Clustering approaches that utilize convex loss functions have recently attracted growing interest in the formation of compact data clusters. Although classical methods like kmeans and its wide family of variants are still widely used, all of them require the number of clusters (k) to be supplied as input, and many are notably sensitive to initialization. Convex clustering provides a more stable alternative by formulating the clustering task as a convex optimization problem, ensuring a unique global solution. However, it faces challenges in handling high dimensional data, especially in the presence of noise and outliers. Additionally, strong fusion regularization, controlled by the tuning parameter, can hinder effective cluster formation within a convex clustering framework. To overcome these challenges, we introduce a robust approach that integrates convex clustering with the Median of Means (MoM) estimator, thus developing an outlier resistant and efficient clustering framework that does not necessitate a prior knowledge of the number of clusters. By leveraging the robustness of MoM alongside the stability of convex clustering, our method enhances both performance and efficiency, especially on large scale datasets. Theoretical analysis demonstrates weak consistency under specific conditions, while experiments on synthetic and real world datasets validate the method’s superior performance compared to existing approaches.

The field of unpaired image-to-image translation has undergone a significant transformation with the introduction of Generative Adversarial Networks (GANs), with CycleGAN and DiscoGAN as prominent variants. While these models show impressive empirical performance, their statistical properties are under-studied. In this paper, we propose a framework for analyzing the generalization error in cross-domain deep generative models. Our findings reveal that when provided with independent and identically distributed (i.i.d.) samples from two domains, the translation error, measured under the Wasserstein-1 loss, scales as $\tilde{\mathcal{O}} \left(\min(n, m)^{-1/\max(d,\tilde{d})}\right)$, provided that the true model possesses sufficient smoothness and the network sizes are chosen appropriately. Here, $n$ and $m$ represent the sizes of the sample sets, while $d$ and $\tilde{d}$ denote the dimensions of the respective data domains. Furthermore, we highlight the importance of a cycle loss term for ensuring distributional cycle consistency. Additionally, we provide insights into the relationship between the network size and the number of data points. Notably, as the true model exhibits greater smoothness, it suffices to work with smaller networks.

Despite the remarkable empirical successes of Generative Adversarial Networks (GANs), the theoretical guarantees for their statistical accuracy remain rather pessimistic. In particular, the data distributions on which GANs are applied, such as natural images, are often hypothesized to have an intrinsic low-dimensional structure in a typically high-dimensional feature space, but this is often not reflected in the derived rates in the state-of-the-art analyses. In this paper, we attempt to bridge the gap between the theory and practice of GANs and their bidirectional variant, Bi-directional GANs (BiGANs), by deriving statistical guarantees on the estimated densities in terms of the intrinsic dimension of the data and the latent space. We analytically show that if one has access to $n$ samples from the unknown target distribution and the network architectures are properly chosen, the expected Wasserstein-1 distance of the estimates from the target scales as $O\left( n^{-1/d_\mu } \right)$ for GANs and $\tilde{O}\left( n^{-1/(d_\mu+\ell)} \right)$ for BiGANs, where $d_\mu$ and $\ell$ are the upper Wasserstein-1 dimension of the data-distribution and latent-space dimension, respectively. The theoretical analyses not only suggest that these methods successfully avoid the curse of dimensionality, in the sense that the exponent of $n$ in the error rates does not depend on the data dimension but also serve to bridge the gap between the theoretical analyses of GANs and the known sharp rates from optimal transport literature. Additionally, we demonstrate that GANs can effectively achieve the minimax optimal rate even for non-smooth underlying distributions, with the use of interpolating generator networks.

We developed an automated photoacoustic and ultrasound breast tomography system that images the patient in the standing pose. The system, named OneTouch-PAT, utilized linear transducer arrays with optical-acoustic combiners for effective dual-modal imaging. During scanning, subjects only need to gently attach their breasts to the imaging window, and co-registered three-dimensional ultrasonic and photoacoustic images of the breast can be obtained within one minute. Our system has a large field of view of 17 cm by 15 cm and achieves an imaging depth of 3 cm with sub-millimeter resolution. A three-dimensional deep-learning network was also developed to further improve the image quality by improving the 3D resolution, enhancing vasculature, eliminating skin signals, and reducing noise. The performance of the system was tested on four healthy subjects and 61 patients with breast cancer. Our results indicate that the ultrasound structural information can be combined with the photoacoustic vascular information for better tissue characterization. Representative cases from different molecular subtypes have indicated different photoacoustic and ultrasound features that could potentially be used for imaging-based cancer classification. Statistical analysis among all patients indicates that the regional photoacoustic intensity and vessel branching points are indicators of breast malignancy. These promising results suggest that our system could significantly enhance breast cancer diagnosis and classification.

Variational Autoencoders (VAEs) have gained significant popularity among researchers as a powerful tool for understanding unknown distributions based on limited samples. This popularity stems partly from their impressive performance and partly from their ability to provide meaningful feature representations in the latent space. Wasserstein Autoencoders (WAEs), a variant of VAEs, aim to not only improve model efficiency but also interpretability. However, there has been limited focus on analyzing their statistical guarantees. The matter is further complicated by the fact that the data distributions to which WAEs are applied - such as natural images - are often presumed to possess an underlying low-dimensional structure within a high-dimensional feature space, which current theory does not adequately account for, rendering known bounds inefficient. To bridge the gap between the theory and practice of WAEs, in this paper, we show that WAEs can learn the data distributions when the network architectures are properly chosen. We show that the convergence rates of the expected excess risk in the number of samples for WAEs are independent of the high feature dimension, instead relying only on the intrinsic dimension of the data distribution.

Principal component analysis (PCA) is a fundamental tool for data visualization, denoising, and dimensionality reduction. It is widely popular in statistics, machine learning, computer vision, and related fields. However, PCA is well-known to fall prey to outliers and often fails to detect the true underlying low-dimensional structure within the dataset. Following the Median of Means (MoM) philosophy, recent supervised learning methods have shown great success in dealing with outlying observations without much compromise to their large sample theoretical properties. This article proposes a PCA procedure based on the MoM principle. Called the MoMPCA, the proposed method is not only computationally appealing but also achieves optimal convergence rates under minimal assumptions. In particular, we explore the nonasymptotic error bounds of the obtained solution via the aid of the Rademacher complexities while granting absolutely no assumption on the outlying observations. The derived concentration results are not dependent on the dimension because the analysis is conducted in a separable Hilbert space, and the results only depend on the fourth moment of the underlying distribution in the corresponding norm. The proposal's efficacy is also thoroughly showcased through simulations and real data applications.

Kernel k-means clustering is a powerful tool for unsupervised learning of non-linearly separable data. Its merits are thoroughly validated on a suite of simulated datasets and real data benchmarks that feature nonlinear and multi-view separation. Since the earliest attempts, researchers have noted that such algorithms often become trapped by local minima arising from the non-convexity of the underlying objective function. In this paper, we generalize recent results leveraging a general family of means to combat sub-optimal local solutions to the kernel and multi-kernel settings. Called Kernel Power k-Means, our algorithm uses majorization-minimization (MM) to better solve this non-convex problem. We show that the method implicitly performs annealing in kernel feature space while retaining efficient, closed-form updates. We rigorously characterize its convergence properties both from computational and statistical points of view. In particular, we characterize the large sample behavior of the proposed method by establishing strong consistency guarantees as well as finite-sample bounds on the excess risk of the estimates through modern tools in learning theory. The proposal's efficacy is demonstrated through an array of simulated and real data experiments.

Clusters in real data are often restricted to low-dimensional subspaces rather than the entire feature space. Recent approaches to circumvent this difficulty are often computationally inefficient and lack theoretical justification in terms of their large-sample behavior. This article deals with the problem by introducing an entropy incentive term to efficiently learn the feature importance within the framework of center-based clustering. A scalable block-coordinate descent algorithm, with closed-form updates, is incorporated to minimize the proposed objective function. We establish theoretical guarantees on our method by Vapnik-Chervonenkis (VC) theory to establish strong consistency along with uniform concentration bounds. The merits of our method are showcased through detailed experimental analysis on toy examples as well as real data clustering benchmarks.

Recent progress in center-based clustering algorithms combats poor local minima by implicit annealing through a family of generalized means. These methods are variations of Lloyd’s celebrated k-means algorithm, and are most appropriate for spherical clusters such as those arising from Gaussian data. In this paper, we bridge these new algorithmic advances to classical work on hard clustering under Bregman divergences, which enjoy a bijection to exponential family distributions and are thus well-suited for clustering objects arising from a breadth of data generating mechanisms. The elegant properties of Bregman divergences allow us to maintain closed form updates in a simple and transparent algorithm, and moreover lead to new theoretical arguments for establishing finite sample bounds that relax the bounded support assumption made in the existing state of the art. Additionally, we consider thorough empirical analyses on simulated experiments and a case study on rainfall data, finding that the proposed method outperforms existing peer methods in a variety of non-Gaussian data settings.

In context to high-dimensional clustering, the concept offeature weightinghas gained considerable importance over the years to capture the relative degrees of importance of different features in revealing the cluster structure of the dataset. However, the popular techniques in this area either fail to perform feature selection or do not preserve the simplicity of Lloyd’s heuristic to solve the$k$-means problem and the like. In this paper, we propose a Lasso Weighted$k$-means ($LW$-$k$-means) algorithm, as a simple yet efficient sparse clustering procedure for high-dimensional data where the number of features ($p$) can be much higher than the number of observations ($n$). The$LW$-$k$-means method imposes an$\ell _1$regularization term involving the feature weights directly to induce feature selection in a sparse clustering framework. We develop a simple block-coordinate descent type algorithm with time-complexity resembling that of Lloyd’s method, to optimize the proposed objective. In addition, we establish the strong consistency of the$LW$-$k$-means procedure. Such an analysis of the large sample properties is not available for the conventional sparse$k$-means algorithms, in general.$LW$-$k$-means is tested on a number of synthetic and real-life datasets and through a detailed experimental analysis, we find that the performance of the method is highly competitive against the baselines as well as the state-of-the-art procedures for center-based high-dimensional clustering, not only in terms of clustering accuracy but also with respect to computational time.

Mean shift is a simple interactive procedure that gradually shifts data points towards the mode which denotes the highest density of data points in the region. Mean shift algorithms have been effectively used for data denoising, mode seeking, and finding the number of clusters in a dataset in an automated fashion. However, the merits of mean shift quickly fade away as the data dimensions increase and only a handful of features contain useful information about the cluster structure of the data. We propose a simple yet elegant feature-weighted variant of mean shift to efficiently learn the feature importance and thus, extending the merits of mean shift to high-dimensional data. The resulting algorithm not only outperforms the conventional mean shift clustering procedure but also preserves its computational simplicity. In addition, the proposed method comes with rigorous theoretical convergence guarantees and a convergence rate of at least a cubic order. The efficacy of our proposal is thoroughly assessed through experimental comparison against baseline and state-of-the-art clustering methods on synthetic as well as real-world datasets.

The concept of Entropy plays a key role in Information Theory, Statistics, and Machine Learning. This paper introduces a new entropy measure, called the t-entropy, which exploits the concavity of the inverse-tan function. We analytically show that the proposed$t$-entropy satisfies the prominent axiomatic properties of an entropy measure. We demonstrate an application of the proposed entropy measure for multi-level thresholding of images. We also propose the entropic-loss as a measure of the divergence between two probability distributions, which leads to robust estimators in the context of parametric statistical inference. The consistency and asymptotic breakdown point of the proposed estimator are mathematically analysed. Finally, we also show an application of the$t$-entropy to feature weighted data clustering.

Recent advances in center-based clustering continue to improve upon the drawbacks of Lloyd's celebrated $k$-means algorithm over $60$ years after its introduction. Various methods seek to address poor local minima, sensitivity to outliers, and data that are not well-suited to Euclidean measures of fit, but many are supported largely empirically. Moreover, combining such approaches in a piecemeal manner can result in ad hoc methods, and the limited theoretical results supporting each individual contribution may no longer hold. Toward addressing these issues in a principled way, this paper proposes a cohesive robust framework for center-based clustering under a general class of dissimilarity measures. In particular, we present a rigorous theoretical treatment within a Median-of-Means (MoM) estimation framework, showing that it subsumes several popular $k$-means variants. In addition to unifying existing methods, we derive uniform concentration bounds that complete their analyses, and bridge these results to the MoM framework via Dudley's chaining arguments. Importantly, we neither require any assumptions on the distribution of the outlying observations nor on the relative number of observations $n$ to features $p$. We establish strong consistency and an error rate of $O(n^{-1/2})$ under mild conditions, surpassing the best-known results in the literature. The methods are empirically validated thoroughly on real and synthetic datasets.

Despite its well-known shortcomings, k-means remains one of the most widely used approaches to data clustering. Current research continues to tackle its flaws while attempting to preserve its simplicity. Recently, the power k-means algorithm was proposed to avoid poor local minima by annealing through a family of smoother surfaces. However, the approach lacks statistical guarantees and fails in high dimensions when many features are irrelevant. This paper addresses these issues by introducing entropy regularization to learn feature relevance while annealing. We prove consistency of the proposed approach and derive a scalable majorization-minimization algorithm that enjoys closed-form updates and convergence guarantees. In particular, our method retains the same computational complexity of k-means and power k-means, but yields significant improvements over both. Its merits are thoroughly assessed on a suite of real and synthetic data.

Large amounts of data generated everyday by different organizations for various purposes have catalyzed research opportunities related to data science. Publishing raw data may raise security concerns among the users or actors who have provided some sensitive information in the raw data. Over the ye ars, it has been observed that attackers can very easily exploit the sensitive information as well as the identity of the users from the raw data. Thus, to protect the identity of the users in the anonymized data, the notion of k-anonymity and its improved version l-diversity have been proposed. Several algorithms have been developed to achieve k-anonymity as well as l-diversity for both relational micro-data and social network data. In this paper, we propose an approach based on the eigenvector centrality value of the individual nodes to achieve k$-anonymity as well as $l$-diversity by adding noise nodes in the raw data. In the process of adding noise nodes, we focused on adding noise nodes in such an intelligent manner so that they have very little influence on the anonymized data. Our proposed algorithm also ensures minimal changes in the social influence of the nodes in the anonymized data which are already present in the raw data. Through various measures and experiments, we establish the effectiveness of our proposed algorithm over the several existing social network anonymization techniques.

We propose an (α, k) anonymity model based on the eigenvector centrality value of the nodes present in the raw graph and further extend it to propose (α, l) diversity model and recursive (α, c, l) diversity model which can handle the protection of the sensitive attributes associated with a particular actor. For anonymization purpose, we applied noise node addition technique to generate the anonymized graphs so that the structural property of the raw graph is preserved. Our proposed methods add noise nodes with very minimal social importance. We applied eigenvector centrality concept over traditional degree centrality concept to prevent mixing of highly influential nodes with less influential nodes in the equivalence groups