edges of this tree are then pruned in order to obtain flat data clusters, which can be naturally nested to form the hierarchical clustering outcome (i.e. hierarchical data clusters). HDBSCAN is a hierarchical extension of DBSCAN [2], in which the flat clusters produced by DBSCAN for all values of the neighbourhood search radius (i.e. ε ) are organised into nested hierarchical clusters. Its efficient implementation, like MST-based SLINK, constructs an MST within a specialised distance space and prunes the edges in the MST [30]. This specific distance space is called the mutual reachability distance space. Considering that the equivalence between HDBSCAN and MST-based SLINK has received little discussion [2], we examine their relationship with awareness of the context of analysing network representations. More specifically, this equivalence requires that the flat representation cluster produced by DBSCAN coincide exactly with that derived from an MST constructed in the mutual reachability distance space. If this condition holds, both algorithms can naturally organise the same flat representation clusters into identical hierarchical representation clusters. Algorithm 1 and Algorithm 2 present side-by-side pseudocode illustrating how DBSCAN and the MST in the mutual reachability distance space identify the same flat cluster. Algorithm 1 is similar to the pesudocode in Fig. 4. In particular, it first selects an initial data point p and determines whether it is a core point that satisfies the density constraint, using the RangeQueryRangeQuery function to find at least minPtsminPts neighbours within the ε search radius in the original distance space defined by distdist distance metric. Alg. 1 then iteratively selects new core points from the reachable neighbours of previously identified core points until no new core points can be found for the same cluster. Notably, in our application, each data point corresponds to a representation extracted from the speaker recognition network. Furthermore, Algorithm 2 also begins from the same initial point p. Specifically, it invokes the RangeQueryRangeQuery function to query the neighbours of point p within the search radius ε , with the distance metric changed to the distmrdist_mr, which is the mutual reachability distance metric derived from distdist. In more detail, given two specific data points x1x_1 and x2x_2 (i.e. representations) in the dataset DBDB, their distance in the original distance space is dist(x1,x2)dist(x_1,x_2), while in the mutual reachability distance space it is defined as: distmr(x1,x2)=max(distcore(x1),distcore(x2),dist(x1,x2)) splitdist_mr(x_1,x_2)= (&dist_core(x_1),\\ &dist_core(x_2),\\ &dist(x_1,x_2) ) split (2) where distcore(xi)dist_core(x_i) is the minimal radius containing minPtsminPts neighbours near xix_i in the original distance space. Taking the maximum value among dist(x1,x2)dist(x_1,x_2), distcore(x1)dist_core(x_1), and distcore(x2)dist_core(x_2) ensures that if either point is not dense enough in the original distance space, the mutual reachability distance between x1x_1 and x2x_2 increases to either distcore(x1)dist_core(x_1) or distcore(x2)dist_core(x_2). Using this mutual reachability distance metric, if RangeQuery finds that x2x_2 lies within a radius ε of x1x_1, or vice versa, the following inequality must hold: ε≥distmr(x1,x2)≥distcore(x1),distcore(x2) ≥ dist_mr(x_1,x_2)≥\,dist_core(x_1),dist_core(x_2) Here, the left-hand side implies that the search radius ε covers distmr(x1,x2)dist_mr(x_1,x_2). From the definition of distmr(x1,x2)dist_mr(x_1,x_2), we have distmr(x1,x2)≥distcore(x1),distcore(x2)dist_mr(x_1,x_2) _core(x_1),dist_core(x_2). Hence, ε≥distcore(x1),distcore(x2) _core(x_1),dist_core(x_2), which means, in the original distance space, both x1x_1 and x2x_2 have at least minPtsminPts neighbours within radius ε , satisfying the density constraint. In summary, the mutual reachability distance metric for neighbour queries ensures that, whenever a neighbour is returned, both the query point and the returned neighbour satisfy the density constraint in the original distance space. Back to line 1 of Algorithm 2, all returned core points queried by RangeQuery(DB,distrm,p,ε)RangeQuery(DB,dist_rm,p, ) are stored in the set S. In the loop from line 5 to 15, new core points reachable from the existing core points in S are obtained by iteratively executing the RangeQueryRangeQuery function based on distmrdist_mr, with each element of S serving as a query point. These newly found core points are collected in the set N. Next, the function NearestLink(S,N,dist)NearestLink(S,N,dist) is defined to identify the pair of core points with the shortest distance, measured by the distmrdist_mr distance metric, between the set of existing core points S and the set of newly found core points N. NearestLinkNearestLink returns the point q∈Nq∈ N that is closest to any point in S, along with the corresponding distance d to its nearest neighbour in S. The point q is then labelled as a new cluster member and added in S, with the shorest distance d added to the set E. Lastly, the loop from line 5 to line 15 iterates to find new core points that can reach the current cluster members in S with the shortest distance, continuing until no additional cluster members are added. Upon termination, S contains all members of a particular flat cluster, and E contains the edges that span all cluster members with the minimal distance (i.e. MST). Comparing Algorithm 1 and Algorithm 2 for finding a flat cluster starting from the same initial point p, the differences are as follows: i) By using the mutual reachability distance metric in RangeQueryRangeQuery, Alg. 2 eliminates the additional core point check required by DBSCAN, which does not affect the selection of cluster members. i) Each cluster member added by Alg. 1 is any reachable core point, whereas each cluster member added by Alg. 2 is the closest core point to the existing cluster members. This distinction affects the order in which cluster members are added, but does not alter the final set of cluster members. Ultimately, running DBSCAN is equivalent to constructing the MST in the mutual reachability distance space when both are configured with the same setup (i.e. search radius ε , density parameter minPtsminPts, and starting point p). Their hierarchical counterparts (i.e. HDBSCAN and MST-based SLINK operating in the mutual reachability distance space) separately organise the same flat clustering result naturally into the hierarchical clustering outcome. The advantage of Algorithm 2 in determining the order of cluster member addition and maintaining the shortest distance path connecting all samples is that, when obtaining flat clustering results at different search radii to construct the hierarchical clustering outcome, DBSCAN must be rerun at each radius. In contrast, MST does not require reconstruction for each radius; instead, it can efficiently obtain the flat clustering result for a new search radius by pruning edges that exceed the current radius. As a result, samples that originally belong to the same cluster may naturally and hierarchically split into new clusters due to the disconnection of certain edges. Today, HDBSCAN is currently implemented as MST-based SLINK operating in the mutual reachability distance space [30, 2] for computational efficiency. IV-B Hierarchical cluster-class matching for interpretation As discussed in Section I-B, in the CCM method [43], the calculation of the overall matching degree aggregates the matching degrees of different best-matched cluster-class pairs. This overall matching degree is treated as the evaluation score of the clustering outcome. In contrast, for our purpose of offering a semantic interpretation of the hierarchical clustering outcome, we are motivated to fully leverage CCM’s partial capability to identify the best-matched cluster–class pairs, revealing which hierarchical data clusters align one-to-one with the predefined data divisions of semantic classes. The more cluster-class pairs we identify, the more hierarchical representation clusters can be interpreted through the semantic classes to which they matched. To this end, we propose the Hierarchical Cluster-Class Matching (HCCM) method. The HCCM algorithm requires a pre-trained neural network, trained for the speaker recognition task, to extract N representations, denoted as DBDB, from a specific network layer after feeding N model inputs (i.s. spectrograms of utterances). These model inputs should be pre-labelled with different semantic classes so that they can be naturally organised into groups. On this basis, given that the representations correspond one-to-one with the model inputs, these representations inherit the same divisions as the model inputs, which are referred to as the predefined representation divisions of semantic classes. We now discuss the semantic classes used in HCCM in greater detail. Specifically, the semantic classes that HCCM uses to label model inputs can be categorised as country (e.g. UK, USA, Ireland, etc.), gender (e.g. male, female, etc.), or speaker identity (e.g. Turing, Shannon, Hinton, etc.). In terms of usage, these semantic classes can function as the individual modifier to modify inputs and representations, for instance, ‘a model input of UK class’ or ‘a representation of male class’. They can also function as the intersective modifier [32], for instance ‘a representation of male and UK class’, which denotes that the representation belongs to two semantic classes simultaneously. HCCM uniformly refers to these semantic classes used as individual modifiers as individual semantic classes, and to those used as intersective modifiers conjoined by the logical connective and as and-logic-based conjunctive semantic classes. In HCCM, both individual semantic classes and and-logic-based conjunctive semantic classes offer different ways to divide representations, as illustrated in Fig. 5. Observing the two second-level subfigures in the first row, the left subfigure depicts that all representations can be divided into two individual semantic classes in the gender category, whereas the right subfigure aims to depict that all representations can be divided into three individual semantic classes in the nationality category. The two gender-related individual classes are mutually independent since no representation can belong to the male and female at the same time. The three nationality-related individual classes are also mutually independent. Figure 5: An illustration of intersecting predefined representation divisions of two gender-related individual classes and that of three nationality-related individual classes From the second-level subfigure in the second row of Fig. 5, it can be observed that some predefined representation partitions of individual semantic classes are not mutually exclusive; intersections may arise between any nationality-related class and any gender-related class. As an example, the intersection between the representation division of ‘male’ and the representation division of ‘UK’ can form a new division in which representations belong to both male and UK simultaneously, denoted as the predefined representation division of ‘UK&Male’ or ‘UK and male’ instead. Formally, the predefined representation divisions of individual semantic classes can be denoted as C=c1,…,cnclassC=\c_1,…,c_n_class\, where nclassn_class is the total number of distinct individual semantic classes regarding speaker nationality, gender, and identity that are used to label model inputs X. cic_i denotes the predefined representation division of a given individual class, containing the indices of those representations in the prepared representation set DBDB that are pre-labelled with the individual semantic class i. In addition, the predefined representation division of and-logic-based conjunctive semantic classes can be denoted as follow: =cT∣T⊆I,|T|=2,cT≠∅,cT=⋂t∈Tct. = \\,c_T T I,\;|T|=2,\;c_T≠ \, c_T= _t∈ Tc_t. (3) Here, I=1,…,nclassI=\1,…,n_class\, denoting the index set of all individual semantic classes associated with speaker nationality, gender, and identity. Eq. (3) selects the indices of a random pair of two individual semantic classes from I as T. cTc_T is a set containing indices of those representations in DBDB that simultaneously belong to each individual semantic class t in T, which is obtained by taking the intersection ⋂t∈Tct _t∈ Tc_t. If the two selected individual semantic classes in T are mutually independent, cTc_T is empty and represents nothing. Otherwise, the non-empty cTc_T represents the predefined representation division of the and-logic-based conjunctive semantic class related to T. Lastly, considering there are different combinations to form T, the collection C gathers representation divisions of diverse conjunctive classes. HCCM then uses both C and C to one-to-one match with hierarchical representation clusters. In particular, hierarchical clusters are produced by applying either SLINK or HDBSCAN to the representations DBDB, denoted as K=k1,…,knclusterK=\k_1,…,k_n_cluster\. Here, kjk_j denotes the hierarchical representation cluster containing the indices of those representations in DBDB that are assigned to the j-th hierarchical cluster. It is worth noting that, since each hierarchical cluster is formed through the natural nesting of a set of flat clusters, kjk_j consists of non-overlapping indices obtained by aggregating the indices of all flat representation clusters that compose the j-th hierarchical representation cluster. In addition, nclustern_cluster denotes the total number of hierarchical representation clusters identified by the algorithm. Typically, ncluster≫nclassn_cluster n_class, which implies that a substantial portion of the hierarchical representation clusters cannot be readily interpreted by a limited number of individual classes alone. To alleviate this interpretability burden, HCCM formulates the one-to-one matching process with the representation divisions of the and-logic-based conjunctive classes involved: ℂ(0) ^(0) =C∪, =C , (4) (cl∗,kl∗) (c^*_l,k^*_l) =argmaxc∈ℂ(l−1),k∈Kfscore(c,k), = _c ^(l-1),\,k∈ Kf_score(c,k), ℂ(l) ^(l) =ℂ(l−1)∖cl∗,l=1,…,|ℂ(0)| =C^(l-1) \c^*_l\,l=1,…,|C^(0)| Here, Eq. (4) describes an iterative greedy matching process between K and ℂ(0)C^(0). Initially, ℂ(0)C^(0) is defined as the union of the predefined representation divisions of all classes (i.e. both individual and conjunctive semantic classes). At the first interation, l=1l=1, a certain best-matched cluster–class pair (cl∗,kl∗)(c^*_l,k^*_l) is determined. Specifically, cl∗c_l^* is selected from the set ℂ(l−1)C^(l-1), and kl∗k_l^* is selected from the set K. This selection ensures that the pair attains the highest F-score-based matching degree (i.e. fscoref_score as defined in Section I-B) among all alternative pairs, which also means that the representation cluster kl∗k^*_l is most appropriately interpreted by the semantic class corresponding to the predefined representation division cl∗c^*_l. Notably, the F-score matching degree fscore(cl∗,kl∗)f_score(c^*_l,k^*_l) is expected to be as high as possible, so that the interpretation of kl∗k^*_l by its matched semantic class is precise and unambiguous. In addition, Eq. (4) removes cl∗c_l^* from the set ℂ(l−1)C^(l-1), ensuring that the same semantic class is not paired with any other clusters in subsequent iterations. As l increases, successive optimal cluster–class pairs are identified, and the iterations continue until l reaches the number of elements in ℂ(0)C^(0). In this way, a total of |ℂ(0)||C^(0)| best-matched cluster–class pairs are obtained. These pairs partially interpret the hierarchical clustering outcome K of representations DBDB, borrowing the predefined grouping information of DBDB (i.e. C and C). IV-C Liebig’s score Section I-B argues that precision and recall metrics provide clear, meaningful interpretations, whereas the F-score offers only a vague value with no clear representational meaning [5]. To provide a more meaningful interpretation of the matching degree in the HCCM method, we propose a new metric, Liebig’s score (L-score), as a replacement for the F-score. The L-score quantifies the matching degree of each cluster-class pair and is characterised by its ability to offer a diagnostic value interpretation of the matching degree. The L-score is inspired by Liebig’s Law of the Minimum [8]. This principle states that the overall performance of an environmental system is determined or constrained by its weakest component or factor. Analogously, we assume that the matching performance between a representation cluster and a predefined representation class division (i.e. a cluster-class pair) is primarily limited by the weakest matching factor. Based on this assumption, drawn from Liebig’s Law of the Minimum, the L-score first measures the precision and recall values for a cluster-class pair, treating these two values as the matching factors that characterise the matching performance. The L-score then determines its value based on the most limited matching factor, which is the smaller of the precision and recall values. Formally, the formula below denotes the calculation of the L-score value in the context of the HCCM method: lscore(c,k)=min(|c∩k||c|,|c∩k||k|)=|c∩k|max(|c|,|k|) splitl_score(c,k)= ( |c∩ k||c|, |c∩ k||k| )\\ = |c∩ k| (|c|,\>|k|) split (5) Figure 6: An overview of experimental procedures Here, |c∩k||k| |c∩ k||k| is the precision for matching representation cluster k with a representation class division c. We define the precision in this context as the proportion of those representations in the cluster k that truly belong to, or are precisely predicted as, the matched semantic class. We define the recall (i.e. |c∩k||c| |c∩ k||c|) in this context as the proportion of those representations within the predefined representation class division c that are correctly assigned to, or retrieved by, the matched cluster. Eq. (5) selects the minimum one between the precision value and the recall value as the L-score value, serving as the matching performance of the cluster-class pair (i.e. (k,c)(k,c)). Considering both precision and recall are defined with clear meanings, we can interpret the L-score value diagnostically by referencing the individual interpretations of either precision or recall. Suppose the L-score-based matching degree for the cluster-class pair (k,c)(k,c) is 0.1. This indicates that the matching performance is limited by either 0.1 precision or 0.1 recall. If precision is the smaller one, the 0.1 L-score-based matching degree is limited by the fact that 90% of the representations within the cluster are not correctly classified, or alternatively, only 10% are accurately predicted as belonging to the matched class. If recall is the smaller one, the 0.1 L-score is limited by the fact that 90% of the relevant representations in the predefined class are not retrieved by the cluster, or alternatively, only 10% of the relevant class members are successfully retrieved. V Experimental Procedures and Setups Fig. 6 provides a visual overview of our experimental procedure, and this section also presents a detailed discussion of each stage with reference to the specific experimental settings. As shown in Fig. 6, the prepared network representations are first subjected to two hierarchical clustering algorithms, namely HDBSCAN and SLINK. The next stage applies the CCM method [43], which involves evaluating the hierarchical clustering outcomes produced by SLINK and HDBSCAN. Subsequently, the HCCM method we designed is performed, which involves interpreting the hierarchical clustering result generated by either SLINK or HDBSCAN. In the final stage, dendrogram visualisation is conducted. This involves constructing a tree-like visual structure for the hierarchical clustering outcome of either SLINK or HDBSCAN and visually presenting the HCCM results on this tree structure. Let us discuss the experimental setups of these stages in more detail. Preparing network representations: A neural network that Chung et al. [7] trained for speaker recognition task is downloaded from https://github.com/clovaai/voxceleb_trainer for subsequent experiments. This network is based on the ResNet34 architecture [16], trained by the prototypical contrastive loss [48, 6] on mel spectrograms of 2-sec audio clips from the Voxceleb 2 dataset [7]. The Equal Error Rate (EER) performance of this model is 2.18 while verifying whether mel spectrograms of 4-sec audio clips from the Voxceleb 1 testset [34] are the correct speaker. Speaker embeddings (i.e. representations) are extracted from the penultimate layer of this well-trained neural network after feeding the network with 20-, 100-, 200-, and 400-frame mel spectrograms, which are calculated from 0.2-, 1-, 2-, and 4-sec audio clips, respectively, from the VoxCeleb1 testset [34]. Hierarchical clustering: The representations are divided into four groups based on audio length (i.e. 0.2-, 1-, 2-, and 4-sec audio clips). SLINK and HDBSCAN are applied independently to representations of each group. Specifically, HDBSCAN is implemented by McInnes et al. [30], available at https://github.com/scikit-learn-contrib/hdbscan. McInnes et al.’s implementation constructs HDBSCAN by applying an MST-based SLINK algorithm in the mutual reachability distance space, which is derived from the original Euclidean distance. Moreover, we do not implement the SLINK algorithm separately, because when the density constraint parameter minPtsminPts in McInnes et al.’s HDBSCAN implementation is set to 0, the mutual reachability distance space coincides with the original Euclidean distance space, in which case McInnes et al.’s HDBSCAN implementation reduces to running SLINK directly in the original Euclidean distance space. More detailed setups of hierarchical clustering algorithms will be discussed in the following Section VI. CCM and HCCM: Given the hierarchical clustering results (i.e. hierarchical representation clusters) obtained under different cases (i.e. performing SLINK and HDBSCAN on representations of different audio length), the CCM [43] is used to evaluate the hierarchical clustering outcome of each case separately. The hierarchical clustering result of the case that achieves the optimal evaluation score (i.e. optimal overall matching degree) is further interpreted by performing the proposed HCCM method. The predefined representation divisions of semantic classes used in both CCM and HCCM come from the data annotation of the VoxCeleb 1 test set [34]. In the VoxCeleb 1 test set, all utterances and their corresponding representations can be divided into groups of 40 identity-related individual classes, 2 gender-related individual classes, and 12 nationality-related individual classes. The CCM method uses predefined representation divisions of 54 (i.e. 40+2+1240+2+12) individual classes for evaluating hierarchical clustering results. In HCCM, the predefined representation divisions of and-logic-based conjunctive semantic classes are constructed by intersecting the representation divisions of 2 gender-related individual classes with those of 12 nationality-related individual classes. The HCCM uses predefined representation divisions of 78 semantic classes (i.e. 54 individual classes and 24 conjunctive classes) for interpreting the hierarchical clustering outcome of the highest CCM evaluation quality. Lastly, HCCM method uses the L-score metric to quantify the matching degree, whereas the CCM method uses both L-score and F-score metrics. Dendrogram visualisation: The hierarchical clustering outcome provided by the algorithm achieving the optimal CCM evaluation score is visualised as a dendrogram using the code provided by McInnes et al. [30], and the best-matched cluster-class pairs yielded by the HCCM method with an L-score-based matching degree of at least 0.25 are manually annotated on this dendrogram. VI Result analysis (a) Matching identity-related individual classes (b) Matching nationality-related individual classes (c) Matching gender-related individual classes Figure 7: Cluster-class matching results [43] for evaluating the hierarchical clustering results obtained by applying SLINK (i.e. minPts=0minPts=0) and HDBSCAN (i.e. minPts=2,4,6,8,12,16,21,27)minPts=2,4,6,8,12,16,21,27)) to representations (i.e. embeddings) of 0.20.2, 11, 22, and 44-sec audios. This section presents results for analysing, visualising, and understanding the inner hierarchical clustering phenomena of the speaker recognition neural network. In particular, Section VI-A presents the results for how well HDBSCAN and SLINK analyse the hierarchical clustering phenomena of network representations, evaluated using the CCM method [43]. Section VI-B presents the results for the dendrogram’s visualisation and the HCCM’s semantic interpretation of the inner hierarchical clustering phenomena. VI-A Analysing inner hierarchical clustering - CCM’s evaluation To analyse the inner hierarchical clustering phenomena of the speaker recognition network, SLINK [14] and HDBSCAN [2] are applied to examine whether representations from a well-trained network form hierarchical representation clusters. The quality of these analyses is evaluated using Rosenberg et al.’s CCM method [43], which quantifies the overall matching degree to which the hierarchical representation clusters produced by SLINK or HDBSCAN align with the predefined representation divisions of semantic classes. Fig. 7(a) presents the CCM’s overall matching degrees for matching hierarchical representation clusters, separately produced by applying SLINK and HDBSCAN to representations of varying audio lengths, with predefined representation divisions of identity-related individual classes. In particular, the four second-level subplots from left to right in Fig. 7(a) correspond to the overall matching degrees evaluated on hierarchical representation clusters of 4-, 2-, 1-, and 0.2-second audio representations, respectively. Within each of the four subplots, the y-axis represents the overall matching degree (i.e. a value range of 0 ∼ 1), and the x-axis represents the minPtsminPts parameter. When minPts=0minPts=0, it indicates that the matching degree is evaluated based on hierarchical representation clusters produced by applying SLINK to representations of a certain audio length. When minPts>0minPts>0, it indicates that the matching degree is evaluated based on hierarchical representation clusters produced by applying HDBSCAN to representations of a certain audio length. Lastly, blue and yellow points denote the usage of F-score and L-score metrics in quantifying the overall matching degree. Observing the four second-level subfigures in Fig. 7(a), the CCM matching degree quantified using both L-score and F-score metrics gradually declines as minPtsminPts increases. Specifically, the highest matching degree in each subfigure is obtained when minPts=0minPts=0, indicating that, compared to HDBSCAN, applying SLINK to 4-, 2-, 1-, and 0.2-second representations yields superior hierarchical representation clusters that align well with the predefined representation divisions of identity-related individual classes. In particular, the optimal hierarchical representation clusters are produced by applying SLINK to 4-second representations, attaining a nearly perfect overall matching degree of 1.0. Furthermore, Fig. 7(b) and Fig. 7(c) present the CCM matching degrees for matching hierarchical representation clusters with predefined representation divisions of nation-related and gender-related individual classes, respectively. The optimal hierarchical representation clusters of both Fig. 7(b) and Fig. 7(c) are consistently produced by applying SLINK to 4-second audio representations, achieving an overall matching degree exceeding 0.6 for nationality-related representation divisions and nearly 1.0 for gender-related representation divisions. In summary, the above observations firstly demonstrate that the predefined grouping information related to speaker identity, nationality, and gender is indeed present within the hierarchical clustering outcomes produced by SLINK and HDBSCAN, which suggests that hierarchical clustering phenomena may exist within the network representation space, though further visualisation and semantic interpretation are required to confirm this; the above observations secondly demonstrate that hierarchical clustering outcomes produced from longer audio representations yield higher overall matching degrees, which may be attributed to the neural network being more adept at processing longer audio samples; the above observations thirdly demonstrate that SLINK achieves higher overall matching degrees than HDBSCAN, and given that our HDBSCAN implementation is essentially SLINK operating in mutual reachability distance space, this indicates that the density constraint introduced by the mutual reachability distance space does not contribute to better alignment with the predefined grouping information. VI-B Visualising and understanding innner hierarchical clustering To further confirm the existence of inner hierarchical clustering phenomena within the speaker recognition network, directly visualising and semantically interpreting the inner hierarchical clustering phenomena could provide more compelling evidence than evaluating how well such phenomena are analysed. In particular, we aim to visualise the hierarchical clustering outcomes of either SLINK or HDBSCAN in the form of a dendrogram, and semantically interpret the hierarchical clustering outcomes of either SLINK or HDBSCAN via the HCCM method we designed. The HCCM method identifies one-to-one correspondence matches between hierarchical representation clusters and predefined grouping information of individual and conjunctive classes, with a diagnostic interpretation provided for the matching performance of each pair under the L-score metric. Fig. 8 depicts the dendrogram visualised using the optimal hierarchical representation clusters, produced by applying SLINK to 4-sec audio representations that achieve the highest CCM overall matching degree. Since the number of hierarchical representation clusters typically reaches the order of thousands, only hierarchical representation clusters containing more than 800 representations are retained in order to keep the visualised dendrogram concise and clear. In the dendrogram, the y-axis represents λ=1ελ= 1 , where ε is the distance threshold used by SLINK to prune the MST. At any fixed λ, pruning the MST at distance threshold ε produces a flat clustering outcome, where the number of representations assigned to each flat cluster corresponds to the length of the horizontal cross-section at that λ. Each icicle-shaped object in the dendrogram represents a hierarchical cluster, which can be understood as the aggregation of a series of flat clusters across a continuous range of λ values. As λ increases (i.e. ε decreases), the edges of the MST linking different cluster members are gradually removed, causing larger icicles to fragment into smaller ones or vanish entirely. Furthermore, in Fig. 8, the HCCM’s one-to-one matching results are annotated using icons and text labels. Specifically, each icon is placed adjacent to a hierarchical representation cluster to indicate the semantic class that the HCCM method identifies as the best match for that cluster. Two types of icons are used: icons without the & symbol denote individual semantic classes, whilst icons with the & symbol denote conjunctive semantic classes formed by an and-logic combination of multiple individual semantic classes. The meaning of each icon can be identified by referring to the three legends provided alongside the dendrogram. Additionally, each icon is accompanied by a text label indicating the L-score-based matching degree of the corresponding best-matched cluster-class pair; when recall is selected as the limiting factor, the label is displayed as ‘rec: degree’, and when precision is selected, it is displayed as ‘pre: degree’. Importantly, only the icons and text labels corresponding to cluster-class pairs that achieve an L-score-based matching degree of at least 0.25 are displayed in Fig. 8. Observing Fig. 8, there is a large icicle at the top of the dendrogram, representing the root hierarchical representation cluster. As λ increases, the width of this largest icicle diminishes, indicating that fewer representations remain in the root cluster. At a certain point, this root hierarchical cluster splits into two sibling hierarchical clusters. By observing the icon annotations around these two sibling clusters, they can be interpreted as the male and female semantic classes, respectively. This is because the HCCM method considers these two representation clusters to be the best matches for the predefined representation divisions of male and female classes, respectively. Based on the text annotations near these two sibling clusters, the L-score identifies precision as the factor that limits the matching performance of both cases. In more detail, the matching performance of one matching case is mainly limited by 2% of representations inside one cluster are not predicted as the individual semantic class of male, and the matching of another matching case is limited by 1% of representations inside another cluster are not predicted as or belongs to the individual semantic class of female. In the following paragraphs, we separately discuss the semantic interpretations HCCM offers for the subtrees derived from the male and female representation clusters. Underneath the male representation cluster, we see that the male representation cluster is further split into two sibling hierarchical clusters as the increase of λ. For the icon annotations around these two sibling clusters, it shows that one is perfectly interpreted as the conjunctive semantic class of India&male, and another one is interpreted as the conjunctive class of USA&male. Observing the USA&male representation cluster, the nearby text annotation shows us that the L-score choose the weaker factor, the precision, as the main determinant or limitation of the matching performance. A detailed diagnostic interpration of this L-score value is: the matching performance of the cluster-class pair is mainly limited by only 65% of the representations within the cluster are actually belong to the individual class of USA and individual class of male at the same time (i.e. belong to the conjunctive class of USA&male), which means 35% remaining representations inside the cluster are imprecisely belong to other semantic classes (e.g. male from other countries). Underneath the USA&male representation cluster, the USA&male representation cluster successively splits into clusters interpreted as Mexico&male, Ireland&male, UK&male, and Canada&male conjunctive semantic classes. Interestingly, the Ireland&male, UK&male, and Canada&male representation clusters can be merged into a larger hierarchical cluster that resides underneath the USA&male representation cluster. Observing the text annotation near the UK&male representation cluster, the L-score-based matching degree is 0.59, and the poorer matching factor chosen by the L-score is the precision. The diagnostic interpretation can be offered to this 0.59 value is: the matching performance is primarily constrained by the fact that only 59% of representations within the cluster precisely belong to both UK and male individual classes, with the remaining 41% belonging to irrelevant semantic classes. Besdies, observing the text annotation near the Canada&male representation cluster, the L-score chooses the recall as the poorer matching factor with a value of 0.96, which can be interpreted as: the matching performance is mainly limited by the fact that only 96% of all representations in the predefined representation divisions of the Canada&male class are retrieved by the matched cluster, with 4% not captured. There is another representation cluster that perfectly captures all representations in the predefined representation division of the Ireland&male conjunctive class, achieving a 1.00 L-score degree, meaning both precision and recall values are also 1.00. Figure 8: Visualising hierarchical representation clusters produced by applying SLINK to 4-sec speaker embeddings as a dendrogram, with icon annotations showing the semantic interpretations that HCCM offered for the unknown representation clusters, and text labels showing the L-score-based matching degree that HCCM measured for each best-matched cluster-class pair. Underneath the female representation cluster, we can observe that one hierarchical representation cluster at a top position is best matched by the predefined representation division of the USA&female conjunctive class. Interestingly, unlike the UK&male representation cluster, which resides underneath the USA&male representation cluster, the UK&female representation cluster does not reside underneath the USA&female representation cluster. According to the text annotation near UK&female representation cluster, the L-score-based matching degree is 0.52, with a poorer recall, which can be interpreted that the matching performance is mainly limited by the recall factor, considering 48% of all representations pre-labelled as the UK&female class are not captured by the matched cluster. About the sibling hierarchical cluster of the UK&female representation cluster, this cluster serves as the optimal match for representation divisions of two conjunctive classes, including division of the French&female conjunctive class and that of the Norway&female conjunctive class. If this cluster is interpreted as the France&female cluster, a considerable number of representations belonging to the irrelevant Norway&female class are imprecisely selected; conversely, if interpreted as the Norway&female cluster, a considerable number of representations belonging to the irrelevant France&female class are imprecisely selected. Therefore, it is reasonable to consider this cluster as a union of the Norway&female and France&female representation clusters. Besides, the Canada&female and Ireland&female conjunctive classes also share the same best-matched representation cluster, implying that this cluster could be considered a union of the Ireland&female and Canada&female representation clusters. In summary, the above observations are further discussed as follows. Firstly, the dendrogram visualisation of the hierarchical clustering outcomes produced by SLINK demonstrates the existence of the inner hierarchical clustering phenomena. This is because, in the absence of this phenomena, the dendrogram produced by SLINK would fail to exhibit meaningful hierarchical structure, with all splits occurring rapidly at low λ values rather than emerging progressively across multiple levels. Secondly, unlike CCM, which broadly demonstrates that predefined grouping information is present within the hierarchical clustering outcomes, HCCM provides a more detailed one-to-one semantic interpretation, explicitly identifying which hierarchical representation clusters can be individually matched to specific semantic classes. More notably, HCCM reveals not only that gender-related individual classes are situated at the upper levels of the visualised dendrogram, but also that the conjunctive semantic classes it introduces, combining gender and nationality information, can be used to interpret the majority of clusters at the middle and lower levels of the dendrogram. Finally, based on the mechanism by which the L-score selects either precision or recall as the matching degree for different cluster-class pairs, we provide a diagnostic interpretation of the underlying reasons for poor matching performance in specific cases. VII Discussion As discussed in Section VI-A, we reason that better hierarchical representation clusters are produced by applying SLINK or HDBSCAN to 4-sec audio representations because the model is more adept at processing longer audio inputs. In particular, when our speaker recognition model evaluates audio clips in the VoxCeleb1 test set, the average EERs for the 4-, 2-, 1-, and 0.2-second audio clips are 2.17%, 2.35%, 2.98%, and 22.22%, respectively, indicating that the model generalises better on longer audio clips. The consistency between the model’s generalisation ability across different audio lengths and the corresponding CCM evaluation results of hierarchical clustering outcomes suggests that better hierarchical clustering phenomena within the network representation space are likely to be observed in cases where the model demonstrates stronger generalisation ability. VIII Conclusion This work proposes to analyse, visualise, and understand the hierarchical clustering phenomena within the representation space of a speaker recognition neural network. Specifically, the inner hierarchical clustering phenomena are analysed by applying SLINK and HDBSCAN to network representations of varying audio lengths. As evaluated by the CCM method, predefined grouping information related to speaker gender, nationality, and identity is present within the hierarchical clustering outcomes of both SLINK and HDBSCAN. Moreover, SLINK produces hierarchical representation clusters that align more closely with predefined grouping information compared to HDBSCAN, with higher alignment observed on longer audio representations. The inner hierarchical clustering phenomena are also visualised and understood. Specifically, the hierarchical representation clusters produced by SLINK are visualised as a dendrogram, which provides evidence for the existence of inner hierarchical clustering phenomena. Moreover, the predefined grouping information present within the visualised dendrogram is interpreted in detail via the proposed HCCM method. In more detail, HCCM one-to-one correspondingly matches some clusters within the visualised dendrogram with predefined representation divisions of gender-related individual classes, while matching a greater number of clusters with predefined representation divisions of conjunctive semantic classes combining speaker gender and nationality information. With the support of the proposed L-score metric to quantify the matching degree, a diagnostic interpretation of the matching performance is provided for each best-matched cluster-class pair. In the near future, we will apply our proposed XAI techniques to enable domain experts (e.g. in psychology [50, 12], linguistics [52], speech pathology [53], and vocal pedagogy [17, 57]) to perceptually understand the representation space of the speaker recognition neural network, and investigate the practical application-level benefits that can be realised once such human understanding is achieved. Acknowledgment Thanks to everyone who remains skeptical of AI systems, and to those who uphold humility and stay curious about the unknown of this world. Special thanks to JS Chung and Arsha Nagrani for releasing the VoxCeleb dataset and for publishing the well-trained speaker recognition neural network. A more detailed discussion of this paper can be found in the PhD thesis of the first author. References [1] W. Cai, J. Chen, and M. Li (2018) Exploring the encoding layer and loss function in end-to-end speaker and language recognition system. arXiv preprint arXiv:1804.05160. Cited by: §I. [2] R. J. G. B. Campello, D. Moulavi, and J. Sander (2013) Density-based clustering based on hierarchical density estimates. In Advances in Knowledge Discovery and Data Mining, J. Pei, V. S. Tseng, L. Cao, H. Motoda, and G. Xu (Eds.), Berlin, Heidelberg, p. 160–172. External Links: ISBN 978-3-642-37456-2 Cited by: §I, Figure 4, Figure 4, §I-A, §I, §IV-A, §IV-A, §IV-A, §IV, §VI-A. [3] S. Carbonnelle and C. De Vleeschouwer (2020) Intraclass clustering: an implicit learning ability that regularizes DNNs. In International Conference on Learning Representations, Cited by: §I, §I-A. [4] M. Caron, P. Bojanowski, A. Joulin, and M. Douze (2018) Deep clustering for unsupervised learning of visual features. In Proceedings of the European Conference on Computer Vision (ECCV), p. 132–149. Cited by: §I, §I-A. [5] P. Christen, D. J. Hand, and N. Kirielle (2023) A review of the f-measure: its history, properties, criticism, and alternatives. ACM Computing Surveys 56 (3), p. 1–24. Cited by: §I-B, §I-B, §IV-C. [6] J. S. Chung, J. Huh, S. Mun, M. Lee, H. S. Heo, S. Choe, C. Ham, S. Jung, B. Lee, and I. Han (2020) In defence of metric learning for speaker recognition. arXiv preprint arXiv:2003.11982. Cited by: §I, §I, §V. [7] J. S. Chung, A. Nagrani, and A. Zisserman (2018) Voxceleb2: deep speaker recognition. arXiv preprint arXiv:1806.05622. Cited by: §V. [8] H. De Baar (1994) Von Liebig’s law of the minimum and plankton ecology (1899–1991). Progress in Oceanography 33 (4), p. 347–386. Cited by: §I, §IV-C. [9] M. Espadoto, R. M. Martins, A. Kerren, N. S. Hirata, and A. C. Telea (2019) Toward a quantitative survey of dimension reduction techniques. IEEE Transactions on Visualization and Computer Graphics 27 (3), p. 2153–2173. Cited by: §I-B. [10] M. Ester, H. Kriegel, J. Sander, and X. Xu (1996) A density-based algorithm for discovering clusters in large spatial databases with noise. Proceedings of the Second International Conference on Knowledge Discovery and Data Mining (KDD’96), p. 226–231. Cited by: §I-A, §I-A, §I. [11] D. Garcia-Romero, D. Snyder, G. Sell, D. Povey, and A. McCree (2017) Speaker diarization using deep neural network embeddings. In 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), p. 4930–4934. Cited by: §I-C. [12] K. J. Gergen (1973) Social psychology as history.. Journal of personality and social psychology 26 (2), p. 309. Cited by: §VIII. [13] L. Goetschalckx, A. Andonian, A. Oliva, and P. Isola (2019) Ganalyze: toward visual definitions of cognitive image properties. In Proceedings of the IEEE/CVF International Conference on Computer Vision, p. 5744–5753. Cited by: §I-A. [14] J. C. Gower and G. J. Ross (1969) Minimum spanning trees and single linkage cluster analysis. Journal of the Royal Statistical Society: Series C (Applied Statistics) 18 (1), p. 54–64. Cited by: §I, §I, §IV-A, §IV, §VI-A. [15] D. Gunning and D. Aha (2019) DARPA’s explainable artificial intelligence (XAI) program. AI magazine 40 (2), p. 44–58. Cited by: §I. [16] K. He, X. Zhang, S. Ren, and J. Sun (2016) Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, p. 770–778. Cited by: §I, §V. [17] K. Heidemann (2016) A system for describing vocal timbre in popular song.. Music Theory Online 22 (1). Cited by: §VIII. [18] D. M. Johnson, C. Xiong, J. Gao, and J. J. Corso (2013) Comprehensive cross-hierarchy cluster agreement evaluation.. In AAAI (Late-Breaking Developments), Cited by: §I-B. [19] W. A. Johnston and V. J. Dark (1986) Selective attention.. Annual Review of Psychology. Cited by: §I. [20] J. B. Kruskal (1956) On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical society 7 (1), p. 48–50. Cited by: §IV-A. [21] Y. LeCun, Y. Bengio, and G. Hinton (2015) Deep learning. nature 521 (7553), p. 436–444. Cited by: §I. [22] Y. Li, W. Guan, H. Huang, S. Miao, Q. Su, L. Li, and Q. Hong (2024) Efficient integrated features based on pre-trained models for speaker verification. In Proc. Interspeech 2024, p. 2140–2144. Cited by: Figure 1, Figure 1, §I, §I-A. [23] P. Linardatos, V. Papastefanopoulos, and S. Kotsiantis (2020) Explainable AI: a review of machine learning interpretability methods. Entropy 23 (1), p. 18. Cited by: §I. [24] Y. Liu, E. Jun, Q. Li, and J. Heer (2019) Latent space cartography: visual analysis of vector space embeddings. In Computer graphics forum, Vol. 38, p. 67–78. Cited by: §I-A. [25] Y. Lukic, C. Vogt, O. Dür, and T. Stadelmann (2016) Speaker identification and clustering using convolutional neural networks. In 2016 IEEE 26th International Workshop on Machine Learning for Signal Processing (MLSP), p. 1–6. Cited by: §I-C. [26] S. M. Lundberg and S. Lee (2017) A unified approach to interpreting model predictions. Advances in Neural Information Processing Systems 30. Cited by: §I-A. [27] C. W. Lynn and D. S. Bassett (2020) How humans learn and represent networks. Proceedings of the National Academy of Sciences 117 (47), p. 29407–29415. Cited by: §I. [28] J. MacQueen et al. (1967) Some methods for classification and analysis of multivariate observations. In Proceedings of the fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, p. 281–297. Cited by: §I. [29] C. D. Manning, P. Raghavan, and H. Schütze (2008) Flat clustering. Introduction to Information Retrieval 356, p. 360. Cited by: §I, §I-B. [30] L. McInnes, J. Healy, and S. Astels (2017) HDBSCAN: hierarchical density based clustering. The Journal of Open Source Software 2 (11), p. 205. Cited by: §IV-A, §IV-A, §V, §V. [31] L. McInnes, J. Healy, and J. Melville (2018) UMAP: uniform manifold approximation and projection for dimension reduction. arXiv preprint arXiv:1802.03426. Cited by: §I. [32] M. Morzycki (2016) Modification. Cambridge University Press. Cited by: §I, §IV-B. [33] F. Murtagh and P. Contreras (2017) Algorithms for hierarchical clustering: an overview, i. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 7 (6), p. e1219. Cited by: §I-C. [34] A. Nagrani, J. S. Chung, and A. Zisserman (2017) Voxceleb: A large-scale speaker identification dataset. arXiv preprint arXiv:1706.08612. Cited by: §I, §V, §V. [35] V. Nasteski (2017) An overview of the supervised machine learning methods. Horizons. b 4 (51-62), p. 56. Cited by: §I. [36] S. Naumov, G. Yaroslavtsev, and D. Avdiukhin (2021-05) Objective-based hierarchical clustering of deep embedding vectors. Proceedings of the AAAI Conference on Artificial Intelligence 35 (10), p. 9055–9063. External Links: Link, Document Cited by: §I-C. [37] G. Peiffer (2021) Investigating deep neural network internal clustering and generalization properties. Master’s thesis, Université catholique de Louvain, Louvain, Belgium. Cited by: §I, §I-A, §I-B. [38] V. Petsiuk, A. Das, and K. Saenko (2018) Rise: randomized input sampling for explanation of black-box models. arXiv preprint arXiv:1806.07421. Cited by: §I-A. [39] V. Prasad, H. van Gorp, C. Humer, A. Vilanova, and N. Pezzotti (2024) The tree of diffusion life: evolutionary embeddings to understand the generation process of diffusion models. arXiv e-prints, p. arXiv–2406. Cited by: §I. [40] R. C. Prim (1957) Shortest connection networks and some generalizations. The Bell System Technical Journal 36 (6), p. 1389–1401. Cited by: §IV-A. [41] X. Ran, Y. Xi, Y. Lu, X. Wang, and Z. Lu (2023) Comprehensive survey on hierarchical clustering algorithms and the recent developments. Artificial Intelligence Review 56 (8), p. 8219–8264. Cited by: §I, §I. [42] M. T. Ribeiro, S. Singh, and C. Guestrin (2016) ‘Why should i trust you?’ explaining the predictions of any classifier. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, p. 1135–1144. Cited by: §I-A. [43] A. Rosenberg and J. Hirschberg (2007) V-measure: a conditional entropy-based external cluster evaluation measure. In Proceedings of the 2007 Joint Conference on Empirical Methods in Natural Language Processing and Computational Natural Language Learning (EMNLP-CoNLL), p. 410–420. Cited by: §I, §I-D, §I-B, §I-B, §I, §IV-B, §V, §V, Figure 7, Figure 7, §VI-A, §VI. [44] R. R. Selvaraju, A. Das, R. Vedantam, M. Cogswell, D. Parikh, and D. Batra (2016) Grad-CAM: why did you say that?. arXiv preprint arXiv:1611.07450. Cited by: §I-A. [45] Y. Shen and B. Zhou (2021) Closed-form factorization of latent semantics in GANs. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, p. 1532–1540. Cited by: §I-A. [46] R. Sibson (1973) SLINK: an optimally efficient algorithm for the single-link cluster method. The Computer Journal 16 (1), p. 30–34. Cited by: §I, §I, §IV-A, §IV. [47] P. Singh, A. Kaul, and S. Ganapathy (2023) Supervised hierarchical clustering using graph neural networks for speaker diarization. In ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), p. 1–5. Cited by: §I-C. [48] J. Snell, K. Swersky, and R. Zemel (2017) Prototypical networks for few-shot learning. Advances in Neural Information Processing Systems 30. Cited by: §V. [49] R. R. Sokal, C. D. Michener, et al. (1958) A statistical method for evaluating systematic relationships. Cited by: §I. [50] R. L. Solso, M. K. MacLin, and O. H. MacLin (2005) Cognitive psychology. Pearson Education New Zealand. Cited by: §VIII. [51] N. Tawara, A. Ogawa, T. Iwata, M. Delcroix, and T. Ogawa (2020) Frame-level phoneme-invariant speaker embedding for text-independent speaker recognition on extremely short utterances. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), p. 6799–6803. Cited by: §I. [52] G. L. Trager (1958) Paralanguage: a first approximation. Stud. Linguist. 13, p. 1–12. Cited by: §VIII. [53] L. E. Travis (1931) Speech pathology; a dynamic neurological treatment of normal speech and speech deviations.. Cited by: §VIII. [54] L. Van der Maaten and G. Hinton (2008) Visualizing data using t-SNE.. Journal of Machine Learning Research 9 (11). Cited by: §I. [55] D. Xu and Y. Tian (2015) A comprehensive survey of clustering algorithms. Annals of Data Science 2, p. 165–193. Cited by: §I-A. [56] F. Xu, H. Uszkoreit, Y. Du, W. Fan, D. Zhao, and J. Zhu (2019) Explainable AI: a brief survey on history, research areas, approaches and challenges. In Natural Language Processing and Chinese Computing: 8th CCF International Conference, NLPCC 2019, Dunhuang, China, October 9–14, 2019, Proceedings, Part I 8, p. 563–574. Cited by: §I. [57] Y. Xu, W. Wang, H. Cui, M. Xu, and M. Li (2022) Paralinguistic singing attribute recognition using supervised machine learning for describing the classical tenor solo singing voice in vocal pedagogy. EURASIP Journal on Audio, Speech, and Music Processing 2022 (1), p. 8. Cited by: §VIII. [58] G. Yang, N. Fei, M. Ding, G. Liu, Z. Lu, and T. Xiang (2021) L2M-gan: learning to manipulate latent space semantics for facial attribute editing. In 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), Vol. , p. 2950–2959. External Links: Document Cited by: §I-A. [59] Y. Zhao, G. Karypis, and U. Fayyad (2005) Hierarchical clustering algorithms for document datasets. Data Mining and Knowledge Discovery 10, p. 141–168. Cited by: §I-B. [60] B. Zhou, A. Khosla, A. Lapedriza, A. Oliva, and A. Torralba (2016) Learning deep features for discriminative localization. In Proceedings of the IEEE conference on computer vision and pattern recognition, p. 2921–2929. Cited by: §I-A. Yanze Xu was admitted to the B.Sc. programme at Sun Yat-sen University as a music-talent student in 2016 and graduated with a degree in Software Engineering in 2020. He then joined Duke Kunshan University as a research assistant, where he developed a paralinguistic singing attribute recognition system capable of automatically analysing the timbral characteristics of the tenor singing voice. In 2022, he was accepted by the University of Surrey as a self-funded PhD student, exploring explainable AI research for speaker recognition neural networks. Wenwu Wang (Fellow, IEEE) was born in Anhui, China. He received the B.Sc., M.E. , and the Ph.D. degrees from Harbin Engineering University, China, in 1997, 2000, and 2002, respectively. He then worked with King’s College London, Cardiff University, Tao Group Ltd. (now Antix Labs Ltd.), and Creative Labs, before joining University of Surrey, U.K., in May 2007, where he is currently a Professor in signal processing and machine learning, and a Co-Director of the Machine Audition Lab within the Centre for Vision Speech and Signal Processing. He is also an AI Fellow at the Surrey Institute for People Centred Artificial Intelligence. His current research interests include signal processing, machine learning and perception, artificial intelligence, machine audition (listening), and statistical anomaly detection. He has (co)-authored over 300 papers in these areas. He has been a (co-)author of various works that have received awards, including the 2022 IEEE Signal Processing Society Young Author Best Paper Award, ICAUS 2021 Best Paper Award, DCASE 2020 and 2023 Judge’s Award, DCASE 2019 and 2020 Reproducible System Award, and LVA/ICA 2018 Best Student Paper Award. He is an Associate Editor (2020-2025) for IEEE/ACM Transactions on Audio Speech and Language Processing. He was a Senior Area Editor (2019-2023) and Associate Editor (2014-2018) for IEEE Transactions on Signal Processing. He is the elected Chair (2023-2024) of IEEE Signal Processing Society (SPS) Machine Learning for Signal Processing Technical Committee, a Board Member (2023-2024) of IEEE SPS Technical Directions Board, the Chair (2025-2027) and Vice Chair (2022-2024) of the EURASIP Technical Area Committee on Acoustic Speech and Music Signal Processing, an elected Member (2021-2026) of the IEEE SPS Signal Processing Theory and Methods Technical Committee. He is or was on the organizing committee of INTERSPEECH 2022, IEEE ICASSP 2019 and 2024, IEEE MLSP 2013, 2024 and 2025, and IEEE SSP 2009. Mark Plumbley (Fellow, IEEE) received the B.A. (Hons.) degree in electrical sciences and the Ph.D. degree in neural networks from University of Cambridge, Cambridge, U.K., in 1984 and 1991, respectively. After his Ph.D. degree, he became a Lecturer with King’s College London, London, before moving to Queen Mary University of London, London, in 2002. He subsequently became a Professor and a Director of the Centre for Digital Music, before joining the University of Surrey in 2015 as a Professor of signal processing. He is known for his work on analysis and processing of audio and music, using a wide range of signal processing techniques, including matrix factorization, sparse representations, and deep learning. He is the co-editor of the recent book on Computational Analysis of Sound Scenes and Events, and Co-Chair of the recent DCASE 2018 Workshop on Detection and Classifications of Acoustic Scenes and Events. He is a Member of the IEEE Signal Processing Society Technical Committee on Signal Processing Theory and Methods, and a Fellow of the IET.