arXiv subjects

Subhroshekhar Ghosh

Publications and source records attributed to Subhroshekhar Ghosh.

At least 19 recordsLinked to original sources

arXiv2026

Gradient Descent with Stochastic Subspaces via Persistence of Memory

Stochastic subspace methods have gained popularity as gradient descent based techniques for large scale optimisation problems, especially in distributed settings. In this paper, we introduce the technique of "persistence of memory" to greatly extend and improve the random subspace methods. To this end, we leverage a vector that is only weakly correlated with the gradient in order to provide a guiding structure to the generative process of the random subspace along which the descent is going to take place. This guidance vector may be fixed for a large number of iterations, only to be refreshed at wide intervals (on whose size we can provide guarantees in terms of problem parameters). In important machine learning settings, such as optimisation problems embodying sparsity or a minibatch structure, we show that the guidance vector can be obtained in an effective and computationally inexpensive manner by leveraging the structured properties of the problem. En route, we establish to our knowledge the first theoretical analysis of classical SSD methods for sparse functions. In a local neighbourhood of the optimum, we demonstrate an alignment phenomenon of our gradient estimates with a low-lying eigenvector of the Hessian, allowing a once-for-all computation of the guidance vector which renders the method computationally favourable even in scenarios with unstructured objectives.

math.OC↗

arXiv2026

A statistical approach to bias in zero-shot learning: the lens of handwriting recognition

Generalized zero-shot learning (GZSL) has emerged as an important paradigm for visual recognition systems that must generalize to classes that were not observed during training. Traditional GZSL techniques are limited by their applicability to a relatively small number of such unseen classes, scalability beyond which is challenging due to its well-known misclassification bias towards classes observed during training. In this work, we investigate the GZSL paradigm through the lens of zero-shot handwritten word recognition over extremely large vocabularies. We propose a statistical approach to rectifying this bias, which views any classical GZSL feature learner as a black box mechanism whose intrinsic bias in identifying the training status (seen vs. unseen) of a typical data point we aim to correct, similar to an out of distribution inferential problem. Our method leverages a simple two-stage hierarchical architecture, combining a classical GZSL blackbox in the first stage and an ensemble of lightweight Monte Carlo bias-correctors in the second. Once debiased, the classification of test data is undertaken only restricted to its predicted training status via well-founded statistical methods (eg nearest neighbour, logistic regression and random forests). We achieve relative accuracy improvements of over 20% in the classification of unseen words compared to established techniques. A key outcome is that word recognition over large scale vocabularies is amenable to a much lower dimensional representation (~15 dimensions). Our approach is underpinned by mathematical analysis that captures the essence of the statistical approach to bias correction. Our approach to bias rectification can be combined in a turn-key fashion with any classical GZSL learner as a blackbox, thereby suggesting a wide scope of applicability of this method for a wide variety of GZSL implementations in different domains.

stat.ML↗

arXiv2026

Gaussian mixtures and non-parametric likelihoods through the lens of statistical mechanics

In this work, we investigate Gaussian Mixture Models ({\it abbrv} GMM) and the related problem of non parametric maximum likelihood estimation ({\it abbrv} NPMLE) from the perspective of statistical mechanics. In particular, we establish stability guarantees for the NPMLE procedure that extend well beyond the state of the art. Crucially, we obtain guarantees on the Kullback-Leibler divergence between NPMLE estimators and the ground truth, a type of result which has been known to be challenging in the literature on this problem. In particular, we provide high probability upper bounds on the KL divergence between the NPMLE and the true density that are of the order of $\min\big\{\frac{(\log n)^{d+2}}{n} , \frac{\log n}{\sqrt n}\big\}$, which cover a wide range of scenarios for the comparative sizes of $n$ and $d$. We obtain similar guarantees for approximate solutions to the NPMLE problem, addressing realistic situations wherein optimization algorithms need to be stopped in finite time, allowing access only to approximations to the true NPMLE. A cornerstone of our approach is an analysis of the function class complexity of logarithms of gaussian mixture densities, which is able to handle their unboundedness, and could be of wider interest. Our methods lead to novel confidence-interval guarantees for entropy estimation in Gaussian mixtures, demonstrating their wider impact. We also establish correspondences between stability phenomena in the NPMLE problem and concepts such as chaos and multiple valleys in random energy landscapes of statistical mechanics models. While these correspondences are largely of a conceptual nature at this point, we believe that these connections, especially those with concentration phenomena and Langevin dynamics, may be developed into a toolbox for studying a wide variety of random optimization problems in statistics and machine learning.

math.ST↗

arXiv2026

Fast determinantal sampling on general spaces and diffusion geometry

Determinantal point processes have recently emerged as a kernel-based alternative to standard independent sampling for constructing efficient minibatches, coresets, and other compact representations of large-scale datasets. In particular, sampling mechanisms based on DPPs are believed to demonstrate better approximation properties compared to classical i.i.d. samplers, even at the scale of the exponent. One of the key strengths of DPP based samplers is that they can be deployed over very general spaces, in contrast to more classical sampling methods beyond i.i.d. which tend to work in very well-structured settings, principally Euclidean spaces. In this work, we establish explicit rate guarantees for determinantal sampling in spaces that extend far beyond known Euclidean setups, focusing on spectral kernels obtained from eigenspaces of naturally associated Laplacian and other Markov diffusion operators. This includes, in particular, Riemannian manifolds and weighted networks. In determinantal sampling from compact Riemannian manifolds, we establish sampling rates that automatically pick up the intrinsic dimensionality $d_{\text{int}}$ of the underlying manifold. In the setting of networks, we investigate DPP-based samplers on the celebrated k-nearest neighbour graphs, as well as weighted random geometric graphs, and demonstrate a similar improved dependence on the intrinsic dimensionality of the data. Overall, our approach achieves guarantees of $\big(\text{sample size}\big)^{-\frac{1}{2}-\frac{1}{2d_{\text{int}}}}$ that match known rates on Euclidean spaces of comparable dimension. In terms of techniques, we connect to the celebrated Weyl's Law for manifold spectra, and leverage tools from the theory of Markov diffusions and Dirichlet forms as well as certain ingredients from the theory of pseudodifferential operators, which could be of independent interest in this area.

stat.ML↗

arXiv2026

State-of-art minibatches via novel DPP kernels: discretization, wavelets, and rough objectives

Determinantal point processes (DPPs) have emerged as a kernelized alternative to vanilla independent sampling for generating efficient minibatches, coresets and other parsimonious representations of large-scale datasets. While theoretical foundations and promising empirical performance have been demonstrated, there are two challenges for current proposals for DPP-based coresets or minibatches. The first is the need for families of DPPs with certain key variance reduction properties, usually constructed in a continuous setting, of which there are few known examples. The second is the need for an ad-hoc construction of a discrete DPP defined on a given dataset, that inherits such variance reduction. In this work, we contribute to the programme of establishing DPPs as a subsampling toolbox for ML by advancing on these two fronts. First, we propose new DPPs on the Euclidean space based on wavelets, with provably better accuracy guarantees than the best known rates. Second, we introduce a general method to convert such continuous DPPs, which are more amenable to proving analytical statements, into discrete kernels, which are pertinent for subsampling tasks such as minibatch and coreset constructions. This conversion mechanism simultaneously preserves the desired variance decay and reveals a low-rank decomposition of the discrete kernel, which makes sampling the corresponding DPP computationally inexpensive. En route, we enlarge the class of ML tasks amenable to improvements via DPP-based minibatches and coresets to include objective functions with arbitrarily low regularity, and rate guarantees that explicitly adapt to this regularity.

stat.ML↗

arXiv2026

Dictionary Learning under Symmetries via Group Representations

The dictionary learning problem can be viewed as a data-driven process to learn a suitable transformation so that data is sparsely represented directly from example data. In this paper, we examine the problem of learning a dictionary that is invariant under a pre-specified group of transformations. Natural settings include Cryo-EM, multi-object tracking, synchronization, pose estimation, etc. We specifically study this problem under the lens of mathematical representation theory. Leveraging the power of non-abelian Fourier analysis for functions over compact groups, we prescribe an algorithmic recipe for learning dictionaries that obey such invariances. We relate the dictionary learning problem in the physical domain, which is naturally modelled as being infinite dimensional, with the associated computational problem, which is necessarily finite dimensional. We establish that the dictionary learning problem can be effectively understood as an optimization instance over certain matrix orbitopes having a particular block-diagonal structure governed by the irreducible representations of the group of symmetries. This perspective enables us to introduce a band-limiting procedure which obtains dimensionality reduction in applications. We provide guarantees for our computational ansatz to provide a desirable dictionary learning outcome. We apply our paradigm to investigate the dictionary learning problem for the groups SO(2) and SO(3). While the SO(2)-orbitope admits an exact spectrahedral description, substantially less is understood about the SO(3)-orbitope. We describe a tractable spectrahedral outer approximation of the SO(3)-orbitope, and contribute an alternating minimization paradigm to perform optimization in this setting. We provide numerical experiments to highlight the efficacy of our approach in learning SO(3)-invariant dictionaries, both on synthetic and on real world data.

math.OC↗

arXiv2025

Negative Dependence as a toolbox for machine learning : review and new developments

Negative dependence is becoming a key driver in advancing learning capabilities beyond the limits of traditional independence. Recent developments have evidenced support towards negatively dependent systems as a learning paradigm in a broad range of fundamental machine learning challenges including optimization, sampling, dimensionality reduction and sparse signal recovery, often surpassing the performance of current methods based on statistical independence. The most popular negatively dependent model has been that of determinantal point processes (DPPs), which have their origins in quantum theory. However, other models, such as perturbed lattice models, strongly Rayleigh measures, zeros of random functions have gained salience in various learning applications. In this article, we review this burgeoning field of research, as it has developed over the past two decades or so. We also present new results on applications of DPPs to the parsimonious representation of neural networks. In the limited scope of the article, we mostly focus on aspects of this area to which the authors contributed over the recent years, including applications to Monte Carlo methods, coresets and stochastic gradient descent, stochastic networks, signal processing and connections to quantum computation. However, starting from basics of negative dependence for the uninitiated reader, extensive references are provided to a broad swath of related developments which could not be covered within our limited scope. While existing works and reviews generally focus on specific negatively dependent models (e.g. DPPs), a notable feature of this article is that it addresses negative dependence as a machine learning methodology as a whole. In this vein, it covers within its span an array of negatively dependent models and their applications well beyond DPPs, thereby putting forward a very general and rather unique perspective.

stat.ML↗

arXiv2025

Gaussian fluctuations for spin systems and point processes: near-optimal rates via quantitative Marcinkiewicz's theorem

We establish asymptotically Gaussian fluctuations for functionals of a large class of spin models and strongly correlated random point fields, achieving near-optimal rates. For spin models, we demonstrate Gaussian asymptotics for the magnetization for a wide class of ferromagnetic spin systems on Euclidean lattices, in particular those with continuous spins. Specific applications include, in particular, the celebrated XY and Heisenberg models under ferromagnetic conditions, and more broadly, systems with very general rotationally invariant spins in arbitrary dimensions. We address both the setting of free boundary conditions and a large class of ferromagnetic boundary conditions, and our CLTs are endowed with near-optimal rate. Our approach leverages the classical Lee-Yang theory for the zeros of partition functions, and subsumes as a special case results of Lebowitz, Ruelle, Pittel and Speer on CLTs in discrete statistical mechanical models for which we obtain sharper convergence rates. In a different direction, we obtain CLTs for linear statistics of a wide class of point processes known as $α$-determinantal point processes which interpolate between negatively and positively associated random point fields. We contribute a unified approach to CLTs in such models; significantly, our approach is able to analyse such processes in dimensions $\ge 3$, where structural alternatives such as connections to random matrix theory are not available. A key ingredient of our approach is a broad, quantitative extension of the classical Marcinkiewicz Theorem that holds under the limited condition that the characteristic function is non-vanishing only on a bounded disk. In spite of the general applicability of the results, our rates for the CLT match the classic Berry-Esseen bounds for independent sums up to a log factor.

math.PR↗

arXiv2025

Filtering through a topological lens: homology for point processes on the time-frequency plane

We introduce a very general approach to the analysis of signals from their noisy measurements from the perspective of Topological Data Analysis (TDA). While TDA has emerged as a powerful analytical tool for data with pronounced topological structures, here we demonstrate its applicability for general problems of signal processing, without any a-priori geometric feature. Our methods are well-suited to a wide array of time-dependent signals in different scientific domains, with acoustic signals being a particularly important application. We invoke time-frequency representations of such signals, focusing on their zeros which are gaining salience as a signal processing tool in view of their stability properties. Leveraging state-of-the-art topological concepts, such as stable and minimal volumes, we develop a complete suite of TDA-based methods to explore the delicate stochastic geometry of these zeros, capturing signals based on the disruption they cause to this rigid, hyperuniform spatial structure. Unlike classical spatial data tools, TDA is able to capture the full spectrum of the stochastic geometry of the zeros, thereby leading to powerful inferential outcomes that are underpinned by a principled statistical foundation. This is reflected in the power and versatility of our applications, which include competitive performance in processing. a wide variety of audio signals (esp. in low SNR regimes), effective detection and reconstruction of gravitational wave signals (a reputed signal processing challenge with non-Gaussian noise), and medical time series data from EEGs, indicating a wide horizon for the approach and methods introduced in this paper.

eess.SP↗

arXiv2025

Learning under Latent Group Sparsity via Diffusion on Networks

Group or cluster structure on explanatory variables in machine learning problems is a very general phenomenon, which has attracted broad interest from practitioners and theoreticians alike. In this work we contribute an approach to sparse learning under such group structure, that does not require prior information on the group identities. Our paradigm is motivated by the Laplacian geometry of an underlying network with a related community structure, and proceeds by directly incorporating this into a penalty that is effectively computed via a heat-flow-based local network dynamics. The proposed penalty interpolates between the lasso and the group lasso penalties, the runtime of the heat-flow dynamics being the interpolating parameter. As such it can automatically default to lasso when the group structure reflected in the Laplacian is weak. In fact, we demonstrate a data-driven procedure to construct such a network based on the available data. Notably, we dispense with computationally intensive pre-processing involving clustering of variables, spectral or otherwise. Our technique is underpinned by rigorous theorems that guarantee its effective performance and provide bounds on its sample complexity. In particular, in a wide range of settings, it provably suffices to run the diffusion for time that is only logarithmic in the problem dimensions. We explore in detail the interfaces of our approach with key statistical physics models in network science, such as the Gaussian Free Field and the Stochastic Block Model. Our work raises the possibility of applying similar diffusion-based techniques to classical learning tasks, exploiting the interplay between geometric, dynamical and stochastic structures underlying the data.

stat.ML↗

arXiv2024

Small coresets via negative dependence: DPPs, linear statistics, and concentration

Determinantal point processes (DPPs) are random configurations of points with tunable negative dependence. Because sampling is tractable, DPPs are natural candidates for subsampling tasks, such as minibatch selection or coreset construction. A \emph{coreset} is a subset of a (large) training set, such that minimizing an empirical loss averaged over the coreset is a controlled replacement for the intractable minimization of the original empirical loss. Typically, the control takes the form of a guarantee that the average loss over the coreset approximates the total loss uniformly across the parameter space. Recent work has provided significant empirical support in favor of using DPPs to build randomized coresets, coupled with interesting theoretical results that are suggestive but leave some key questions unanswered. In particular, the central question of whether the cardinality of a DPP-based coreset is fundamentally smaller than one based on independent sampling remained open. In this paper, we answer this question in the affirmative, demonstrating that \emph{DPPs can provably outperform independently drawn coresets}. In this vein, we contribute a conceptual understanding of coreset loss as a \emph{linear statistic} of the (random) coreset. We leverage this structural observation to connect the coresets problem to a more general problem of concentration phenomena for linear statistics of DPPs, wherein we obtain \emph{effective concentration inequalities that extend well-beyond the state-of-the-art}, encompassing general non-projection, even non-symmetric kernels. The latter have been recently shown to be of interest in machine learning beyond coresets, but come with a limited theoretical toolbox, to the extension of which our result contributes. Finally, we are also able to address the coresets problem for vector-valued objective functions, a novelty in the coresets literature.

stat.ML↗

arXiv2024

Bulk Spectra of Truncated Sample Covariance Matrices

Determinantal Point Processes (DPPs), which originate from quantum and statistical physics, are known for modelling diversity. Recent research [Ghosh and Rigollet (2020)] has demonstrated that certain matrix-valued $U$-statistics (that are truncated versions of the usual sample covariance matrix) can effectively estimate parameters in the context of Gaussian DPPs and enhance dimension reduction techniques, outperforming standard methods like PCA in clustering applications. This paper explores the spectral properties of these matrix-valued $U$-statistics in the null setting of an isotropic design. These matrices may be represented as $X L X^\top$, where $X$ is a data matrix and $L$ is the Laplacian matrix of a random geometric graph associated to $X$. The main mathematically interesting twist here is that the matrix $L$ is dependent on $X$. We give complete descriptions of the bulk spectra of these matrix-valued $U$-statistics in terms of the Stieltjes transforms of their empirical spectral measures. The results and the techniques are in fact able to address a broader class of kernelised random matrices, connecting their limiting spectra to generalised Marčenko-Pastur laws and free probability.

math.ST↗

arXiv2024

On an Empirical Likelihood based Solution to the Approximate Bayesian Computation Problem

Approximate Bayesian Computation (ABC) methods are applicable to statistical models specified by generative processes with analytically intractable likelihoods. These methods try to approximate the posterior density of a model parameter by comparing the observed data with additional process-generated simulated datasets. For computational benefit, only the values of certain well-chosen summary statistics are usually compared, instead of the whole dataset. Most ABC procedures are computationally expensive, justified only heuristically, and have poor asymptotic properties. In this article, we introduce a new empirical likelihood-based approach to the ABC paradigm called ABCel. The proposed procedure is computationally tractable and approximates the target log posterior of the parameter as a sum of two functions of the data -- namely, the mean of the optimal log-empirical likelihood weights and the estimated differential entropy of the summary functions. We rigorously justify the procedure via direct and reverse information projections onto appropriate classes of probability densities. Past applications of empirical likelihood in ABC demanded constraints based on analytically tractable estimating functions that involve both the data and the parameter; although by the nature of the ABC problem such functions may not be available in general. In contrast, we use constraints that are functions of the summary statistics only. Equally importantly, we show that our construction directly connects to the reverse information projection. We show that ABCel is posterior consistent and has highly favourable asymptotic properties. Its construction justifies the use of simple summary statistics like moments, quantiles, etc, which in practice produce an accurate approximation of the posterior density. We illustrate the performance of the proposed procedure in a range of applications.

stat.ME↗

arXiv2024

Hole event for random holomorphic sections on compact Riemann surfaces

Let $X$ be a compact Riemann surface and $\mathcal L$ be a positive line bundle on it. We study the conditional zero expectation of all the holomorphic sections of $\mathcal L^n$ which do not vanish on $D$ for some fixed open subset $D$ of $X$. We prove that as $n$ tends to infinity, the zeros of these sections are equidistributed outside $D$ with respect to a probability measure $ν$. This gives rise to a surprising forbidden set.

math.CV↗

arXiv2022

Generative Principal Component Analysis

In this paper, we study the problem of principal component analysis with generative modeling assumptions, adopting a general model for the observed matrix that encompasses notable special cases, including spiked matrix recovery and phase retrieval. The key assumption is that the underlying signal lies near the range of an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs. We propose a quadratic estimator, and show that it enjoys a statistical rate of order $\sqrt{\frac{k\log L}{m}}$, where $m$ is the number of samples. We also provide a near-matching algorithm-independent lower bound. Moreover, we provide a variant of the classic power method, which projects the calculated data onto the range of the generative model during each iteration. We show that under suitable conditions, this method converges exponentially fast to a point achieving the above-mentioned statistical rate. We perform experiments on various image datasets for spiked matrix and phase retrieval models, and illustrate performance gains of our method to the classic power method and the truncated power method devised for sparse principal component analysis.

stat.ML↗

arXiv2022

Signal Analysis via the Stochastic Geometry of Spectrogram Level Sets

Spectrograms are fundamental tools in time-frequency analysis, being the squared magnitude of the so-called short time Fourier transform (STFT). Signal analysis via spectrograms has traditionally explored their peaks, i.e. their maxima. This is complemented by a recent interest in their zeros or minima, following seminal work by Flandrin and others, which exploits connections with Gaussian analytic functions (GAFs). However, the zero sets (or extrema) of GAFs have a complicated stochastic structure, complicating any direct theoretical analysis. Standard techniques largely rely on statistical observables from the analysis of spatial data, whose distributional properties for spectrograms are mostly understood only at an empirical level. In this work, we investigate spectrogram analysis via an examination of the stochastic geometric properties of their level sets. We obtain rigorous theorems demonstrating the efficacy of a spectrogram level sets based approach to the detection and estimation of signals, framed in a concrete inferential set-up. Exploiting these ideas as theoretical underpinnings, we propose a level sets based algorithm for signal analysis that is intrinsic to given spectrogram data, and substantiate its effectiveness via extensive empirical studies. Our results also have theoretical implications for spectrogram zero based approaches to signal analysis. To our knowledge, these results are arguably among the first to provide a rigorous statistical understanding of signal detection and reconstruction in this set up, complemented with provable guarantees on detection thresholds and rates of convergence.

eess.SP↗

arXiv2022

Fluctuation and Entropy in Spectrally Constrained random fields

We investigate the statistical properties of translation invariant random fields (including point processes) on Euclidean spaces (or lattices) under constraints on their spectrum or structure function. An important class of models that motivate our study are hyperuniform and stealthy hyperuniform systems, which are characterised by the vanishing of the structure function at the origin (resp., vanishing in a neighbourhood of the origin). We show that many key features of two classical statistical mechanical measures of randomness - namely, fluctuations and entropy, are governed only by some particular local aspects of their structure function. We obtain exponents for the fluctuations of the local mass in domains of growing size, and show that spatial geometric considerations play an important role - both the shape of the domain and the mode of spectral decay. In doing so, we unveil intriguing oscillatory behaviour of spatial correlations of local masses in adjacent box domains. We describe very general conditions under which we show that the field of local masses exhibit Gaussian asymptotics, with an explicitly described limit. We further demonstrate that stealthy hyperuniform systems with joint densities exhibit degeneracy in their asymptotic entropy per site. In fact, our analysis shows that entropic degeneracy sets in under much milder conditions than stealthiness, as soon as the structure function fails to be logarithmically integrable.

math.PR↗

arXiv2022

Learning with latent group sparsity via heat flow dynamics on networks

Group or cluster structure on explanatory variables in machine learning problems is a very general phenomenon, which has attracted broad interest from practitioners and theoreticians alike. In this work we contribute an approach to learning under such group structure, that does not require prior information on the group identities. Our paradigm is motivated by the Laplacian geometry of an underlying network with a related community structure, and proceeds by directly incorporating this into a penalty that is effectively computed via a heat flow-based local network dynamics. In fact, we demonstrate a procedure to construct such a network based on the available data. Notably, we dispense with computationally intensive pre-processing involving clustering of variables, spectral or otherwise. Our technique is underpinned by rigorous theorems that guarantee its effective performance and provide bounds on its sample complexity. In particular, in a wide range of settings, it provably suffices to run the heat flow dynamics for time that is only logarithmic in the problem dimensions. We explore in detail the interfaces of our approach with key statistical physics models in network science, such as the Gaussian Free Field and the Stochastic Block Model. We validate our approach by successful applications to real-world data from a wide array of application domains, including computer science, genetics, climatology and economics. Our work raises the possibility of applying similar diffusion-based techniques to classical learning tasks, exploiting the interplay between geometric, dynamical and stochastic structures underlying the data.

stat.ME↗