# interpretable_deep_clustering_for_tabular_data__91a41f89.pdf

Interpretable Deep Clustering for Tabular Data

Jonathan Svirsky 1 Ofir Lindenbaum 1

Clustering is a fundamental learning task widely used as a first step in data analysis. For example, biologists use cluster assignments to analyze genome sequences, medical records, or images. Since downstream analysis is typically performed at the cluster level, practitioners seek reliable and interpretable clustering models. We propose a new deep-learning framework for general domain tabular data that predicts interpretable cluster assignments at the instance and cluster levels. First, we present a self-supervised procedure to identify the subset of the most informative features from each data point. Then, we design a model that predicts cluster assignments and a gate matrix that provides cluster-level feature selection. Overall, our model provides cluster assignments with an indication of the driving feature for each sample and each cluster. We show that the proposed method can reliably predict cluster assignments in biological, text, image, and physics tabular datasets. Furthermore, using previously proposed metrics, we verify that our model leads to interpretable results at a sample and cluster level. Our code is available on Github.

1. Introduction

Clustering is a crucial task in data science that helps researchers uncover and study latent structures in complex data. By grouping related data points into clusters, researchers can gain insights into the underlying characteristics of the data and identify relationships between samples and variables. Clustering is used in various scientific fields, including biology (Reddy et al., 2018), physics (Mikuni & Canelli, 2021), and social sciences (Varghese et al., 2010). For instance, in biology, clustering can identify different

*Equal contribution 1Department of Engineering, Bar Ilan University, Ramat-Gan, Israel. Correspondence to: Jonathan Svirsky <svirskj@biu.ac.il>, Ofir Lindenbaum <ofir.lindenbaum@biu.ac.il>.

Proceedings of the 41 st International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).

disease subtypes based on molecular or genetic data. In psychology, based on survey data, clustering can identify different types of behavior or personality traits.

Clustering is a common technique used in bio-medicine to analyze gene expression data. It involves identifying groups of genes that have similar expression patterns across different samples. Scientists often cluster high-dimensional points corresponding to individual cells to recover known cell populations and discover new, potentially rare cell types. However, bio-med gene expression data is generally represented in a tabular, high-dimensional format, making it difficult to obtain accurate clusters with meaningful structures. In addition, interpretability is a crucial requirement for real-world bio-med datasets since it is essential to understand the biological meaning behind the identified clusters. As a result, there is an increasing demand in bio-medicine for clustering models that offer interpretability for tabular data.

In bio-medicine, clustering is widely used to analyze gene expression data, where cluster assignments can identify groups of genes with similar expression patterns across different samples (Armingol et al., 2021). When applied to single-cell omics data (Wang & Bodovitz, 2010), clustering recovers known cell populations while discovering new and perhaps rare cell types (Deprez et al., 2020). Unfortunately, such bio-med gene expression data types are generally represented in a tabular, high-dimensional format, making it difficult to obtain accurate clusters with meaningful structures. In addition, interpretability is a crucial requirement for realworld bio-med datasets since it is essential to understand the biological meaning behind the identified clusters (Yang et al., 2021). As a result, there is an increasing demand in bio-medicine for clustering models that offer interpretability for tabular data.

Interpretability in machine learning refers to the ability to understand and explain the predictions and decisions made by predictive models. It is critical for the proper deployment of machine learning systems, especially in applications where transparency and accountability are essential. Interpretability can take different forms, including interpretable model structure, identification of feature importance for model predictions, visualization of data, and generation of explanations for the prediction. In this work,

Interpretable Deep Clustering for Tabular Data

we aim to design a model that achieves interpretability by sample-wise feature selection and generating cluster-level interpretations of model results. This type of interpretability is crucial for biomedical applications, for example, when seeking marker genes that are "typical" for different clusters in high-dimensional biological measurements.

In recent years, the use of deep learning models for clustering has been gaining interest (Shen et al., 2021; Li et al., 2022; Cai et al., 2022; Niu et al., 2021). These models offer better clustering capabilities by providing an improved embedding of data points. However, most existing schemes focus on image data, require domain-specific augmentations, and are not interpretable. Interpretability has also been gaining attention in deep learning, but most models focus on supervised learning (Alvarez Melis & Jaakkola, 2018; Yoon et al., 2019; Yang et al., 2022). We aim to extend these ideas to unsupervised learning by developing a deep clustering model that is interpretable by design and can be applied to general domain data. In this context, interpretability means being able to identify variables that drive the formation of clusters in the data (Bertsimas et al., 2021). We demonstrate the effectiveness of our method in terms of clustering accuracy and interpretability in biomed, physics, text, and image data represented in tabular format.

This work introduces Interpretable Deep Clustering (IDC), an unsupervised two-stage clustering method for tabular data. The method first selects informative features for reliable data reconstruction, a task that was demonstrated to be correlated with clustering capabilities (Song et al., 2013; Han et al., 2018). Then, using the sparsified data, we learn the cluster assignments by optimizing neural network parameters subject to a clustering objective function (Yu et al., 2020). In addition, the method provides both instance-level and cluster-level explanations represented by the selected feature set. The model learns instance-level local gates that select a subset of features using an autoencoder (AE) trained to reconstruct the original sample. The global gates for cluster-level interpretability are derived from the cluster label assignments learned by the model. To ensure sample-level interpretations, the gates are encouraged by the recently proposed discrimination constraint denoted as the total coding rate. Using synthetic data and MNIST, we demonstrate the interpretability quality of our model. Then, we use real-world data from biomed, images, text, and physics to demonstrate that our model can find meaningful clusters while using only a small subset of informative features. In the following sections, we provide a detailed description of our approach.

2. Related Work

Unsupervised Feature Selection The problem of unsupervised feature selection (UFS) involves identifying vari-

ables useful for downstream tasks such as data clustering. Towards this goal, several works have exploited regularized AEs (Han et al., 2018; Lee et al., 2022; Sokar et al., 2022; Balın et al., 2019), which identify a global subset of features that are sufficient for data reconstruction. Another line of UFS schemes relies on the graph Laplacian (Von Luxburg, 2007) to identify subsets of smooth features with respect to the core structure in the data (He et al., 2005; Zhao & Liu, 2012; Lindenbaum et al., 2021; Shaham et al., 2022). Both types of UFS frameworks can help improve downstream clustering capabilities; however, existing global schemes do not provide sample-level or cluster-level interpretability. Although there are recent works on supervised local feature selection (Yoon et al., 2019; Yang et al., 2022) that provide interpretability, we are not aware of any sample-level unsupervised feature selection schemes. Therefore, we present, for the first time, an end-to-end clustering scheme with local feature selection capabilities.

Interpretable Clustering Guan et al. (2011) presented a pioneering work for simultaneous unsupervised feature selection and clustering. The authors proposed a probabilistic model that performs feature selection by using beta Bernoulli prior in the context of a Dirichlet process mixture for clustering. However, their model can only select clusterlevel and dataset-level informative features, whereas our approach offers interpretability with sample-level granularity. In (Frost et al., 2020), the authors proposed tree-based K-means clustering as a part of other works in the same direction (Lawless et al., 2022; Gabidolla & Carreira-Perpiñán, 2022; Cohen, 2023; Kauffmann et al., 2022). However, the explanations are global only for a given dataset and rely on the whole set of points. In contrast, our approach learns local gates for each sample in the dataset by optimizing a neural network that performs local feature selection, thus producing sample-level interpretations. Since our method is fully parametric, it also offers enhanced generalization capabilities compared to existing schemes. Precisely, our model can predict cluster assignments and informative features for samples not seen during training.

Deep clustering Recently, several methods were proposed for NN-based clustering, to name a few: (Gao et al., 2020; Niu et al., 2021; Li et al., 2022; Shaham et al., 2018; Cai et al., 2022; Lv et al., 2021; Shen et al., 2021; Peng et al., 2022). However, these methods primarily focus on vision and rely on domain-specific augmentations and, therefore, can not be applied to tabular datasets. Therefore, we introduce an interpretable NN model for general highdimensional datasets, such as biomedical tabular data.

Li et al. (2022) have generalized the maximum coding rate reduction loss (MCR2) (Yu et al., 2020) for embedding and clustering. The model aims to separate clusters while mak-

Interpretable Deep Clustering for Tabular Data

Figure 1. Illustration of the proposed model. The first step involves self-supervision for learning a meaningful latent representation and sample-level informative features. During this stage, we optimize the parameters of the Gating Network (green) and the autoencoder (blue) that reconstructs ˆx from latent embedding h. The gating Network learns a sample-specific sparse gate vector z for input sample x such that x z is sufficient for reconstruction via an autoencoder. We train a clustering head (orange) to predict cluster assignments ˆy from the latent embedding h by minimizing the mean cluster coding rate loss (see Eq. 4). This loss is designed to push clusters apart while making each cluster more compact. The Auxiliary Classifier (yellow) is trained on sparse representations x z ZG to predict cluster labels and optimizes the cluster level gating matrix ZG.

ing them denser. However, this scheme also requires domain knowledge to design semantic preserving augmentations to train the contrastive loss. While such augmentations could be designed effectively for the visual domain, creating such semantic preserving augmentations of tabular data remains an open challenge (Qian et al., 2023). To overcome this limitation, we propose a two-step procedure that involves (i) self-supervision with locally sparse reconstruction that improves interpretability and induces a spectral bias that enhances performance on tabular data, and (ii) an adapted MCR2 (Li et al., 2022) objective for identifying clusters based on diverse features.

3. Problem Setup

We want to cluster a set of data points X = {xi}N i=1 into K clusters. Each data point xi RD is a D-dimensional vector-valued observation of general type, meaning it does not follow a particular feature structure. Our goal is to learn an interpretable clustering model defined by the tuple fΘ, Sglob such that fΘ(xi) = {ˆyi, Sloc i } where ˆyi {1, 2, ..., K} is an accurate clustering assignment, and Sloc i is a local feature importance set for sample i and defined by Sloc i = {sj i [0, 1]}D j=1. Sglob [0, 1]K D is a global feature importance matrix where each gating vector of size D is learned for K clusters. By forcing |Sloc i | << D, we can attenuate nuisance features and identify (samplespecific) subsets of informative features, thus improving the interpretability of the clustering model.

Our research is driven by the critical task in biomedicine of cell clustering and identifying marker genes that are associated with each cluster (Kiselev et al., 2017). This task involves clustering a high-dimensional dataset while identifying genes with unique patterns in each cluster. Therefore, our goal is to develop an interpretable model that performs well in the clustering task, i.e., can identify groups with semantically related samples while focusing on local subsets of features. We want to highlight that in the supervised setting, this kind of local feature selection has been linked to interpretability by various authors (Alvarez Melis & Jaakkola, 2018; Yoon et al., 2019; Yang et al., 2022).

We have extended the concept of supervised interpretability to the unsupervised domain by introducing a new model that combines clustering and local feature selection. This model identifies and eliminates irrelevant input features that do not contribute to the clustering task. Moreover, the model generates unique explanations for each sample and cluster-level interpretations, which can help comprehend the predictions. We have used various metrics to demonstrate that our model enhances interpretability and produces accurate clustering results.

We suggest a two-step approach involving training selfsupervised local (sample-specific) gates and latent representations to enhance cluster separation. This approach pro-

Interpretable Deep Clustering for Tabular Data

vides interpretability through local feature selection, which improves performance on tabular data by allowing for the learning of high-frequency prediction functions using Neural Networks (NNs). In contrast to image or audio data, tabular data requires a prediction function with higher frequency components (Beyazit et al., 2023). However, the implicit bias of NNs towards learning low-frequency functions, as described in (Basri et al., 2020), hinders the performance of vanilla fully connected models on tabular data. We empirically demonstrate that local sparsification (gates) can aid NNs in learning high-frequency functions and improving predictive capabilities, as shown in Section 6.3.

In the second step of our proposed method, we identify the cluster assignments and global gates which focus on the driving features at the cluster level. The global gates are learned based on the clustering assignments, which are determined by minimizing the coding rate (Eq. 4) for each cluster. The result of this step is the clustering assignments for each sample, local gates for sample-level interpretation, and global gates for cluster-level explanations. The following subsection will present the proposed framework and architecture design in more detail.

4.1. Local Self Supervised Feature Selection

Given unlabeled observations {xi}N i=1, where xi RD, our goal is to learn a prediction function fθ (parametrized using a NN) and a feature importance vectors zi [0, 1]D. The feature importance vector will highlight which subset of variables the model should rely on for clustering each sample xi. This will allow the model to use fewer features for each sample and reduce overfitting when predicting the clusters of unseen samples.

Towards our goal, we expand upon the recently proposed stochastic gates (Lindenbaum et al., 2021; Yamada et al., 2020) by allowing them to learn the importance vector locally (for each sample). These stochastic gates are continuously relaxed Bernoulli variables defined based on the hard thresholding function for feature d and sample i:

zd i = max(0, min(1, 0.5 + µd i + ϵd i )), (1)

where ϵd i is drawn from N(0, σ2). The value µd i is the logit output of the network before being passed through the hard thresholding function (in Eq. 1). In our model, σ is fixed to 0.5 throughout training, as suggested in (Yamada et al., 2020). The injected noise is controlled by σ and helps push the converged values of zd i towards 0 or 1. For more information, please see (Yamada et al., 2020) and (Yang et al., 2022). The parameters µi RD, i = 1, ..., N are specific to each sample and are predicted using a gating network fθ1. In other words, µi = f(xi|θ1), where θ1 are the weights of the gating network. These weights are learned simultaneously with the weights of the prediction

network θ2 by minimizing the following loss:

Lsparse = Ezi,xi L(fθ2(xi zi)) + λ Lreg(zi) ,

where L is a desired prediction loss, for example, a clustering objective function or reconstruction error. The Hadamard product is simply an element-wise multiplication and is denoted by . We compute the empirical expectation over xi and zi, where i is an index that ranges over a dataset of size N. The term Lreg(zi) is a regularizer designed to sparsify the gates. It is defined as follows: Lreg = zi 0. After taking the expectation over zi and the samples xi, E[Lreg] can be rewritten using a double sum in terms of the Gaussian error function (erf):

2 erf µd i + 0.5

here, we calculate the expectation using the parametric definition of zi.

We have chosen a denoising autoencoder (Vincent et al., 2008) for our prediction network. It aims to identify and select only the informative features required for reconstruction by disregarding nuisance features. The network is trained through self-supervision using a reconstruction loss with domain-agnostic augmentations. This helps the network learn a latent embedding of the input sample and prompts the gating network to open only the gates needed to reconstruct data. The model consists of the following components:

Gating Network: fθG(xi) = zi, is a hypernetwork that predicts the gates zi vector for sample xi, where zi [0, 1]D (each element is defined based on Eq.1). Encoder: fθE(x i) = hi, is a mapping function that learns an embedding hi based on the element-wise gated sample x i = xi zi. Decoder: fθD(hi) = ˆxi, that reconstructs xi based on the embedding hi.

We utilize an autoencoder with parameters θE θD and employ gated input reconstruction loss Lrecon(fθD(fθE(fθG(xi) xi)), xi) to measure the deviation between estimated ˆxi = fθD(fθE(fθG(xi) xi)) and input sample xi. We introduce input (Vincent et al., 2008) and latent data augmentations (Doi et al., 2007) to learn semantically informative features. Additional details about these augmentations are in Appendix F. Additionally, we introduce an extra gates total coding rate loss, Lgtcr(Z) that encourages the selection of unique gates for each sample. The following equation defines this loss:

2 logdet(I + D NB ϵ (ZT Z)),

which is approximately the negative Shannon coding rate of a multivariate Gaussian distribution (Yu et al., 2020) up

Interpretable Deep Clustering for Tabular Data

to precision ϵ, and is defined for a mini-batch of size NB of normalized local gates Z {zi}N 1 . This component, which is inspired by (Li et al., 2022), aims to decrease the correlation between gate vectors zi.

The final loss for predicting sparse data is calculated as follows:

Lsparse = Lrecon + Lgtcr + λ Lreg. (3)

The weight term for regularization, represented by λ, is gradually increased in training using a cosine function scheduler. The autoencoder s weights are initially set through standard reconstruction objective training. In Section 6.4, we conduct an ablation study to validate the importance of each component of our loss.

Our model is designed to minimize the number of features required for data reconstruction. This is achieved by sparsifying the input samples. Additionally, the gated input reconstruction loss helps the model learn local masks, which are used to attenuate noisy features and improve interpretability. The gates total coding rate loss encourages the selection of diverse gates across samples, ensuring that informative features are not constant across the data.

4.2. Cluster assignments with Global Interpretations

Clustering Head During the clustering phase, our goal is to compress the learned representations of gated samples of X into K clusters and predict cluster-level gates. To accomplish this, we train a clustering head fθC(hi) = ˆyi, which learns cluster one-hot assignments ˆyi {1, ..., K}. The model outputs logits values π1 K i for each sample. These logits values are then converted to cluster assignment probabilities ˆyi by using a Gumbel-Softmax (Jang et al., 2016) reparameterization:

ˆyk i = exp((log(πk i ) + gk i )/τ) PK j=1 exp((log(πk i ) + gk i )/τ) , for k = 1, ..., K.

where g1 K i are i.i.d. samples drawn from a Gumbel(0, 1) distribution. The variable τ represents a temperature hyperparameter, where larger values of τ produce a uniform distribution of ˆy. On the other hand, decreasing the temperature leads to one-hot vectors. This neural network is optimized with the following loss

1 2 logdet I + dh NB ϵ (HT k Hk) , (4)

where Hk = {hi : fθC(hi) = k} {hi = fθE(xi zi)}N i=1 is a mini-bach of size NB embedding vectors for all samples xi s which were assigned to cluster k. dh is the embedding dimension and ϵ is the hyperparameter that denotes the desired precision of coding rate (Yu et al., 2020). The aim is to decrease the average coding rate for each cluster of embeddings Hk to make them more compact.

Global Interpretations To provide global cluster-level interpretations, we develop a gate matrix called ZG {0, 1}K D. Each row in the matrix corresponds to a cluster and each column to an input variable. We use an auxiliary classifier, fθAC, to train this gate matrix. The classifier accepts a gated representation of xi, defined by xi zi zk G, where zk G is a global gates vector learned for cluster k = argmax1,...,K ˆyi. The locally sparse samples xi zi learned with the autoencoder are multiplied by global gates ZG and passed through a single-hidden-layer classifier. This classifier is trained to output cluster assignments ˆyi identical to those predicted with clustering head fθC using a cross-entropy loss. During inference, only ZG is used, and fθAC is not needed.

In our method, we use a regularization loss term called Lreg with an increasing weight λg to sparsify the gates in the global gates matrix. This is added to the clustering loss term Lclust, which is the sum of the clustering-head term Lhead and the cross-entropy loss term LCE. To summarize, we train the clustering head to predict assignments along with global gates. We use an auxiliary classifier to optimize the global gates while being trained in a self-supervised manner on the pseudo labels predicted by the clustering head.

5. Interpretability

Practitioners may need interpretability at different levels of granularity. At a coarser level, it is useful to identify which features are common to a group of semantically related samples (or clusters) (Guan et al., 2011). At a finer level of detail, we aim to find unique explanations for each data point. Specifically, we want to know which features drive the model to make specific predictions (Alvarez Melis & Jaakkola, 2018).

Although the internal workings of a model may remain a black box to the user, the relationship between the input features and the model s predictions can provide some insight into its interpretability. Recently, several criteria have been proposed to assess the interpretability of supervised models (Alvarez Melis & Jaakkola, 2018; Yang et al., 2022), as outlined in the following paragraphs. We will also discuss how these criteria can be adapted for an unsupervised setting.

Diversity We expect a good interpretability model to identify different sets of variables as driving factors for explaining distinct clusters. This is measured by the diversity metric, which is calculated by finding the negative mean Jaccard similarity between cluster-level informative features across all pairs of clusters. In simple terms, given a set of indices Sci 1, ..., D that indicate the selected informative variables of cluster ci, where i = 1, ..., K, the diversity is defined as 1 P i =j J(Sci,Scj ) K (K 1)/2. Here, J is the Jaccard similarity between two sets. A score of 1 is obtained when

Interpretable Deep Clustering for Tabular Data

there is no overlap between cluster-level features, indicating perfect diversity.

Faithfulness An interpretation is faithful if it accurately represents the reasoning behind the model s prediction function. To evaluate this quantity, faithfulness measures the correlation between the predictivity of the model and the feature importance. To calculate this, we first compute a feature s importance value, for instance, the value of our corresponding predicted gate averaged over all samples. We then sort the vector of importance values, remove features individually starting from the most important one, and measure the clustering model s performance. If the model s performance decreases in a monotonic manner with the importance of the removed features, we will get a high correlation, indicating that the model s prediction is faithful to the learned feature importance values. An example of this metric is presented in Fig. 3.

Uniqueness Since we are interested in sample-level interpretation, we extend the diversity to a metric that compares samples instead of clusters. Specifically, we propose to measure the uniqueness of the selected features for similar samples, or in other words, how granular our explanations are. We define uniqueness of an explanation zi by: 1 |Ii| P ℓ Ii min zi zℓ 2

xi xℓ 2 , where Ii is the set of indices of the r nearest neighbors of xi, and zi, zℓare feature weights for samples xi, xℓ. The smaller this value is, the less samplespecific the interpretation of the model is. Therefore, we want our model to obtain high uniqueness values. In case the uniqueness is not a desired property and could raise confusion in the produces interpretations, the loss term Lgtcr in Eq. 3 should be removed.

Generalizability We want the interpretation of the prediction to generalize to other simple prediction models. Specifically, if the selected features also lead to high predictive capabilities across different models, this may indicate that the interpretability is not an artifact of a specific model architecture or training instance. To evaluate this quantity, termed generalizability, we follow (Yang et al., 2022) and measure the accuracy of SVM when applied to the data, masked by the most informative features as identified by each method.

6. Experiments

We performed six different types of experiments in our study. Firstly, we tested our method s clustering and local feature selection capabilities on a synthetic dataset. Then, we evaluated the interpretability of our model using MNIST. The main experiment focused on assessing our method s ability to cluster real-world tabular datasets, including those with small sample sizes, high-dimensional biomed datasets, and physics and text datasets. It s worth noting that the

real-world datasets we used are still considered challenging by several clustering studies that have used tabular data (Shaham et al., 2022; Xu et al., 2023).

In the fourth experiment, we demonstrated that our model can also perform well on image data without needing domain-specific augmentations. The datasets we used for our experiments have been summarized in Table E in the Appendix. In the fifth experiment, we showed that our local gates help the network learn a prediction function that includes high-frequency components. Finally, we conducted an ablation study to demonstrate the importance of each component in our model.

6.1. Evaluation

Interpretability Quality We use the metrics detailed in Section 5 to compare the interpretability of our method to the popular SHAP feature importance detection method (Lundberg & Lee, 2017) implemented here 1. In addition, we compare Gradient SHAP 2 and Integrated Gradients (Sundararajan et al., 2017), both trained on IDC clustering model predictions. Additionally, both Gradient SHAP and Integrated Gradients explainers are evaluated while being applied on the TELL (Peng et al., 2022) model predictions. For uniqueness, we set the number of neighbors to 2 (r = 2).

Clustering Accuracy We use three popular clustering evaluation metrics: Clustering Accuracy (ACC), The adjusted Rand index score (ARI), and Normalized Mutual Information (NMI).

6.2. Results

Synthetic Dataset We started by testing our model s performance with a synthetic dataset inspired by (Armanfard et al., 2015). The dataset includes three informative features, denoted as X[j] [ 1, 1], j = 1, .., 3, in which the samples are generated using Gaussian blobs 3. We added ten background features, resulting in a total of 13 features.

The samples are equally distributed between 4 clusters, with 800 samples in each cluster. Given the first two dimensions {X[1], X[2]}, only 3 clusters are separable, and the same property holds for dimensions pair {X[1], X[3]}. A visualization of these leading features appears in Figure 2. A good interpretable clustering model should correctly identify the four clusters while selecting the informative features for each cluster. For a more detailed explanation of how the dataset was generated, please refer to Appendix H.

We compute clustering accuracy (ACC) and F1-score to

1https://github.com/slundberg/shap 2https://captum.ai/api/gradient_shap.html 3https://scikit-learn.org/stable/modules/generated/ sklearn.datasets.make_blobs.html

Interpretable Deep Clustering for Tabular Data

Table 1. Evaluating the interpretability quality of our model on the MNIST10K data. Our IDC model improved clustering accuracy compared to baselines that do not use semantic-preserving image augmentations. We focus on the |S| = 15 most informative features as provided by each interpretation model. We compare the interpretability of our model, as predicted by the gates (IDC), to (i) the top features explained by SHAP trained based on a K-means model, (ii) Integrated Gradients, and (iii) Gradient SHAP applied as explainers to our model and TELL. Our local gating network is generalizable (88.5) by selecting faithful (0.96) and unique (0.69) features while providing comparable diversity values (94.8).

Method ACC |S| Uniqueness Diversity Faithfulness Generalizability K-means + SHAP 53.34 15 0.12 100.0 0.79 29.1 TELL + Integ Grads 74.79 15 0.03 89.1 0.67 75.0 TELL + Grad SHAP 74.79 15 0.15 92.5 0.63 78.9 IDC w/o gates + Integ Grads 82.32 15 0.02 95.8 0.78 80.3 IDC w/o gates + Grad SHAP 82.32 15 0.08 100.0 0.86 59.9 IDC + Integ Grads 83.45 15 0.01 95.3 0.94 63.7 IDC + Grad SHAP 83.45 15 0.02 97.0 0.93 66.0 IDC 83.45 15 0.69 94.8 0.96 88.5

assess the model s effectiveness, which measures the feature selection quality. Since we know which are the informative features for each sample, we can determine the precision and recall for gate-level feature selection. We compare our results with those of K-means with informative features identified by SHAP. Our method leads to 99.91% clustering accuracy, thus significantly outperforming K-means with 25.72%. Moreover, IDC yields an average F1-score of 88.95 (compared to 49.65 by K-means), demonstrating its ability to identify the correct informative features.

Interpretability Results on MNIST10K Our next interpretability evaluation involves MNIST10K images, with 1K images for each category. We train our model with and without (w/o) gates and evaluate its clustering accuracy. These are compared to TELL (Peng et al., 2022), an interpretable deep clustering model, and to K-means. As demonstrated in Table 1, our model with the gates leads to improved clustering capabilities. It is important to note that we do not use any domain-specific image augmentations and that existing schemes that do use will typically lead to higher clustering accuracies on this data.

In terms of interpretability, we compare the most informative features identified by each method based on the metrics described in Section 5. As indicated in Table 1, IDC with our gates (bottom row) improves faithfulness, uniqueness, and generalizability while maintaining competitive diversity. In the left panel of Fig. 3, we plot the model s accuracy against the feature importance. Providing a more detailed view on the faithfulness of the interpretation. Finally, in the right panel of this figure, the top 15 features selected by each model. It can be seen that IDC selects more informative features that are local for each sample.

Real Tabular Data We conducted a benchmark of our method using 11 real tabular datasets. Most of these datasets are from different biological domains, such as Tox-171, ALLAML, PROSTATE, SRBCT, BIASE, INTESTINE, and PBMC-2. These datasets usually have more features than

Figure 2. Visualization of the synthetic dataset. To separate between clusters, the model should select one of the pairs {X[1], X[2]} or {X[1], X[3]} of non-background features.

samples, making it challenging for clustering models to predict cluster assignments accurately. Additionally, we used image (MFEATZ), text (CNAE-9, ALBERT) and physical measurements (Mini MBoo NE) datasets, all of which were treated as tables.

Since most datasets are tabular and high dimensional, we compare our clustering capabilities to several unsupervised feature selection (UFS) models, which have been demonstrated to lead to state-of-the-art clustering results on these datasets (Lindenbaum et al., 2021). Here, we follow the evaluation protocol of (Lindenbaum et al., 2021) when using the UFS methods. Table 2 presents clustering accuracy of different methods: K-means on the full set of features (KM), TELL an interpretable deep clustering model (Peng et al., 2022), variational autoencoder based k-DVAE (Caciularu & Goldberger, 2023) and different leading UFS models followed by K-means. These include SRCFS (Huang et al., 2019), Concrete Autoencoders (CAE) (Abid et al., 2019), DUFS (Lindenbaum et al., 2021), and our model IDC. This table shows that our model leads to the best or second-best clustering accuracy across all datasets. We run every model 10 times for each dataset and report the mean accuracy with standard deviation.

Extension to Image Domain We conducted usability tests on our model using popular image datasets. Initially, we

Interpretable Deep Clustering for Tabular Data

Figure 3. Left: Faithfulness plot of the proposed method (green) and K-Means+SHAP (blue) on MNIST10K subset. Accuracy drop and feature importance are well correlated for our approach 0.96 (see green dots) while less correlated for SHAP features with K-means clustering 0.79 (see blue dots). Furthermore, notice that K-means accuracy only reaches 53% while our method 83%. Right: Features selected by different interpretability model using MNIST10K. The features are learned during clustering training as proposed by our approach (top), features selected by SHAP with K-means predictor (KM-SHAP), features obtained from Gradient-SHAP (Grad SHAP), and features selected by Integrated Gradients explainer (bottom).

Table 2. Clustering Accuracy on Real Datasets. We compare the proposed method against leading unsupervised feature selection methods followed by K-means clustering, the interpretable deep clustering model TELL (Peng et al., 2022) and recently published k-DVAE (Caciularu & Goldberger, 2023). Our model produces superior results on 9 of 11 datasets and second-rank accuracy on two datasets.

Method/Data TOX-171 ALLAML PROSTATE SRBCT BIASE INTESTINE PBMC-2 CNAE-9 MFEATZ Mini Boo NE ALBERT

KM 41.5 2 67.3 3 58.1 0 39.6 3 41.8 8 54.8 3 52.4 0 44.9 5 56.2 0 71.6 0 55.7 0 LS 47.5 1 73.2 0 58.6 0 41.1 3 83.8 0 43.2 3 57.2 0 35.9 3 56.6 3 71.9 0 64.2 0 MCFS 42.5 3 72.9 2 57.3 0 43.7 3 95.5 3 48.2 4 60.6 12 43.4 9 39.4 9 71.6 0 60.9 0 SRCFS 45.8 6 67.7 6 60.6 2 33.5 5 50.8 5 58.1 10 58.5 0 34.3 3 58.5 1 71.6 0 56.8 0 CAE 47.7 1 73.5 0 56.9 0 62.6 7 85.1 2 51.9 3 59.1 6 45.3 2 70.0 0 71.6 0 64.1 0 DUFS 49.1 3 74.5 1 64.7 0 51.7 1 100 0 71.9 7 57.6 9 46.3 0 57.3 9 71.6 0 62.3 0 TELL 28.7 3 66.7 14 63.6 0 38.9 8 84.6 2 52.1 28 52.1 1 11.1 0 63.8 36 75.6 5 60.6 3 k-DVAE 49.1 5 72.6 5 59.5 2 49.4 12 39.3 6 55.7 7 50.1 0 62.8 7 66.5 5 71.7 0 51.5 1 IDC 50.6 3 77.5 6 65.3 3 55.4 5 95.7 1 74.2 2 65.1 5 66.0 7 86.81 4 77.0 4 64.2 4

assessed the performance of IDC both with and without the gates on MNIST and Fashion MNIST. The results, as presented in Table 3, indicate that the gating network improves the model s ability to cluster data accurately and select only a few relevant features, thereby enhancing interpretability.

In our second experiment, we utilized CIFAR10 and compared our model to other deep clustering models that do not utilize image augmentations. These models include TELL (Peng et al., 2022), Va DE (Jiang et al., 2016), and DEC (Xie et al., 2016). The results of the experiment are presented in Table 4. Our model produced competitive results while only selecting approximately 20% of the input features. It is worth noting that while these results are not state-of-the-art in deep image clustering, they were achieved without using strong image augmentations like resizing, translation, or rotations. Such augmentations are incompatible with our gating procedure, as the informative features may vary from one augmentation to the next. Additionally, these augmentations do not apply to tabular data..

6.3. Sparsity and Inductive Frequency Bias

According to a study by (Beyazit et al., 2023), tabular datasets generally require higher frequency target functions than images. However, several authors have shown that neural networks tend to learn low-frequency functions faster than higher ones. This might explain why tree-based models often outperform neural networks. For more information, please refer to the appendix section on inductive learning (Section I).

In this section, we analyze whether our local gates, which sparsify the weights of our learned prediction function, can provide an inductive bias that helps the network learn highfrequency functions. To test this, we trained our model on the tabular ALLAML data and evaluated the absolute value of Fourier amplitudes of predictions of our model |NUDFT(f IDC)| at different frequencies |k|. In Figure 4, we compare these amplitudes to those learned without the gates (IDCw/o_gates) and to those learned by the interpretable

Interpretable Deep Clustering for Tabular Data

Table 3. Clustering evaluation and the number of features selected (last column). IDC with the proposed gates improves clustering accuracy by using only 16 from MNIST60K and 69 features from Fashion MNIST datasets.

Dataset Method ACC ARI NMI |S| MNIST60K IDC (w/o gates) 81.1 75.9 80.3 784 IDC 87.9 82.8 85.1 15.81 Fashion MNIST IDC (w/o gates) 61.0 49.3 62.7 784 IDC 61.9 49.1 63.3 68.6

Table 4. Clustering performance on CIFAR10 dataset. IDC model selects 586 features (on average) out of 3,072.

Model ACC ARI NMI TELL 25.65 5.96 10.41 Va DE 20.87 3.95 7.20 DEC 18.09 2.47 4.56 IDC 25.01 6.16 11.96

Table 5. Ablation study on MNIST60K dataset Model ACC ARI NMI IDC 87.9 82.8 85.1 IDC w/o Lreg 85.9 (-2.0) 81.2(-1.6) 84.7 (-0.4) IDC w/o latent denoising 86.5 (-1.4) 80.9 (-1.9) 83.2 (-1.9) IDC w/o input denoising 84.3 (-3.6) 80.0 (-2.8) 83.9 (-1.2) IDC features + K-Means 65.5 (-22.4) 49.3 (-33.5) 57.6 (-27.5) IDC w/o Lrecon 18.0 (-69.9) 2.6 (-80.2) 4.3 (-80.8)

Figure 4. Spectral properties of the learned predictive function using ALLAML dataset. The model trained with the gating network (IDC) has higher Fourier amplitudes at all frequency levels than without gates (IDCw/o_gates) the baseline (TELL). This suggests that IDC can better handle the inductive bias of tabular data.

deep clustering model TELL (Peng et al., 2022). This figure demonstrates that the sparsification helps the model learn high-frequency components. The bias induced by the gating network makes our model more accurate on tabular data while being interpretable.

6.4. Ablation Study

We conducted an ablation study to determine if all the components of the loss are necessary for the model to converge.

The evaluation was carried out on the MNIST60K dataset, and we ran each experiment 10 times. The results are presented in Table 4. We train our model without regularization term in Eq.2 (row 2), without latent or input augmentations (rows 3-4), then we replace the clustering head with K-Means (row 5) and finally train without reconstruction term Lrecon. Our findings indicate that all the proposed components significantly contribute to the performance of the model.

7. Conclusions

We present a deep clustering model that accurately assigns clusters on tabular data and predicts informative features at both the sample and cluster levels, providing interpretability. We tested our model on 15 datasets including synthetic datasets, high-dimensional tabular datasets, and several image datasets (treated as tables). A main limitation of our scheme is dealing with correlated variables. This is a known caveat of the reconstruction loss (Abid et al., 2019). One way to alleviate this limitation is to incorporate a group sparsity loss as presented by (Imrie et al., 2022). Our method also struggles to handle datasets with a large number of clusters. This problem could be mitigated using weak supervision or by introducing several clustering heads to prevent collapse to a subset of clusters. We hope our work will be beneficial to scientists in the biomedical field.

Impact Statement

The proposed framework for deep learning clustering in general domain tabular data has wide applications in scientific research, healthcare, and biomedicine. It addresses a critical need in data analysis across diverse domains by delivering reliable and interpretable cluster assignments. Biologists can use the framework to analyze genome sequences, medical records, and images with precision. Our hope is that it will increase trust in deep learning-based clustering models for multidisciplinary datasets, such as biological, text, image, and physics tabular datasets. Nonetheless, the framework can suffer from fairness biases at a societal level since it is fully unsupervised and could reflect data biases. We note that this concern is not different from any other unsupervised model, and we leave this as an open issue for future work.

Acknowledgements

We thank the anonymous reviewers for constructive comments during the reviewing process. In addition, we would like to express our deepest gratitude to Uri Shaham (Bar Ilan University) for insightful and constructive discussions significantly enriched this research.

Interpretable Deep Clustering for Tabular Data

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Interpretable Deep Clustering for Tabular Data

A. Implementation details

We implement our model in Pytorch and run experiments on Nvidia A100 GPU server with Intel(R) Xeon(R) Gold 6338 CPU @ 2.00GHz.

B. Model Architecture

The model is trained with a single hidden layer in the Clustering Head and the Gating NN. We use up to 4 hidden layers for the Encoder, and The Decoder is a mirrored version of the encoder. The dimensions of hidden layers are detailed in the supplementary included code and will be released to Git Hub.

C. Training Setup

We train all models with a two-stage approach - we train Encoder, Decoder, and Gating NN in the first stage and then train Clustering Head in the second stage. For interpretabiltiy experiments, we train K-means 4 and TELL (Peng et al., 2022) 5. Both methods are trained without additional augmentations for fair comparison to our method with the provided default experimental parameters. For small-sample datasets we increase the number of epochs in TELL model up to 10K for fair comparison. In method k-DVAE 6 (Caciularu & Goldberger, 2023) is trained with default parameters provided in the shared code with only incresed number of epochs to 200 (initial step) and to 100 (other steps) for small-sample datasets.

Table 6. The number of epochs and batch size for different datasets.

Dataset Epochs Stage 1 Epochs Stage 2 Batch size Synthetic 50 2000 800 MNIST60K 300 600 256 MNIST10K 300 700 100 Fashion MNIST 100 500 256 TOX-171 1000 1000 16 ALLAML 1000 1000 36 PROSTATE 1000 1000 102 SRBCT 2000 1000 83 BIASE 10000 1000 56 INTESTINE 5000 1000 238 PBMC-2 100 100 256 CNAE-9 1000 1000 500 MFEATZ 1000 1000 500 Mini Boo NE 20 30 512 ALBERT 10 40 1024 CIFAR-10 600 700 256 D. Regularization Term

Figure 5. The value of µEZ||Z||0|µ=0.5 = 1

2πσ2 e 1 8σ2 for σ = [0.001, 2].

The leading term in our regularizer is expressed by :

4https://scikit-learn.org/stable/modules/generated/sklearn.cluster.KMeans.html 5The implementation was found here: https://github.com/XLearning-SCU/2022-JMLR-TELL/tree/main, accessed on 2023-09-28. 6https://github.com/aviclu/k-DVAE

Interpretable Deep Clustering for Tabular Data

d=1 P[zd > 0] =

d=1 P[µd + σϵd + 0.5 > 0]

d=1 {1 P[µd + σϵd + 0.5 0]}

d=1 {1 Φ( µd 0.5

d=1 Φ µd + 0.5

2 erf µd + 0.5

To tune σ, we follow the suggestion in (Yamada et al., 2020). Specifically, the effect of σ can be understood by looking at the value of µd EZ||Z||0. In the first training step, µd is 0. Therefore, at initial training phase, µd EZ||Z||0 is close

2πσ2 d e 1 8σ2 d . To enable sparsification, this term (multiplied by the regularization parameter λ) has to be greater than

the derivative of the loss with respect to µd because otherwise µd is updated in the incorrect direction. To encourage such behavior, we tune σ to the value that maximizes the gradient of the regularization term. As demonstrated in Fig. 5 this is obtained when σ = 0.5. Therefore, we keep σ = 0.5 throughout our experiments unless specifically noted.

E. Datasets Properties and References

In Table E we add the references of the datasets used in the experiments. We provide here a short description for each dataset:

MNIST10K and MNIST60K are the subsets of MNIST (lec) dataset, the first one of size 10,000 samples and the second one is the full trainset. Additionally, we use the MNIST test set split for evaluations of the model on unseen data. The images include hand-written digits from 10 categories. Fashion MNIST60K is a train set of (git). The images include 10 categories of clothers. TOX-171 (Piloto & Schilling, 2010) dataset is an example of the use of toxicology to integrate diverse biological data, such as clinical chemistry, expression, and other types of data. The database contains the profiles resulting from the three toxicants: alpha-naphthyl-isothiocyanate, dimethylnitrosamine, and N-methylformamide administered to rats. The classification task is to identify whether the samples are toxic, non toxic or control. ALLAML dataset (Golub et al., 1999) consists of gene expression profiles of two acute cases of leukemia: acute lymphoblastic leukemia (ALL) and acute myeloblastic leukemia (AML). The ALL part of the dataset comes from two types, B-cell and T-cell, while the AML part is split into two types, bone marrow samples and peripheral blood samples. It contains in total 72 samples in 2 classes, ALL and AML, which have 47 and 25 samples, respectively. Every sample contains 7,129 gene expression values. PROSTATE dataset (Singh et al., 2002) has in total 102 samples in two classes tumor and normal, which have 52 and 50 samples, respectively. After preprocessing described in (Nie et al., 2010), a data set contains 102 samples and 5966 genes. SRBCT dataset (Kar et al., 2015) consists of four classes of cancers in 83 samples. These four classes were ewing sarcoma (EWS), non-Hodgkin lymphoma (NHL), neuroblastoma (NB), and rhabdomyosarcoma (RMS). BIASE dataset (Biase et al., 2014) is bimodal m RNA expressions to embryonic genome activation and it contains 56 samples each one of dimension 25,683. INTESTINE (Sato et al., 2009) is a mouse intestine dataset with 238 samples. PBMC-2 dataset is a binary-class subset of the original PBMC (Zheng et al., 2017) dataset. We select two categories that have the most number of samples in the original set. In addition, we remove all zero columns from the data resulting in 17,126 featurees 20,742 samples size. CIFAR10 dataset (Krizhevsky et al., 2009) contains 60,000 small images of size 32 32 with 3 color channels. In total there are 3,072 features.

Interpretable Deep Clustering for Tabular Data

cnae-9 7 (Bischl et al., 2017; Ciarelli & Oliveira, 2009) contains 1080 documents of free text business descriptions of Brazilian companies categorized into a subset of 9 categories. The original texts were preprocessed to obtain the current data set: initially, it was kept only letters and then it was removed prepositions of the texts. Next, the words were transformed to their canonical form. Finally, each document was represented as a vector, where the weight of each word is its frequency in the document. This data set is highly sparse (99.22% of the matrix is filled with zeros). MFEATZ is known as mfeat-zernike 8 (van Breukelen et al., 1998; Bischl et al., 2017) dataset describes features of handwritten numerals (0 - 9) extracted from a collection of Dutch utility maps. Corresponding patterns in different datasets correspond to the same original character. 200 instances per class (for a total of 2,000 instances) have been digitized in binary images. These digits are represented in terms of 47 Zernike moments. Mini Boo NE is a physical dataset for particle identification task (Roe et al., 2005). ALBERT is a text dataset from Auto ML challenge (Guyon et al., 2019). Since the dataset comes with mnostly categorical variables, we follow the work (Beyazit et al., 2023) and preprocess the dataset with target-encoder9.

Table 7. Properties and references for the dataset used in the experiments. Dataset Features Samples Clusters Reference MNIST10K 784 10,000 10 (lec) MNIST60K 784 60,000 10 (lec) Fashion MNIST60K 784 60,000 10 (git) TOX-171 5,748 171 4 (jun) ALLAML 7,192 72 2 (jun) PROSTATE 5,966 102 2 (jun) SRBCT 2,308 83 4 (Khan et al., 2001) BIASE 25,683 56 4 (Biase et al., 2014) INTESTINE 3,775 238 13 (Sato et al., 2009) PBMC-2 17,126 20,742 2 (Zheng et al., 2017) CIFAR10 3,072 60,000 10 (Krizhevsky et al., 2009) CNAE-9 857 1080 9 (Ciarelli & Oliveira, 2009; Bischl et al., 2017) MFEATZ 48 2000 2 (van Breukelen et al., 1998; Bischl et al., 2017) Mini Boo NE 50 130064 2 (Roe et al., 2005; Bischl et al., 2017) ALBERT 78 425240 2 (Guyon et al., 2019; Bischl et al., 2017)

F. Train Loss Augmentations

In addition to the loss presented in Section 4.2 we exploit the next dataset-agnostic augmentations during model training. The first one is the standard reconstruction loss that is calculated between input samples and reconstructed samples. Input denoising is based on (Vincent et al., 2008) and latent denoising on (Doi et al., 2007):

Clean reconstruction loss, ||fθD fθE(xi) xi||1, which measures the deviation of estimated ˆxi from the input sample xi. Denoising reconstruction loss (Vincent et al., 2008), ||fθD fθE(xi mrand) xi||1, where mrand {0, 1}D is a random binary mask generated for each sample xi. We generate a mask such that about mrand% of the input features are multiplied by zero value, which indicates that the gate is closed. The loss pushes the method to pay less attention to unnecessary features for the reconstruction. Latent denoising reconstruction loss, ||fθD(hi hnoise) xi||1, where hnoise N(1, σh) is a noise generated from a normal distribution with mean one and scale σh which is a dataset-specific hyperparameter (Doi et al., 2007). This term aims to improve latent embedding representation by small perturbation augmentation to treat small sample-size datasets.

G. Model Training Scalability

In Figure 9 we show the training time as a function of number of data samples. It could be seen that training time scales linearly with an increase in dataset length. In addition, in Table 8 we present the total training time (in seconds) measured for the baselines and our method. The experiments were done on a single A100 GPU. The measurements were done for different dataset (MNIST) sizes and averaged over 5 trials with different random seeds (700 epochs, batch size 128, time in seconds).

7https://www.openml.org/search?type=data&sort=runs&id=1468. Accessed on 2024-01-23 8https://www.openml.org/search?type=data&status=active&id=22. Accessed on 2024-01-23 9https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.Target Encoder.html

Interpretable Deep Clustering for Tabular Data

Table 8. Training time in seconds for different MNIST subset sizes.

MNIST subset IDC TELL k-DVAE 100 181.4 1.8 487.1 1000 401.3 69.3 1647.9 10000 2016.2 735.5 9705.6

Table 9. Training time in seconds measured for different numbers of samples. H. Synthetic Dataset Generation

The dataset consists of three informative features xi[j] [ 1, 1], j = 1, .., 3 for each sample xi and is generated as isotropic Gaussian blobs 10 with standard deviation of each cluster of 0.5. The detailed description of the dataset generation could be found in H. Then we add ten nuisance background features with values drawn from N(0, σ2 n) (with σn = 0.1) resulting in 13 total features. The samples are equally distributed between 4 clusters, with 800 samples in each cluster. Given the first two dimensions {x[1], x[2]}, only 3 clusters are separable, and the same property holds for dimensions pair {x[1], x[3]}.

I. Indactive Bias Analysis

We present the original plot from the paper (Beyazit et al., 2023) that emphasizes the difference in target labels Fourier amplitudes distribution Py across different frequencies. The authors claim that distributiuon of Foueirer amplitueds obtaned on ground truth targets on tabular datasets has higher values than those of image datasets. We support this claim by testing it from the learned function - does our model learns a bias for tabular domain by using gating network. To produce the anaylsis plot we predict ˆyj i = argmaxj [1,...,K](ˆpi) where ˆpi = fθC(fθE(f G(xi))) and is calculated for each sample xi and for each feature value xd i xi. Then we calculate the non uniform discrete fourier transform (NUDFT) for 1000 frequencies values in range [0.1, 20] which accepts a vector of N values of xd i , i = 1, ...N and corresponding binary prediction values yi. NUDFT is calculated by the pytorch code:

def spectrum_NUDFT ( x , y , kmax=20 , nk =1000): kvals = np . l i n s p a c e ( 0 . 1 , kmax , nk +1) n u f f t = (1 / len ( x ) ) * n f f t _ a d j o i n t ( ( x * kmax / nk ) , y , 2 * ( nk + 1 ) ) [ nk + 1 : ] return [ kvals , np . a r r a y ( n u f f t , dtype=" complex_ " ) ]

where nfft_adjoint function is part of the nfft package 11. Then we take the absolute value of NUDFT amplitudes. By repeating the process for each feature d we obtain D vectors with |NUDFT| values and we plot them with 1000 frequency steps |k| in logarithmic scale axes. In this way we obtain the plot in Figure 4.

J. The number of selected features

Table 10. Selected and total features for various datasets Dataset TOX-171 ALLAML PROSTATE SRBCT BIASE INTESTINE PBMC-2 CNAE-9 MFEATZ ALBERT Mini Boon E Selected 49.6 527.6 170.9 46.7 210 65 137.8 360.5 26.2 59.4 29.4 Total 5748 7192 5966 2308 25683 3775 17126 856 47 78 50

10https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_blobs.html 11https://github.com/jakevdp/nfft

Interpretable Deep Clustering for Tabular Data

Figure 6. (Beyazit et al., 2023): Due to their heterogeneous nature, tabular datasets tend to describe higher frequency target functions compared to images. The spectra corresponding to image datasets (curves in color) tend to feature lower Fourier amplitudes at higher frequencies than hetergoneous tabular datasets (cyan region). K. Sensitivity analysis of hyper parameters

We present the sensitivity analysis of the clustering as a function of hyperparameter changes. The analysis is done on MNIST subset (10K samples), we present the mean values obtained from 5 trials for each parameter change. Please, note that in same cases we get even better results than were presented in the main part of this paper:

Table 11. Sensitivity to ϵgtcr values ϵgtcr ACC ARI NMI # open gates 100 0.8318 0.7533 0.7856 16 10 0.7732 0.6897 0.7386 17 1 0.834 0.7479 0.7807 18 0.1 0.8067 0.7251 0.7641 18 0.01 0.8081 0.7256 0.7667 18

Table 12. Sensitivity to σh values σh ACC ARI NMI # open gates 0.01 0.8104 0.7249 0.7683 18 0.1 0.8183 0.7377 0.7736 18 0.2 0.8531 0.7646 0.7841 18 0.5 0.6158 0.4707 0.5661 18 1 0.7336 0.6115 0.6745 18

Table 13. Sensitivity to ϵhead values ϵhead ACC ARI NMI # open gates 100 0.6645 0.5339 0.6294 18 10 0.6025 0.503 0.6206 18 1 0.7941 0.6842 0.7298 18 0.1 0.834 0.7479 0.7807 18 0.01 0.708 0.6123 0.7122 18

Table 14. Sensitivity to λreg values λreg ACC ARI NMI # open gates 0 0.8643 0.7803 0.8021 435 0.1 0.8227 0.7471 0.7804 111 1 0.834 0.7479 0.7807 18 10 0.7795 0.6921 0.7421 4 100 0.1374 0.0028 0.053 0

Table 15. Sensitivity to mrand values

mrand ACC ARI NMI # open gates 0% 0.7969 0.7131 0.7585 21 10% 0.8381 0.7539 0.7864 20 30% 0.8812 0.7915 0.8049 19 50% 0.8334 0.7491 0.7844 19 70% 0.8188 0.7464 0.7864 18 90% 0.8104 0.7249 0.7683 18