arXiv subjects

Subhabrata Sen

Publications and source records attributed to Subhabrata Sen.

At least 37 records · Page 2Linked to original sources

arXiv2023

Spectral Universality of Regularized Linear Regression with Nearly Deterministic Sensing Matrices

It has been observed that the performances of many high-dimensional estimation problems are universal with respect to underlying sensing (or design) matrices. Specifically, matrices with markedly different constructions seem to achieve identical performance if they share the same spectral distribution and have ``generic'' singular vectors. We prove this universality phenomenon for the case of convex regularized least squares (RLS) estimators under a linear regression model with additive Gaussian noise. Our main contributions are two-fold: (1) We introduce a notion of universality classes for sensing matrices, defined through a set of deterministic conditions that fix the spectrum of the sensing matrix and precisely capture the previously heuristic notion of generic singular vectors; (2) We show that for all sensing matrices that lie in the same universality class, the dynamics of the proximal gradient descent algorithm for solving the regression problem, as well as the performance of RLS estimators themselves (under additional strong convexity conditions) are asymptotically identical. In addition to including i.i.d. Gaussian and rotational invariant matrices as special cases, our universality class also contains highly structured, strongly correlated, or even (nearly) deterministic matrices. Examples of the latter include randomly signed versions of incoherent tight frames and randomly subsampled Hadamard transforms. As a consequence of this universality principle, the asymptotic performance of regularized linear regression on many structured matrices constructed with limited randomness can be characterized by using the rotationally invariant ensemble as an equivalent yet mathematically more tractable surrogate.

cs.IT↗

arXiv2023

Universality of Approximate Message Passing with Semi-Random Matrices

Approximate Message Passing (AMP) is a class of iterative algorithms that have found applications in many problems in high-dimensional statistics and machine learning. In its general form, AMP can be formulated as an iterative procedure driven by a matrix $\mathbf{M}$. Theoretical analyses of AMP typically assume strong distributional properties on $\mathbf{M}$ such as $\mathbf{M}$ has i.i.d. sub-Gaussian entries or is drawn from a rotational invariant ensemble. However, numerical experiments suggest that the behavior of AMP is universal, as long as the eigenvectors of $\mathbf{M}$ are generic. In this paper, we take the first step in rigorously understanding this universality phenomenon. In particular, we investigate a class of memory-free AMP algorithms (proposed by Çakmak and Opper for mean-field Ising spin glasses), and show that their asymptotic dynamics is universal on a broad class of semi-random matrices. In addition to having the standard rotational invariant ensemble as a special case, the class of semi-random matrices that we define in this work also includes matrices constructed with very limited randomness. One such example is a randomly signed version of the Sine model, introduced by Marinari, Parisi, Potters, and Ritort for spin glasses with fully deterministic couplings.

math.PR↗

arXiv2022

TAP equations for orthogonally invariant spin glasses at high temperature

We study the high-temperature regime of a mean-field spin glass model whose couplings matrix is orthogonally invariant in law. The magnetization of this model is conjectured to satisfy a system of TAP equations, originally derived by Parisi and Potters using a diagrammatic expansion of the Gibbs free energy. We prove that this TAP description is correct in an $L^2$ sense, in a regime of sufficiently high temperature. Our approach develops a novel geometric argument for proving the convergence of an Approximate Message Passing (AMP) algorithm to the magnetization vector, which is applicable in models without i.i.d. couplings. This convergence is shown via a conditional second moment analysis of the free energy restricted to a thin band around the output of the AMP algorithm, in a system of many "orthogonal" replicas.

math.PR↗

arXiv2022

Random linear estimation with rotationally-invariant designs: Asymptotics at high temperature

We study estimation in the linear model $y=Aβ^\star+ε$, in a Bayesian setting where $β^\star$ has an entrywise i.i.d. prior and the design $A$ is rotationally-invariant in law. In the large system limit as dimension and sample size increase proportionally, a set of related conjectures have been postulated for the asymptotic mutual information, Bayes-optimal mean squared error, and TAP mean-field equations that characterize the Bayes posterior mean of $β^\star$. In this work, we prove these conjectures for a general class of signal priors and for arbitrary rotationally-invariant designs $A$, under a "high-temperature" condition that restricts the range of eigenvalues of $A^\top A$. Our proof uses a conditional second-moment method argument, where we condition on the iterates of a version of the Vector AMP algorithm for solving the TAP mean-field equations.

cs.IT↗

arXiv2022

A New Central Limit Theorem for the Augmented IPW Estimator: Variance Inflation, Cross-Fit Covariance and Beyond

Estimation of the average treatment effect (ATE) is a central problem in causal inference. In recent times, inference for the ATE in the presence of high-dimensional covariates has been extensively studied. Among the diverse approaches that have been proposed, augmented inverse probability weighting (AIPW) with cross-fitting has emerged a popular choice in practice. In this work, we study this cross-fit AIPW estimator under well-specified outcome regression and propensity score models in a high-dimensional regime where the number of features and samples are both large and comparable. Under assumptions on the covariate distribution, we establish a new central limit theorem for the suitably scaled cross-fit AIPW that applies without any sparsity assumptions on the underlying high-dimensional parameters. Our CLT uncovers two crucial phenomena among others: (i) the AIPW exhibits a substantial variance inflation that can be precisely quantified in terms of the signal-to-noise ratio and other problem parameters, (ii) the asymptotic covariance between the pre-cross-fit estimators is non-negligible even on the root-n scale. These findings are strikingly different from their classical counterparts. On the technical front, our work utilizes a novel interplay between three distinct tools--approximate message passing theory, the theory of deterministic equivalents, and the leave-one-out approach. We believe our proof techniques should be useful for analyzing other two-stage estimators in this high-dimensional regime. Finally, we complement our theoretical results with simulations that demonstrate both the finite sample efficacy of our CLT and its robustness to our assumptions.

math.ST↗

arXiv2022

High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models

We study Langevin dynamics for recovering the planted signal in the spiked matrix model. We provide a "path-wise" characterization of the overlap between the output of the Langevin algorithm and the planted signal. This overlap is characterized in terms of a self-consistent system of integro-differential equations, usually referred to as the Crisanti-Horner-Sommers-Cugliandolo-Kurchan (CHSCK) equations in the spin glass literature. As a second contribution, we derive an explicit formula for the limiting overlap in terms of the signal-to-noise ratio and the injected noise in the diffusion. This uncovers a sharp phase transition -- in one regime, the limiting overlap is strictly positive, while in the other, the injected noise overcomes the signal, and the limiting overlap is zero.

math.ST↗

arXiv2022

The TAP free energy for high-dimensional linear regression

We derive a variational representation for the log-normalizing constant of the posterior distribution in Bayesian linear regression with a uniform spherical prior and an i.i.d. Gaussian design. We work under the "proportional" asymptotic regime, where the number of observations and the number of features grow at a proportional rate. This rigorously establishes the Thouless-Anderson-Palmer (TAP) approximation arising from spin glass theory, and proves a conjecture of Krzakala et. al. (2014) in the special case of the spherical prior.

math.PR↗

arXiv2021

Regret Minimization in Isotonic, Heavy-Tailed Contextual Bandits via Adaptive Confidence Bands

In this paper we initiate a study of non parametric contextual bandits under shape constraints on the mean reward function. Specifically, we study a setting where the context is one dimensional, and the mean reward function is isotonic with respect to this context. We propose a policy for this problem and show that it attains minimax rate optimal regret. Moreover, we show that the same policy enjoys automatic adaptation; that is, for subclasses of the parameter space where the true mean reward functions are also piecewise constant with $k$ pieces, this policy remains minimax rate optimal simultaneously for all $k \geq 1.$ Automatic adaptation phenomena are well-known for shape constrained problems in the offline setting; %The phenomenon of automatic adaptation of shape constrained methods is known to occur in offline problems; we show that such phenomena carry over to the online setting. The main technical ingredient underlying our policy is a procedure to derive confidence bands for an underlying isotonic function using the isotonic quantile estimator. The confidence band we propose is valid under heavy tailed noise, and its average width goes to $0$ at an adaptively optimal rate. We consider this to be an independent contribution to the isotonic regression literature.

math.ST↗

arXiv2021

Variational Inference in high-dimensional linear regression

We study high-dimensional Bayesian linear regression with product priors. Using the nascent theory of non-linear large deviations (Chatterjee and Dembo,2016), we derive sufficient conditions for the leading-order correctness of the naive mean-field approximation to the log-normalizing constant of the posterior distribution. Subsequently, assuming a true linear model for the observed data, we derive a limiting infinite dimensional variational formula for the log normalizing constant of the posterior. Furthermore, we establish that under an additional "separation" condition, the variational problem has a unique optimizer, and this optimizer governs the probabilistic properties of the posterior distribution. We provide intuitive sufficient conditions for the validity of this "separation" condition. Finally, we illustrate our results on concrete examples with specific design matrices.

math.ST↗

arXiv2020

The Overlap Gap Property in Principal Submatrix Recovery

We study support recovery for a $k \times k$ principal submatrix with elevated mean $λ/N$, hidden in an $N\times N$ symmetric mean zero Gaussian matrix. Here $λ>0$ is a universal constant, and we assume $k = N ρ$ for some constant $ρ\in (0,1)$. We establish that {there exists a constant $C>0$ such that} the MLE recovers a constant proportion of the hidden submatrix if $λ{\geq C} \sqrt{\frac{1}ρ \log \frac{1}ρ}$, {while such recovery is information theoretically impossible if $λ= o( \sqrt{\frac{1}ρ \log \frac{1}ρ} )$}. The MLE is computationally intractable in general, and in fact, for $ρ>0$ sufficiently small, this problem is conjectured to exhibit a \emph{statistical-computational gap}. To provide rigorous evidence for this, we study the likelihood landscape for this problem, and establish that for some $\varepsilon>0$ and $\sqrt{\frac{1}ρ \log \frac{1}ρ } \ll λ\ll \frac{1}{ρ^{1/2 + \varepsilon}}$, the problem exhibits a variant of the \emph{Overlap-Gap-Property (OGP)}. As a direct consequence, we establish that a family of local MCMC based algorithms do not achieve optimal recovery. Finally, we establish that for $λ> 1/ρ$, a simple spectral method recovers a constant proportion of the hidden submatrix.

math.PR↗

arXiv2020

Large deviation for uniform graphs with given degrees

Consider the random graph sampled uniformly from the set of all simple graphs with a given degree sequence. Under mild conditions on the degrees, we establish a Large Deviation Principle (LDP) for these random graphs, viewed as elements of the graphon space. As a corollary of our result, we obtain LDPs for functionals continuous with respect to the cut metric, and obtain an asymptotic enumeration formula for graphs with given degrees, subject to an additional constraint on the value of a continuous functional. Our assumptions on the degrees are identical to those of Chatterjee, Diaconis and Sly (2011), who derived the almost sure graphon limit for these random graphs.

math.PR↗

arXiv2020

Contextual Stochastic Block Model: Sharp Thresholds and Contiguity

We study community detection in the contextual stochastic block model arXiv:1807.09596 [cs.SI], arXiv:1607.02675 [stat.ME]. In arXiv:1807.09596 [cs.SI], the second author studied this problem in the setting of sparse graphs with high-dimensional node-covariates. Using the non-rigorous cavity method from statistical physics, they conjectured the sharp limits for community detection in this setting. Further, the information theoretic threshold was verified, assuming that the average degree of the observed graph is large. It is expected that the conjecture holds as soon as the average degree exceeds one, so that the graph has a giant component. We establish this conjecture, and characterize the sharp threshold for detection and weak recovery.

cs.SI↗

arXiv2020

On the unbalanced cut problem and the generalized Sherrington-Kirkpatrick model

We establish a strict asymptotic inequality between a class of graph partition problems on the sparse End\H{o]s-Rényi and random regular graph ensembles with the same average degree. Along the way, we establish a variational representation for the ground state energy for generalized mixed $p$-spin glasses and derive strict comparison inequalities for such models as the alphabet changes.

math.PR↗

arXiv2020

On Minimax Exponents of Sparse Testing

We consider exact asymptotics of the minimax risk for global testing against sparse alternatives in the context of high dimensional linear regression. Our results characterize the leading order behavior of this minimax risk in several regimes, uncovering new phase transitions in its behavior. This complements a vast literature characterizing asymptotic consistency in this problem, and provides a useful benchmark, against which the performance of specific tests may be compared. Finally, we provide some preliminary evidence that popular sparsity adaptive procedures might be sub-optimal in terms of the minimax risk.

math.ST↗

arXiv2019

Testing Degree Corrections in Stochastic Block Models

We study sharp detection thresholds for degree corrections in Stochastic Block Models in the context of a goodness of fit problem, and explore the effect of the unknown community assignment (a high dimensional nuisance parameter) and the graph density on testing for degree corrections. When degree corrections are relatively dense, a simple test based on the total number of edges is asymptotically optimal. For sparse degree corrections, the results undergo several changes in behavior depending on density of the underlying Stochastic Block Model. For graphs which are not extremely sparse, optimal tests are based on Higher Criticism or Maximum Degree type tests based on a linear combination of within and across (estimated) community degrees. In the special case of balanced communities, a simple degree based Higher Criticism Test (Mukherjee, Mukherjee, Sen 2016) is optimal in case the graph is not completely dense, while the more complicated linear combination based procedure is required in the completely dense setting. The ``necessity" of the two step procedure is demonstrated for the case of balanced communities by the failure of the ordinary Maximum Degree Test in achieving sharp constants. Finally for extremely sparse graphs the optimal rates change, and a version of the maximum degree test with a different rejection region is shown to be optimal.

math.ST↗

arXiv2019

Limits of Sparse Configuration Models and Beyond: Graphexes and Multi-Graphexes

We investigate structural properties of large, sparse random graphs through the lens of "sampling convergence" (Borgs et. al. (2017)). Sampling convergence generalizes left convergence to sparse graphs, and describes the limit in terms of a "graphex". We introduce a notion of sampling convergence for sequences of multigraphs, and establish the graphex limit for the configuration model, a preferential attachment model, the generalized random graph, and a bipartite variant of the configuration model. The results for the configuration model, preferential attachment model and bipartite configuration model provide necessary and sufficient conditions for these random graph models to converge. The limit for the configuration model and the preferential attachment model is an augmented version of an exchangeable random graph model introduced by Caron and Fox (2017).

math.PR↗

arXiv2019

A correction to Kallenberg's theorem for jointly exchangeable random measures

Kallenberg (2005) provided a necessary and sufficient condition for the local finiteness of a jointly exchangeable random measure on $\R_+^2$. Here we note an additional condition that was missing in Kallenberg's theorem, but was implicitly used in the proof. We also provide a counter-example when the additional condition does not hold.

math.PR↗

arXiv2018

Contextual Stochastic Block Models

We provide the first information theoretic tight analysis for inference of latent community structure given a sparse graph along with high dimensional node covariates, correlated with the same latent communities. Our work bridges recent theoretical breakthroughs in the detection of latent community structure without nodes covariates and a large body of empirical work using diverse heuristics for combining node covariates with graphs for inference. The tightness of our analysis implies in particular, the information theoretical necessity of combining the different sources of information. Our analysis holds for networks of large degrees as well as for a Gaussian version of the model.

cs.SI↗