Swagatam Das

Yanjie Song, Yutong Wu, Yangyang Guo et al.

Evolutionary algorithms (EA), a class of stochastic search methods based on the principles of natural evolution, have received widespread acclaim for their exceptional performance in various real-world optimization problems. While researchers worldwide have proposed a wide variety of EAs, certain limitations remain, such as slow convergence speed and poor generalization capabilities. Consequently, numerous scholars actively explore improvements to algorithmic structures, operators, search patterns, etc., to enhance their optimization performance. Reinforcement learning (RL) integrated as a component in the EA framework has demonstrated superior performance in recent years. This paper presents a comprehensive survey on integrating reinforcement learning into the evolutionary algorithm, referred to as reinforcement learning-assisted evolutionary algorithm (RL-EA). We begin with the conceptual outlines of reinforcement learning and the evolutionary algorithm. We then provide a taxonomy of RL-EA. Subsequently, we discuss the RL-EA integration method, the RL-assisted strategy adopted by RL-EA, and its applications according to the existing literature. The RL-assisted procedure is divided according to the implemented functions including solution generation, learnable objective function, algorithm/operator/sub-population selection, parameter adaptation, and other strategies. Additionally, different attribute settings of RL in RL-EA are discussed. In the applications of RL-EA section, we also demonstrate the excellent performance of RL-EA on several benchmarks and a range of public datasets to facilitate a quick comparative study. Finally, we analyze potential directions for future research.

Abhishek Kumar, Oladayo S. Ajani, Swagatam Das et al.

In machine learning and computer vision, mean shift (MS) qualifies as one of the most popular mode-seeking algorithms used for clustering and image segmentation. It iteratively moves each data point to the weighted mean of its neighborhood data points. The computational cost required to find the neighbors of each data point is quadratic to the number of data points. Consequently, the vanilla MS appears to be very slow for large-scale datasets. To address this issue, we propose a mode-seeking algorithm called GridShift, with significant speedup and principally based on MS. To accelerate, GridShift employs a grid-based approach for neighbor search, which is linear in the number of data points. In addition, GridShift moves the active grid cells (grid cells associated with at least one data point) in place of data points towards the higher density, a step that provides more speedup. The runtime of GridShift is linear in the number of active grid cells and exponential in the number of features. Therefore, it is ideal for large-scale low-dimensional applications such as object tracking and image segmentation. Through extensive experiments, we showcase the superior performance of GridShift compared to other MS-based as well as state-of-the-art algorithms in terms of accuracy and runtime on benchmark datasets for image segmentation. Finally, we provide a new object-tracking algorithm based on GridShift and show promising results for object tracking compared to CamShift and meanshift++.

Vojtěch Novák, Tomáš Bezděk, Ivan Zelinka et al.

This paper provides a historical analysis of the IEEE CEC Single Objective Optimization competition results (2010-2024). We analyze how benchmark functions shaped winning algorithms, identifying the 2014 introduction of dense rotation matrices as a key performance filter. This design choice introduced parameter non-separability, reduced effectiveness of coordinate-dependent methods (PSO, GA), and established the dominance of Differential Evolution variants capable of preserving the rotational invariance of their difference vectors, specifically L-SHADE. Post-2020 analysis reveals a shift towards high complexity hybrid optimizers that combine different mechanisms (e.g., Eigenvector Crossover, Societal Sharing, Reinforcement Learning) to maximize ranking stability. We conclude by identifying structural similarities between these modern benchmarks and Variational Quantum Algorithm landscapes, suggesting that evolved CEC solvers possess the specific adaptive capabilities required for quantum control.

Omatharv Bharat Vaidya, Rithvik Terence DSouza, Snehanshu Saha et al.

We introduce the Hamiltonian Monte Carlo Particle Swarm Optimizer (HMC-PSO), an optimization algorithm that reaps the benefits of both Exponentially Averaged Momentum PSO and HMC sampling. The coupling of the position and velocity of each particle with Hamiltonian dynamics in the simulation allows for extensive freedom for exploration and exploitation of the search space. It also provides an excellent technique to explore highly non-convex functions while ensuring efficient sampling. We extend the method to approximate error gradients in closed form for Deep Neural Network (DNN) settings. We discuss possible methods of coupling and compare its performance to that of state-of-the-art optimizers on the Golomb's Ruler problem and Classification tasks.

Pourya Shamsolmoali, Masoumeh Zareapoor, Swagatam Das et al.

Capsule networks (CapsNets) aim to parse images into a hierarchy of objects, parts, and their relations using a two-step process involving part-whole transformation and hierarchical component routing. However, this hierarchical relationship modeling is computationally expensive, which has limited the wider use of CapsNet despite its potential advantages. The current state of CapsNet models primarily focuses on comparing their performance with capsule baselines, falling short of achieving the same level of proficiency as deep CNN variants in intricate tasks. To address this limitation, we present an efficient approach for learning capsules that surpasses canonical baseline models and even demonstrates superior performance compared to high-performing convolution models. Our contribution can be outlined in two aspects: firstly, we introduce a group of subcapsules onto which an input vector is projected. Subsequently, we present the Hybrid Gromov-Wasserstein framework, which initially quantifies the dissimilarity between the input and the components modeled by the subcapsules, followed by determining their alignment degree through optimal transport. This innovative mechanism capitalizes on new insights into defining alignment between the input and subcapsules, based on the similarity of their respective component distributions. This approach enhances CapsNets' capacity to learn from intricate, high-dimensional data while retaining their interpretability and hierarchical structure. Our proposed model offers two distinct advantages: (i) its lightweight nature facilitates the application of capsules to more intricate vision tasks, including object detection; (ii) it outperforms baseline approaches in these demanding tasks.

Supratik Basu, Jyotishka Ray Choudhury, Debolina Paul et al.

Clustering stands as one of the most prominent challenges in unsupervised machine learning. Among centroid-based methods, the classic $k$-means algorithm, based on Lloyd's heuristic, is widely used. Nonetheless, it is a well-known fact that $k$-means and its variants face several challenges, including heavy reliance on initial cluster centroids, susceptibility to converging into local minima of the objective function, and sensitivity to outliers and noise in the data. When data contains noise or outliers, the Median-of-Means (MoM) estimator offers a robust alternative for stabilizing centroid-based methods. On a different note, another limitation in many commonly used clustering methods is the need to specify the number of clusters beforehand. Model-based approaches, such as Bayesian nonparametric models, address this issue by incorporating infinite mixture models, which eliminate the requirement for predefined cluster counts. Motivated by these facts, in this article, we propose an efficient and automatic clustering technique by integrating the strengths of model-based and centroid-based methodologies. Our method mitigates the effect of noise on the quality of clustering; while at the same time, estimates the number of clusters. Statistical guarantees on an upper bound of clustering error, and rigorous assessment through simulated and real datasets, suggest the advantages of our proposed method over existing state-of-the-art clustering algorithms.

Sharanya Dasgupta, Sujoy Nath, Arkaprabha Basu et al.

Large Language Models (LLMs) have recently garnered widespread attention due to their adeptness at generating innovative responses to the given prompts across a multitude of domains. However, LLMs often suffer from the inherent limitation of hallucinations and generate incorrect information while maintaining well-structured and coherent responses. In this work, we hypothesize that hallucinations stem from the internal dynamics of LLMs. Our observations indicate that, during passage generation, LLMs tend to deviate from factual accuracy in subtle parts of responses, eventually shifting toward misinformation. This phenomenon bears a resemblance to human cognition, where individuals may hallucinate while maintaining logical coherence, embedding uncertainty within minor segments of their speech. To investigate this further, we introduce an innovative approach, HalluShift, designed to analyze the distribution shifts in the internal state space and token probabilities of the LLM-generated responses. Our method attains superior performance compared to existing baselines across various benchmark datasets. Our codebase is available at https://github.com/sharanya-dasgupta001/hallushift.

Debanjan Dutta, Faizanuddin Ansari, Swagatam Das

Generating rational and generally accurate responses to tasks, often accompanied by example demonstrations, highlights Large Language Model's (LLM's) remarkable In-Context Learning (ICL) capabilities without requiring updates to the model's parameter space. Despite having an ongoing exploration focused on the inference from a document-level concept, its behavior in learning well-defined functions or relations in context needs a careful investigation. In this article, we present the performance of ICL on partially ordered relation by introducing the notion of inductively increasing complexity in prompts. In most cases, the saturated performance of the chosen metric indicates that while ICL offers some benefits, its effectiveness remains constrained as we increase the complexity in the prompts even in presence of sufficient demonstrative examples. The behavior is evident from our empirical findings and has further been theoretically justified in term of its implicit optimization process. The code is available \href{https://anonymous.4open.science/r/ICLonPartiallyOrderSet}{here}.

Sagar Ghosh, Swagatam Das

Clustering, as an unsupervised technique, plays a pivotal role in various data analysis applications. Among clustering algorithms, Spectral Clustering on Euclidean Spaces has been extensively studied. However, with the rapid evolution of data complexity, Euclidean Space is proving to be inefficient for representing and learning algorithms. Although Deep Neural Networks on hyperbolic spaces have gained recent traction, clustering algorithms or non-deep machine learning models on non-Euclidean Spaces remain underexplored. In this paper, we propose a spectral clustering algorithm on Hyperbolic Spaces to address this gap. Hyperbolic Spaces offer advantages in representing complex data structures like hierarchical and tree-like structures, which cannot be embedded efficiently in Euclidean Spaces. Our proposed algorithm replaces the Euclidean Similarity Matrix with an appropriate Hyperbolic Similarity Matrix, demonstrating improved efficiency compared to clustering in Euclidean Spaces. Our contributions include the development of the spectral clustering algorithm on Hyperbolic Spaces and the proof of its weak consistency. We show that our algorithm converges at least as fast as Spectral Clustering on Euclidean Spaces. To illustrate the efficacy of our approach, we present experimental results on the Wisconsin Breast Cancer Dataset, highlighting the superior performance of Hyperbolic Spectral Clustering over its Euclidean counterpart. This work opens up avenues for utilizing non-Euclidean Spaces in clustering algorithms, offering new perspectives for handling complex data structures and improving clustering efficiency.

Arindam Majee, Avisek Gupta, Sourav Raha et al.

Alzheimer's disease (AD), characterized by progressive cognitive decline and memory loss, presents a formidable global health challenge, underscoring the critical importance of early and precise diagnosis for timely interventions and enhanced patient outcomes. While MRI scans provide valuable insights into brain structures, traditional analysis methods often struggle to discern intricate 3D patterns crucial for AD identification. Addressing this challenge, we introduce an alternative end-to-end deep learning model, the 3D Hybrid Compact Convolutional Transformers 3D (HCCT). By synergistically combining convolutional neural networks (CNNs) and vision transformers (ViTs), the 3D HCCT adeptly captures both local features and long-range relationships within 3D MRI scans. Extensive evaluations on prominent AD benchmark dataset, ADNI, demonstrate the 3D HCCT's superior performance, surpassing state of the art CNN and transformer-based methods in classification accuracy. Its robust generalization capability and interpretability marks a significant stride in AD classification from 3D MRI scans, promising more accurate and reliable diagnoses for improved patient care and superior clinical outcomes.

Sandipan Dhar, Nanda Dulal Jana, Swagatam Das

Voice conversion (VC) stands as a crucial research area in speech synthesis, enabling the transformation of a speaker's vocal characteristics to resemble another while preserving the linguistic content. This technology has broad applications, including automated movie dubbing, speech-to-singing conversion, and assistive devices for pathological speech rehabilitation. With the increasing demand for high-quality and natural-sounding synthetic voices, researchers have developed a wide range of VC techniques. Among these, generative adversarial network (GAN)-based approaches have drawn considerable attention for their powerful feature-mapping capabilities and potential to produce highly realistic speech. Despite notable advancements, challenges such as ensuring training stability, maintaining linguistic consistency, and achieving perceptual naturalness continue to hinder progress in GAN-based VC systems. This systematic review presents a comprehensive analysis of the voice conversion landscape, highlighting key techniques, key challenges, and the transformative impact of GANs in the field. The survey categorizes existing methods, examines technical obstacles, and critically evaluates recent developments in GAN-based VC. By consolidating and synthesizing research findings scattered across the literature, this review provides a structured understanding of the strengths and limitations of different approaches. The significance of this survey lies in its ability to guide future research by identifying existing gaps, proposing potential directions, and offering insights for building more robust and efficient VC systems. Overall, this work serves as an essential resource for researchers, developers, and practitioners aiming to advance the state-of-the-art (SOTA) in voice conversion technology.

Swagatam Das

We present a formal operator-theoretic framework for analyzing Transformer-based language models using free probability theory. By modeling token embeddings and attention mechanisms as self-adjoint operators in a tracial \( W^* \)-probability space, we reinterpret attention as non-commutative convolution and describe representation propagation via free additive convolution. This leads to a spectral dynamic system interpretation of deep Transformers. We derive entropy-based generalization bounds under freeness assumptions and provide insight into positional encoding, spectral evolution, and representational complexity. This work offers a principled, though theoretical, perspective on structural dynamics in large language models.

Kushal Bose, Swagatam Das

Graph Transformers (GTs) facilitate the comprehension of graph-structured data by calculating the self-attention of node pairs without considering node position information. To address this limitation, we introduce an innovative and efficient framework that introduces Positional Encodings (PEs) into the Transformer, generating a set of learnable positional encodings in the hyperbolic space, a non-Euclidean domain. This approach empowers us to explore diverse options for optimal selection of PEs for specific downstream tasks, leveraging hyperbolic neural networks or hyperbolic graph convolutional networks. Additionally, we repurpose these positional encodings to mitigate the impact of over-smoothing in deep Graph Neural Networks (GNNs). Comprehensive experiments on molecular benchmark datasets, co-author, and co-purchase networks substantiate the effectiveness of hyperbolic positional encodings in enhancing the performance of deep GNNs.

Sagar Ghosh, Kushal Bose, Swagatam Das

The emergence of Deep Convolutional Neural Networks (DCNNs) has been a pervasive tool for accomplishing widespread applications in computer vision. Despite its potential capability to capture intricate patterns inside the data, the underlying embedding space remains Euclidean and primarily pursues contractive convolution. Several instances can serve as a precedent for the exacerbating performance of DCNNs. The recent advancement of neural networks in the hyperbolic spaces gained traction, incentivizing the development of convolutional deep neural networks in the hyperbolic space. In this work, we propose Hyperbolic DCNN based on the Poincaré Disc. The work predominantly revolves around analyzing the nature of expansive convolution in the context of the non-Euclidean domain. We further offer extensive theoretical insights pertaining to the universal consistency of the expansive convolution in the hyperbolic space. Several simulations were performed not only on the synthetic datasets but also on some real-world datasets. The experimental results reveal that the hyperbolic convolutional architecture outperforms the Euclidean ones by a commendable margin.

Arkaprabha Basu, Kushal Bose, Sankha Subhra Mullick et al.

Super-Resolution (SR) is a time-hallowed image processing problem that aims to improve the quality of a Low-Resolution (LR) sample up to the standard of its High-Resolution (HR) counterpart. We aim to address this by introducing Super-Resolution Generator (SuRGe), a fully-convolutional Generative Adversarial Network (GAN)-based architecture for SR. We show that distinct convolutional features obtained at increasing depths of a GAN generator can be optimally combined by a set of learnable convex weights to improve the quality of generated SR samples. In the process, we employ the Jensen-Shannon and the Gromov-Wasserstein losses respectively between the SR-HR and LR-SR pairs of distributions to further aid the generator of SuRGe to better exploit the available information in an attempt to improve SR. Moreover, we train the discriminator of SuRGe with the Wasserstein loss with gradient penalty, to primarily prevent mode collapse. The proposed SuRGe, as an end-to-end GAN workflow tailor-made for super-resolution, offers improved performance while maintaining low inference time. The efficacy of SuRGe is substantiated by its superior performance compared to 18 state-of-the-art contenders on 10 benchmark datasets.

Asmit Bandyopadhyay, Rohit Basu, Tanmay Sen et al.

Human Activity Recognition (HAR) plays a vital role in applications such as fitness tracking, smart homes, and healthcare monitoring. Traditional HAR systems often rely on single modalities, such as motion sensors or cameras, limiting robustness and accuracy in real-world environments. This work presents FedTime-MAGNET, a novel multimodal federated learning framework that advances HAR by combining heterogeneous data sources: depth cameras, pressure mats, and accelerometers. At its core is the Multimodal Adaptive Graph Neural Expert Transformer (MAGNET), a fusion architecture that uses graph attention and a Mixture of Experts to generate unified, discriminative embeddings across modalities. To capture complex temporal dependencies, a lightweight T5 encoder only architecture is customized and adapted within this framework. Extensive experiments show that FedTime-MAGNET significantly improves HAR performance, achieving a centralized F1 Score of 0.934 and a strong federated F1 Score of 0.881. These results demonstrate the effectiveness of combining multimodal fusion, time series LLMs, and federated learning for building accurate and robust HAR systems.

Priyobrata Mondal, Faizanuddin Ansari, Swagatam Das

Class imbalance, where certain classes have insufficient data, poses a critical challenge for robust classification, often biasing models toward majority classes. Distribution calibration offers a promising avenue to address this by estimating more accurate class distributions. In this work, we propose Rebalancing with Calibrated Sub-classes (RCS) - a novel distribution calibration framework for robust imbalanced classification. RCS aims to fuse statistical information from the majority and intermediate class distributions via a weighted mixture of Gaussian components to estimate minority class parameters more accurately. An encoder-decoder network is trained to preserve structural relationships in imbalanced datasets and prevent feature disentanglement. Post-training, encoder-extracted feature vectors are leveraged to generate synthetic samples guided by the calibrated distributions. This fusion-based calibration effectively mitigates overgeneralization by incorporating neighborhood distribution information rather than relying solely on majority-class statistics. Extensive experiments on diverse image, text, and tabular datasets demonstrate that RCS consistently outperforms several baseline and state-of-the-art methods, highlighting its effectiveness and broad applicability in addressing real-world imbalanced classification challenges.

Priyobrata Mondal, Faizanuddin Ansari, Swagatam Das

Ensuring fairness in machine learning models is critical, especially when biases compound across intersecting protected attributes like race, gender, and age. While existing methods address fairness for single attributes, they fail to capture the nuanced, multiplicative biases faced by intersectional subgroups. We introduce Adaptive Pareto Front Explorer (APFEx), the first framework to explicitly model intersectional fairness as a joint optimization problem over the Cartesian product of sensitive attributes. APFEx combines three key innovations- (1) an adaptive multi-objective optimizer that dynamically switches between Pareto cone projection, gradient weighting, and exploration strategies to navigate fairness-accuracy trade-offs, (2) differentiable intersectional fairness metrics enabling gradient-based optimization of non-smooth subgroup disparities, and (3) theoretical guarantees of convergence to Pareto-optimal solutions. Experiments on four real-world datasets demonstrate APFEx's superiority, reducing fairness violations while maintaining competitive accuracy. Our work bridges a critical gap in fair ML, providing a scalable, model-agnostic solution for intersectional fairness.

Kushal Bose, Swagatam Das

Graph heterophily poses a formidable challenge to the performance of Message-passing Graph Neural Networks (MP-GNNs). The familiar low-pass filters like Graph Convolutional Networks (GCNs) face performance degradation, which can be attributed to the blending of the messages from dissimilar neighboring nodes. The performance of the low-pass filters on heterophilic graphs still requires an in-depth analysis. In this context, we update the heterophilic graphs by adding a number of self-loops and parallel edges. We observe that eigenvalues of the graph Laplacian decrease and increase respectively by increasing the number of self-loops and parallel edges. We conduct several studies regarding the performance of GCN on various benchmark heterophilic networks by adding either self-loops or parallel edges. The studies reveal that the GCN exhibited either increasing or decreasing performance trends on adding self-loops and parallel edges. In light of the studies, we established connections between the graph spectra and the performance trends of the low-pass filters on the heterophilic graphs. The graph spectra characterize the essential intrinsic properties of the input graph like the presence of connected components, sparsity, average degree, cluster structures, etc. Our work is adept at seamlessly evaluating graph spectrum and properties by observing the performance trends of the low-pass filters without pursuing the costly eigenvalue decomposition. The theoretical foundations are also discussed to validate the impact of adding self-loops and parallel edges on the graph spectrum.

Kushal Bose, Swagatam Das

Graph Neural Networks (GNNs) suffer from Oversquashing, which occurs when tasks require long-range interactions. The problem arises from the presence of bottlenecks that limit the propagation of messages among distant nodes. Recently, graph rewiring methods modify edge connectivity and are expected to perform well on long-range tasks. Yet, graph rewiring compromises the inductive bias, incurring significant information loss in solving the downstream task. Furthermore, increasing channel capacity may overcome information bottlenecks but enhance the parameter complexity of the model. To alleviate these shortcomings, we propose an efficient model-agnostic framework that asynchronously updates node features, unlike traditional synchronous message passing GNNs. Our framework creates node batches in every layer based on the node centrality values. The features of the nodes belonging to these batches will only get updated. Asynchronous message updates process information sequentially across layers, avoiding simultaneous compression into fixed-capacity channels. We also theoretically establish that our proposed framework maintains higher feature sensitivity bounds compared to standard synchronous approaches. Our framework is applied to six standard graph datasets and two long-range datasets to perform graph classification and achieves impressive performances with a $5\%$ and $4\%$ improvements on REDDIT-BINARY and Peptides-struct, respectively.

Dhruv Sarkar, Aprameyo Chakrabartty, Anish Chakrabarty et al.

The Sliced Gromov-Wasserstein (SGW) distance, aiming to relieve the computational cost of solving a non-convex quadratic program that is the Gromov-Wasserstein distance, utilizes projecting directions sampled uniformly from unit hyperspheres. This slicing mechanism incurs unnecessary computational costs due to uninformative directions, which also affects the representative power of the distance. However, finding a more appropriate distribution over the projecting directions (slicing distribution) is often an optimization problem in itself that comes with its own computational cost. In addition, with more intricate distributions, the sampling itself may be expensive. As a remedy, we propose an optimization-free slicing distribution that provides fast sampling for the Monte Carlo approximation. We do so by introducing the Relation-Aware Projecting Direction (RAPD), effectively capturing the pairwise association of each of two pairs of random vectors, each following their ambient law. This enables us to derive the Relation-Aware Slicing Distribution (RASD), a location-scale law corresponding to sampled RAPDs. Finally, we introduce the RASGW distance and its variants, e.g., IWRASGW (Importance Weighted RASGW), which overcome the shortcomings experienced by SGW. We theoretically analyze its properties and substantiate its empirical prowess using extensive experiments on various alignment tasks.

Sagar Ghosh, Kushal Bose, Swagatam Das

Despite their central role in the success of foundational models and large-scale language modeling, the theoretical foundations governing the operation of Transformers remain only partially understood. Contemporary research has largely focused on their representational capacity for language comprehension and their prowess in in-context learning, frequently under idealized assumptions such as linearized attention mechanisms. Initially conceived to model sequence-to-sequence transformations, a fundamental and unresolved question is whether Transformers can robustly perform functional regression over sequences of input tokens. This question assumes heightened importance given the inherently non-Euclidean geometry underlying real-world data distributions. In this work, we establish that Transformers equipped with softmax-based nonlinear attention are uniformly consistent when tasked with executing Ordinary Least Squares (OLS) regression, provided both the inputs and outputs are embedded in hyperbolic space. We derive deterministic upper bounds on the empirical error which, in the asymptotic regime, decay at a provable rate of $\mathcal{O}(t^{-1/2d})$, where $t$ denotes the number of input tokens and $d$ the embedding dimensionality. Notably, our analysis subsumes the Euclidean setting as a special case, recovering analogous convergence guarantees parameterized by the intrinsic dimensionality of the data manifold. These theoretical insights are corroborated through empirical evaluations on real-world datasets involving both continuous and categorical response variables.

Arghya Pratihar, Kushal Bose, Swagatam Das

Clustering aims to form groups of similar data points in an unsupervised regime. Yet, clustering complex datasets containing critically intertwined shapes poses significant challenges. The prevailing clustering algorithms widely depend on evaluating similarity measures based on Euclidean metrics. Exploring topological characteristics to perform clustering of complex datasets inevitably presents a better scope. The topological clustering algorithms predominantly perceive the point set through the lens of Simplicial complexes and Persistent homology. Despite these approaches, the existing topological clustering algorithms cannot somehow fully exploit topological structures and show inconsistent performances on some highly complicated datasets. This work aims to mitigate the limitations by identifying topologically similar neighbors through the Vietoris-Rips complex and Betti number filtration. In addition, we introduce the concept of the Betti sequences to capture flexibly essential features from the topological structures. Our proposed algorithm is adept at clustering complex, intertwined shapes contained in the datasets. We carried out experiments on several synthetic and real-world datasets. Our algorithm demonstrated commendable performances across the datasets compared to some of the well-known topology-based clustering algorithms.

Swagato Das, Arghya Pratihar, Swagatam Das

Clustering algorithms play a pivotal role in unsupervised learning by identifying and grouping similar objects based on shared characteristics. Although traditional clustering techniques, such as hard and fuzzy center-based clustering, have been widely used, they struggle with complex, high-dimensional, and non-Euclidean datasets. In particular, the fuzzy $C$-Means (FCM) algorithm, despite its efficiency and popularity, exhibits notable limitations in non-Euclidean spaces. Euclidean spaces assume linear separability and uniform distance scaling, limiting their effectiveness in capturing complex, hierarchical, or non-Euclidean structures in fuzzy clustering. To overcome these challenges, we introduce Filtration-based Hyperbolic Fuzzy C-Means (HypeFCM), a novel clustering algorithm tailored for better representation of data relationships in non-Euclidean spaces. HypeFCM integrates the principles of fuzzy clustering with hyperbolic geometry and employs a weight-based filtering mechanism to improve performance. The algorithm initializes weights using a Dirichlet distribution and iteratively refines cluster centroids and membership assignments based on a hyperbolic metric in the Poincaré Disc model. Extensive experimental evaluations on $6$ synthetic and $12$ real-world datasets demonstrate that HypeFCM significantly outperforms conventional fuzzy clustering methods in non-Euclidean settings, underscoring its robustness and effectiveness.

Anish Chakrabarty, Arkaprabha Basu, Swagatam Das

The Gromov-Wasserstein (GW) distance is an effective measure of alignment between distributions supported on distinct ambient spaces. Calculating essentially the mutual departure from isometry, it has found vast usage in domain translation and network analysis. It has long been shown to be vulnerable to contamination in the underlying measures. All efforts to introduce robustness in GW have been inspired by similar techniques in optimal transport (OT), which predominantly advocate partial mass transport or unbalancing. In contrast, the cross-domain alignment problem being fundamentally different from OT, demands specific solutions to tackle diverse applications and contamination regimes. Deriving from robust statistics, we discuss three contextually novel techniques to robustify GW and its variants. For each method, we explore metric properties and robustness guarantees along with their co-dependencies and individual relations with the GW distance. For a comprehensive view, we empirically validate their superior resilience to contamination under real machine learning tasks against state-of-the-art methods.

Srinjoy Roy, Swagatam Das

Accounting for the uncertainty of value functions boosts exploration in Reinforcement Learning (RL). Our work introduces Maximum Mean Discrepancy Q-Learning (MMD-QL) to improve Wasserstein Q-Learning (WQL) for uncertainty propagation during Temporal Difference (TD) updates. MMD-QL uses the MMD barycenter for this purpose, as MMD provides a tighter estimate of closeness between probability measures than the Wasserstein distance. Firstly, we establish that MMD-QL is Probably Approximately Correct in MDP (PAC-MDP) under the average loss metric. Concerning the accumulated rewards, experiments on tabular environments show that MMD-QL outperforms WQL and other algorithms. Secondly, we incorporate deep networks into MMD-QL to create MMD Q-Network (MMD-QN). Making reasonable assumptions, we analyze the convergence rates of MMD-QN using function approximation. Empirical results on challenging Atari games demonstrate that MMD-QN performs well compared to benchmark deep RL algorithms, highlighting its effectiveness in handling large state-action spaces.

Anish Chakrabarty, Arkaprabha Basu, Swagatam Das

Variational Autoencoders (VAEs) have been a pioneering force in the realm of deep generative models. Amongst its legions of progenies, Wasserstein Autoencoders (WAEs) stand out in particular due to the dual offering of heightened generative quality and a strong theoretical backbone. WAEs consist of an encoding and a decoding network forming a bottleneck with the prime objective of generating new samples resembling the ones it was catered to. In the process, they aim to achieve a target latent representation of the encoded data. Our work is an attempt to offer a theoretical understanding of the machinery behind WAEs. From a statistical viewpoint, we pose the problem as concurrent density estimation tasks based on neural network-induced transformations. This allows us to establish deterministic upper bounds on the realized errors WAEs commit. We also analyze the propagation of these stochastic errors in the presence of adversaries. As a result, both the large sample properties of the reconstructed distribution and the resilience of WAE models are explored.

Saptarshi Chakraborty, Debolina Paul, Swagatam Das

The problem of linear predictions has been extensively studied for the past century under pretty generalized frameworks. Recent advances in the robust statistics literature allow us to analyze robust versions of classical linear models through the prism of Median of Means (MoM). Combining these approaches in a piecemeal way might lead to ad-hoc procedures, and the restricted theoretical conclusions that underpin each individual contribution may no longer be valid. To meet these challenges coherently, in this study, we offer a unified robust framework that includes a broad variety of linear prediction problems on a Hilbert space, coupled with a generic class of loss functions. Notably, we do not require any assumptions on the distribution of the outlying data points ($\mathcal{O}$) nor the compactness of the support of the inlying ones ($\mathcal{I}$). Under mild conditions on the dual norm, we show that for misspecification level $ε$, these estimators achieve an error rate of $O(\max\left\{|\mathcal{O}|^{1/2}n^{-1/2}, |\mathcal{I}|^{1/2}n^{-1} \right\}+ε)$, matching the best-known rates in literature. This rate is slightly slower than the classical rates of $O(n^{-1/2})$, indicating that we need to pay a price in terms of error rates to obtain robust estimates. Additionally, we show that this rate can be improved to achieve so-called "fast rates" under additional assumptions.

Debolina Paul, Saptarshi Chakraborty, Swagatam Das et al.

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.

Anish Chakrabarty, Swagatam Das

The introduction of Variational Autoencoders (VAE) has been marked as a breakthrough in the history of representation learning models. Besides having several accolades of its own, VAE has successfully flagged off a series of inventions in the form of its immediate successors. Wasserstein Autoencoder (WAE), being an heir to that realm carries with it all of the goodness and heightened generative promises, matching even the generative adversarial networks (GANs). Needless to say, recent years have witnessed a remarkable resurgence in statistical analyses of the GANs. Similar examinations for Autoencoders, however, despite their diverse applicability and notable empirical performance, remain largely absent. To close this gap, in this paper, we investigate the statistical properties of WAE. Firstly, we provide statistical guarantees that WAE achieves the target distribution in the latent space, utilizing the Vapnik Chervonenkis (VC) theory. The main result, consequently ensures the regeneration of the input distribution, harnessing the potential offered by Optimal Transport of measures under the Wasserstein metric. This study, in turn, hints at the class of distributions WAE can reconstruct after suffering a compression in the form of a latent law.

Sandipan Dhar, Nanda Dulal Jana, Swagatam Das

Voice Conversion (VC) emerged as a significant domain of research in the field of speech synthesis in recent years due to its emerging application in voice-assisting technology, automated movie dubbing, and speech-to-singing conversion to name a few. VC basically deals with the conversion of vocal style of one speaker to another speaker while keeping the linguistic contents unchanged. VC task is performed through a three-stage pipeline consisting of speech analysis, speech feature mapping, and speech reconstruction. Nowadays the Generative Adversarial Network (GAN) models are widely in use for speech feature mapping from source to target speaker. In this paper, we propose an adaptive learning-based GAN model called ALGAN-VC for an efficient one-to-one VC of speakers. Our ALGAN-VC framework consists of some approaches to improve the speech quality and voice similarity between source and target speakers. The model incorporates a Dense Residual Network (DRN) like architecture to the generator network for efficient speech feature learning, for source to target speech feature conversion. We also integrate an adaptive learning mechanism to compute the loss function for the proposed model. Moreover, we use a boosted learning rate approach to enhance the learning capability of the proposed model. The model is trained by using both forward and inverse mapping simultaneously for a one-to-one VC. The proposed model is tested on Voice Conversion Challenge (VCC) 2016, 2018, and 2020 datasets as well as on our self-prepared speech dataset, which has been recorded in Indian regional languages and in English. A subjective and objective evaluation of the generated speech samples indicated that the proposed model elegantly performed the voice conversion task by achieving high speaker similarity and adequate speech quality.

Debolina Paul, Saptarshi Chakraborty, Swagatam Das

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 paper proposes a PCA procedure based on the MoM principle. Called the \textbf{M}edian of \textbf{M}eans \textbf{P}rincipal \textbf{C}omponent \textbf{A}nalysis (MoMPCA), the proposed method is not only computationally appealing but also achieves optimal convergence rates under minimal assumptions. In particular, we explore the non-asymptotic 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.

Saptarshi Chakraborty, Debolina Paul, Swagatam Das

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.

Debolina Paul, Saptarshi Chakraborty, Swagatam Das et al.

Kernel $k$-means clustering is a powerful tool for unsupervised learning of non-linearly separable data. Since the earliest attempts, researchers have noted that such algorithms often become trapped by local minima arising from 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 makes use of majorization-minimization (MM) to better solve this non-convex problem. We show the method implicitly performs annealing in kernel feature space while retaining efficient, closed-form updates, and 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. Its merits are thoroughly validated on a suite of simulated datasets and real data benchmarks that feature non-linear and multi-view separation.

Sankha Subhra Mullick, Shounak Datta, Sourish Gunesh Dhekane et al.

Indices quantifying the performance of classifiers under class-imbalance, often suffer from distortions depending on the constitution of the test set or the class-specific classification accuracy, creating difficulties in assessing the merit of the classifier. We identify two fundamental conditions that a performance index must satisfy to be respectively resilient to altering number of testing instances from each class and the number of classes in the test set. In light of these conditions, under the effect of class imbalance, we theoretically analyze four indices commonly used for evaluating binary classifiers and five popular indices for multi-class classifiers. For indices violating any of the conditions, we also suggest remedial modification and normalization. We further investigate the capability of the indices to retain information about the classification performance over all the classes, even when the classifier exhibits extreme performance on some classes. Simulation studies are performed on high dimensional deep representations of subset of the ImageNet dataset using four state-of-the-art classifiers tailored for handling class imbalance. Finally, based on our theoretical findings and empirical evidence, we recommend the appropriate indices that should be used to evaluate the performance of classifiers in presence of class-imbalance.

Arka Ghosh, Sankha Subhra Mullick, Shounak Datta et al.

Constructing adversarial perturbations for deep neural networks is an important direction of research. Crafting image-dependent adversarial perturbations using white-box feedback has hitherto been the norm for such adversarial attacks. However, black-box attacks are much more practical for real-world applications. Universal perturbations applicable across multiple images are gaining popularity due to their innate generalizability. There have also been efforts to restrict the perturbations to a few pixels in the image. This helps to retain visual similarity with the original images making such attacks hard to detect. This paper marks an important step which combines all these directions of research. We propose the DEceit algorithm for constructing effective universal pixel-restricted perturbations using only black-box feedback from the target network. We conduct empirical investigations using the ImageNet validation set on the state-of-the-art deep neural classifiers by varying the number of pixels to be perturbed from a meagre 10 pixels to as high as all pixels in the image. We find that perturbing only about 10% of the pixels in an image using DEceit achieves a commendable and highly transferable Fooling Rate while retaining the visual quality. We further demonstrate that DEceit can be successfully applied to image dependent attacks as well. In both sets of experiments, we outperformed several state-of-the-art methods.

Saptarshi Chakraborty, Debolina Paul, Swagatam Das et al.

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 \textit{power $k$-means} algorithm was proposed to avoid trapping in local minima by annealing through a family of smoother surfaces. However, the approach lacks theoretical justification and fails in high dimensions when many features are irrelevant. This paper addresses these issues by introducing \textit{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 experiments.

Emille E. O. Ishida, Matwey V. Kornilov, Konstantin L. Malanchev et al.

We present the first evidence that adaptive learning techniques can boost the discovery of unusual objects within astronomical light curve data sets. Our method follows an active learning strategy where the learning algorithm chooses objects which can potentially improve the learner if additional information about them is provided. This new information is subsequently used to update the machine learning model, allowing its accuracy to evolve with each new information. For the case of anomaly detection, the algorithm aims to maximize the number of scientifically interesting anomalies presented to the expert by slightly modifying the weights of a traditional Isolation Forest (IF) at each iteration. In order to demonstrate the potential of such techniques, we apply the Active Anomaly Discovery (AAD) algorithm to 2 data sets: simulated light curves from the PLAsTiCC challenge and real light curves from the Open Supernova Catalog. We compare the AAD results to those of a static IF. For both methods, we performed a detailed analysis for all objects with the ~2% highest anomaly scores. We show that, in the real data scenario, AAD was able to identify ~80\% more true anomalies than the IF. This result is the first evidence that AAD algorithms can play a central role in the search for new physics in the era of large scale sky surveys.

Yi Chen, Aimin Zhou, Swagatam Das

Several real-world applications could be modeled as Mixed-Integer Non-Linear Programming (MINLP) problems, and some prominent examples include portfolio optimization, remote sensing technology, and so on. Most of the models for these applications are non-convex and always involve some conflicting objectives. The mathematical and heuristic methods have their advantages in solving this category of problems. In this work, we turn to Multi-Objective Evolutionary Algorithms (MOEAs) for finding elegant solutions for such problems. In this framework, we investigate a multi-objective constrained portfolio optimization problem, which can be cast as a classical financial problem and can also be naturally modeled as an MINLP problem. Consequently, we point out one challenge, faced by a direct coding scheme for MOEAs, to this problem. It is that the dependence among variables, like the selection and weights for one same asset, will likely make the search difficult. We thus, propose a Compressed Coding Scheme (CCS), compressing the two dependent variables into one variable to utilize the dependence and thereby meeting this challenge. Subsequently, we carry out a detailed empirical study on two sets of instances. The first part consists of 5 instances from OR-Library, which is solvable for the general mathematical optimizer, like CPLEX, while the remaining 15 instances from NGINX are addressed only by MOEAs. The two benchmarks, involving the number of assets from 31 to 2235, consistently indicate that CCS is not only efficient but also robust for dealing with the constrained multi-objective portfolio optimization.

Saptarshi Chakraborty, Swagatam Das

We propose the 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 larger compared to the number of observations ($n$). In the $LW$-$k$-means algorithm, we introduce a lasso-based penalty term, directly on the feature weights to incorporate feature selection in the framework of sparse clustering. $LW$-$k$-means does not make any distributional assumption of the given dataset and thus, induces a non-parametric method for feature selection. We also analytically investigate the convergence of the underlying optimization procedure in $LW$-$k$-means and establish the strong consistency of our algorithm. $LW$-$k$-means is tested on several real-life and synthetic datasets and through detailed experimental analysis, we find that the performance of the method is highly competitive against some state-of-the-art procedures for clustering and feature selection, not only in terms of clustering accuracy but also with respect to computational time.

Sankha Subhra Mullick, Shounak Datta, Swagatam Das

Class imbalance is a long-standing problem relevant to a number of real-world applications of deep learning. Oversampling techniques, which are effective for handling class imbalance in classical learning systems, can not be directly applied to end-to-end deep learning systems. We propose a three-player adversarial game between a convex generator, a multi-class classifier network, and a real/fake discriminator to perform oversampling in deep learning systems. The convex generator generates new samples from the minority classes as convex combinations of existing instances, aiming to fool both the discriminator as well as the classifier into misclassifying the generated samples. Consequently, the artificial samples are generated at critical locations near the peripheries of the classes. This, in turn, adjusts the classifier induced boundaries in a way which is more likely to reduce misclassification from the minority classes. Extensive experiments on multiple class imbalanced image datasets establish the efficacy of our proposal.

Avisek Gupta, Shounak Datta, Swagatam Das

Fuzzy clustering methods identify naturally occurring clusters in a dataset, where the extent to which different clusters are overlapped can differ. Most methods have a parameter to fix the level of fuzziness. However, the appropriate level of fuzziness depends on the application at hand. This paper presents Entropy $c$-Means (ECM), a method of fuzzy clustering that simultaneously optimizes two contradictory objective functions, resulting in the creation of fuzzy clusters with different levels of fuzziness. This allows ECM to identify clusters with different degrees of overlap. ECM optimizes the two objective functions using two multi-objective optimization methods, Non-dominated Sorting Genetic Algorithm II (NSGA-II), and Multiobjective Evolutionary Algorithm based on Decomposition (MOEA/D). We also propose a method to select a suitable trade-off clustering from the Pareto front. Experiments on challenging synthetic datasets as well as real-world datasets show that ECM leads to better cluster detection compared to the conventional fuzzy clustering methods as well as previously used multi-objective methods for fuzzy clustering.

Shounak Datta, Sayak Nag, Sankha Subhra Mullick et al.

The diversification (generating slightly varying separating discriminators) of Support Vector Machines (SVMs) for boosting has proven to be a challenge due to the strong learning nature of SVMs. Based on the insight that perturbing the SVM kernel may help in diversifying SVMs, we propose two kernel perturbation based boosting schemes where the kernel is modified in each round so as to increase the resolution of the kernel-induced Reimannian metric in the vicinity of the datapoints misclassified in the previous round. We propose a method for identifying the disjuncts in a dataset, dispelling the dependence on rule-based learning methods for identifying the disjuncts. We also present a new performance measure called Geometric Small Disjunct Index (GSDI) to quantify the performance on small disjuncts for balanced as well as class imbalanced datasets. Experimental comparison with a variety of state-of-the-art algorithms is carried out using the best classifiers of each type selected by a new approach inspired by multi-criteria decision making. The proposed method is found to outperform the contending state-of-the-art methods on different datasets (ranging from mildly imbalanced to highly imbalanced and characterized by varying number of disjuncts) in terms of three different performance indices (including the proposed GSDI).

Shounak Datta, Sayak Nag, Swagatam Das

A large amount of research effort has been dedicated to adapting boosting for imbalanced classification. However, boosting methods are yet to be satisfactorily immune to class imbalance, especially for multi-class problems. This is because most of the existing solutions for handling class imbalance rely on expensive cost set tuning for determining the proper level of compensation. We show that the assignment of weights to the component classifiers of a boosted ensemble can be thought of as a game of Tug of War between the classes in the margin space. We then demonstrate how this insight can be used to attain a good compromise between the rare and abundant classes without having to resort to cost set tuning, which has long been the norm for imbalanced classification. The solution is based on a lexicographic linear programming framework which requires two stages. Initially, class-specific component weight combinations are found so as to minimize a hinge loss individually for each of the classes. Subsequently, the final component weights are assigned so that the maximum deviation from the class-specific minimum loss values (obtained in the previous stage) is minimized. Hence, the proposal is not only restricted to two-class situations, but is also readily applicable to multi-class problems. Additionally,we also derive the dual formulation corresponding to the proposed framework. Experiments conducted on artificial and real-world imbalanced datasets as well as on challenging applications such as hyperspectral image classification and ImageNet classification establish the efficacy of the proposal.

Shounak Datta, Supritam Bhattacharjee, Swagatam Das

Many real-world clustering problems are plagued by incomplete data characterized by missing or absent features for some or all of the data instances. Traditional clustering methods cannot be directly applied to such data without preprocessing by imputation or marginalization techniques. In this article, we overcome this drawback by utilizing a penalized dissimilarity measure which we refer to as the Feature Weighted Penalty based Dissimilarity (FWPD). Using the FWPD measure, we modify the traditional k-means clustering algorithm and the standard hierarchical agglomerative clustering algorithms so as to make them directly applicable to datasets with missing features. We present time complexity analyses for these new techniques and also undertake a detailed theoretical analysis showing that the new FWPD based k-means algorithm converges to a local optimum within a finite number of iterations. We also present a detailed method for simulating random as well as feature dependent missingness. We report extensive experiments on various benchmark datasets for different types of missingness showing that the proposed clustering techniques have generally better results compared to some of the most well-known imputation methods which are commonly used to handle such incomplete data. We append a possible extension of the proposed dissimilarity measure to the case of absent features (where the unobserved features are known to be undefined).

Satrajit Mukherjee, Bodhisattwa Prasad Majumder, Aritran Piplai et al.

The paper proposes a novel Kernelized image segmentation scheme for noisy images that utilizes the concept of Smallest Univalue Segment Assimilating Nucleus (SUSAN) and incorporates spatial constraints by computing circular colour map induced weights. Fuzzy damping coefficients are obtained for each nucleus or center pixel on the basis of the corresponding weighted SUSAN area values, the weights being equal to the inverse of the number of horizontal and vertical moves required to reach a neighborhood pixel from the center pixel. These weights are used to vary the contributions of the different nuclei in the Kernel based framework. The paper also presents an edge quality metric obtained by fuzzy decision based edge candidate selection and final computation of the blurriness of the edges after their selection. The inability of existing algorithms to preserve edge information and structural details in their segmented maps necessitates the computation of the edge quality factor (EQF) for all the competing algorithms. Qualitative and quantitative analysis have been rendered with respect to state-of-the-art algorithms and for images ridden with varying types of noises. Speckle noise ridden SAR images and Rician noise ridden Magnetic Resonance Images have also been considered for evaluating the effectiveness of the proposed algorithm in extracting important segmentation information.

Debdipta Goswami, Chiranjib Saha, Kunal Pal et al.

Principle of Swarm Intelligence has recently found widespread application in formation control and automated tracking by the automated multi-agent system. This article proposes an elegant and effective method inspired by foraging dynamics to produce geometric-patterns by the search agents. Starting from a random initial orientation, it is investigated how the foraging dynamics can be modified to achieve convergence of the agents on the desired pattern with almost uniform density. Guided through the proposed dynamics, the agents can also track a moving point by continuously circulating around the point. An analytical treatment supported with computer simulation results is provided to better understand the convergence behaviour of the system.