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Subhroshekhar Ghosh

Publications and source records attributed to Subhroshekhar Ghosh.

28 records · Page 2Linked to original sources
arXiv2022

Fractal Gaussian Networks: A sparse random graph model based on Gaussian Multiplicative Chaos

We propose a novel stochastic network model, called Fractal Gaussian Network (FGN), that embodies well-defined and analytically tractable fractal structures. Such fractal structures have been empirically observed in diverse applications. FGNs interpolate continuously between the popular purely random geometric graphs (a.k.a. the Poisson Boolean network), and random graphs with increasingly fractal behavior. In fact, they form a parametric family of sparse random geometric graphs that are parametrized by a fractality parameter which governs the strength of the fractal structure. FGNs are driven by the latent spatial geometry of Gaussian Multiplicative Chaos (GMC), a canonical model of fractality in its own right. We asymptotically characterize the expected number of edges, triangles, cliques and hub-and-spoke motifs in FGNs, unveiling a distinct pattern in their scaling with the size parameter of the network. We then examine the natural question of detecting the presence of fractality and the problem of parameter estimation based on observed network data, in addition to fundamental properties of the FGN as a random graph model. We also explore fractality in community structures by unveiling a natural stochastic block model in the setting of FGNs. Finally, we substantiate our results with phenomenological analysis of the FGN in the context of available scientific literature for fractality in networks, including applications to real-world massive network data.

stat.ML↗
arXiv2021

Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors

Compressive phase retrieval is a popular variant of the standard compressive sensing problem in which the measurements only contain magnitude information. In this paper, motivated by recent advances in deep generative models, we provide recovery guarantees with near-optimal sample complexity for phase retrieval with generative priors. We first show that when using i.i.d. Gaussian measurements and an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs, roughly $O(k \log L)$ samples suffice to guarantee that any signal minimizing an amplitude-based empirical loss function is close to the true signal. Attaining this sample complexity with a practical algorithm remains a difficult challenge, and finding a good initialization for gradient-based methods has been observed to pose a major bottleneck. To partially address this, we further show that roughly $O(k \log L)$ samples ensure sufficient closeness between the underlying signal and any {\em globally optimal} solution to an optimization problem designed for spectral initialization (though finding such a solution may still be challenging). We also adapt this result to sparse phase retrieval, and show that $O(s \log n)$ samples are sufficient for a similar guarantee when the underlying signal is $s$-sparse and $n$-dimensional, matching an information-theoretic lower bound. While these guarantees do not directly correspond to a practical algorithm, we propose a practical spectral initialization method motivated by our findings, and experimentally observe performance gains over various existing spectral initialization methods for sparse phase retrieval.

stat.ML↗
arXiv2021

Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priors

In 1-bit compressive sensing, each measurement is quantized to a single bit, namely the sign of a linear function of an unknown vector, and the goal is to accurately recover the vector. While it is most popular to assume a standard Gaussian sensing matrix for 1-bit compressive sensing, using structured sensing matrices such as partial Gaussian circulant matrices is of significant practical importance due to their faster matrix operations. In this paper, we provide recovery guarantees for a correlation-based optimization algorithm for robust 1-bit compressive sensing with randomly signed partial Gaussian circulant matrices and generative models. Under suitable assumptions, we match guarantees that were previously only known to hold for i.i.d.~Gaussian matrices that require significantly more computation. We make use of a practical iterative algorithm, and perform numerical experiments on image datasets to corroborate our theoretical results.

cs.LG↗
arXiv2020

On a Variational Approximation based Empirical Likelihood ABC Method

Many scientifically well-motivated statistical models in natural, engineering, and environmental sciences are specified through a generative process. However, in some cases, it may not be possible to write down the likelihood for these models analytically. Approximate Bayesian computation (ABC) methods allow Bayesian inference in such situations. The procedures are nonetheless typically computationally intensive. Recently, computationally attractive empirical likelihood-based ABC methods have been suggested in the literature. All of these methods rely on the availability of several suitable analytically tractable estimating equations, and this is sometimes problematic. We propose an easy-to-use empirical likelihood ABC method in this article. First, by using a variational approximation argument as a motivation, we show that the target log-posterior can be approximated as a sum of an expected joint log-likelihood and the differential entropy of the data generating density. The expected log-likelihood is then estimated by an empirical likelihood where the only inputs required are a choice of summary statistic, it's observed value, and the ability to simulate the chosen summary statistics for any parameter value under the model. The differential entropy is estimated from the simulated summaries using traditional methods. Posterior consistency is established for the method, and we discuss the bounds for the required number of simulated summaries in detail. The performance of the proposed method is explored in various examples.

stat.ME↗
arXiv2020

Learning from DPPs via Sampling: Beyond HKPV and symmetry

Determinantal point processes (DPPs) have become a significant tool for recommendation systems, feature selection, or summary extraction, harnessing the intrinsic ability of these probabilistic models to facilitate sample diversity. The ability to sample from DPPs is paramount to the empirical investigation of these models. Most exact samplers are variants of a spectral meta-algorithm due to Hough, Krishnapur, Peres and Virág (henceforth HKPV), which is in general time and resource intensive. For DPPs with symmetric kernels, scalable HKPV samplers have been proposed that either first downsample the ground set of items, or force the kernel to be low-rank, using e.g. Nyström-type decompositions. In the present work, we contribute a radically different approach than HKPV. Exploiting the fact that many statistical and learning objectives can be effectively accomplished by only sampling certain key observables of a DPP (so-called linear statistics), we invoke an expression for the Laplace transform of such an observable as a single determinant, which holds in complete generality. Combining traditional low-rank approximation techniques with Laplace inversion algorithms from numerical analysis, we show how to directly approximate the distribution function of a linear statistic of a DPP. This distribution function can then be used in hypothesis testing or to actually sample the linear statistic, as per requirement. Our approach is scalable and applies to very general DPPs, beyond traditional symmetric kernels.

stat.ML↗
arXiv2020

Transmission and navigation on disordered lattice networks, directed spanning forests and Brownian web

Stochastic networks based on random point sets as nodes have attracted considerable interest in many applications, particularly in communication networks, including wireless sensor networks, peer-to-peer networks and so on. The study of such networks generally requires the nodes to be independently and uniformly distributed as a Poisson point process. In this work, we venture beyond this standard paradigm and investigate the stochastic geometry of networks obtained from \textit{directed spanning forests} (DSF) based on randomly perturbed lattices, which have desirable statistical properties as a models of spatially dependent point fields. In the regime of low disorder, we show in 2D and 3D that the DSF almost surely consists of a single tree. In 2D, we further establish that the DSF, as a collection of paths, converges under diffusive scaling to the Brownian web.

math.PR↗
arXiv2018

Gaussian complex zeros on the hole event: the emergence of a forbidden region

We consider the Gaussian Entire Function (GEF) whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the kth coefficient is 1/k!. This random Taylor series is distinguished by the invariance of its zero set with respect to the isometries of the complex plane. We show that the law of the zero set, conditioned on the GEF having no zeros in a disk of radius r, and properly normalized, converges to an explicit limiting Radon measure in the plane, as r goes to infinity. A remarkable feature of this limiting measure is the existence of a large 'forbidden region' between a singular part supported on the boundary of the (scaled) hole and the equilibrium measure far from the hole.

math.CV↗
arXiv2016

Multivariate CLT follows from strong Rayleigh property

Let $(X_1 , \ldots , X_d)$ be random variables taking nonnegative integer values and let $f(z_1, \ldots , z_d)$ be the probability generating function. Suppose that $f$ is real stable; equivalently, suppose that the polarization of this probability distribution is strong Rayleigh. In specific examples, such as occupation counts of disjoint sets by a determinantal point process, it is known~\cite{soshnikov02} that the joint distribution must approach a multivariate Gaussian distribution. We show that this conclusion follows already from stability of $f$.

math.PR↗
arXiv2005

Quantum algorithm to distinguish Boolean functions of different weights

We exploit Grover operator of database search algorithm for weight decision algorithm. In this research, weight decision problem is to find an exact weight w from given two weights as w1 and w2 where w1+w2=1 and 0 2. Hence the quantum algorithm achieves at least quadratic speedup.

quant-ph↗