Topology-Driven Clustering: Enhancing Performance with Betti Number Filtration

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.

Hyperbolic Fuzzy $C$-Means with Adaptive Weight-based Filtering for Clustering in Non-Euclidean Spaces

Clustering algorithms play a pivotal role in unsupervised learning by identifying and grouping similar objects based on shared characteristics. While 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 Poincar\'e Disc model. Extensive experimental evaluations demonstrate that HypeFCM significantly outperforms conventional fuzzy clustering methods in non-Euclidean settings, underscoring its robustness and effectiveness.

Generative Adversarial Network based Voice Conversion: Techniques, Challenges, and Recent Advancements

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.

Collective Learning Mechanism based Optimal Transport Generative Adversarial Network for Non-parallel Voice Conversion

After demonstrating significant success in image synthesis, Generative Adversarial Network (GAN) models have likewise made significant progress in the field of speech synthesis, leveraging their capacity to adapt the precise distribution of target data through adversarial learning processes. Notably, in the realm of State-Of-The-Art (SOTA) GAN-based Voice Conversion (VC) models, there exists a substantial disparity in naturalness between real and GAN-generated speech samples. Furthermore, while many GAN models currently operate on a single generator discriminator learning approach, optimizing target data distribution is more effectively achievable through a single generator multi-discriminator learning scheme. Hence, this study introduces a novel GAN model named Collective Learning Mechanism-based Optimal Transport GAN (CLOT-GAN) model, incorporating multiple discriminators, including the Deep Convolutional Neural Network (DCNN) model, Vision Transformer (ViT), and conformer. The objective of integrating various discriminators lies in their ability to comprehend the formant distribution of mel-spectrograms, facilitated by a collective learning mechanism. Simultaneously, the inclusion of Optimal Transport (OT) loss aims to precisely bridge the gap between the source and target data distribution, employing the principles of OT theory. The experimental validation on VCC 2018, VCTK, and CMU-Arctic datasets confirms that the CLOT-GAN-VC model outperforms existing VC models in objective and subjective assessments.

HalluShift: Measuring Distribution Shifts towards Hallucination Detection in LLMs

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.

On Robust Cross Domain Alignment

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.

On the Universal Statistical Consistency of Expansive Hyperbolic Deep Convolutional Neural Networks

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 Poincar\'{e} 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.

Consistent Spectral Clustering in Hyperbolic Spaces

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.

Fortifying Fully Convolutional Generative Adversarial Networks for Image Super-Resolution Using Divergence Measures

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.

Utilizing Maximum Mean Discrepancy Barycenter for Propagating the Uncertainty of Value Functions in Reinforcement Learning

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.