arXiv subjects

Sohail Bahmani

Publications and source records attributed to Sohail Bahmani.

At least 19 recordsLinked to original sources

arXiv2025

A Fundamental Accuracy--Robustness Trade-off in Regression and Classification

We derive a fundamental trade-off between standard and adversarial risk in a rather general situation that formalizes the following simple intuition: "If no (nearly) optimal predictor is smooth, adversarial robustness comes at the cost of accuracy." As a concrete example, we evaluate the derived trade-off in regression with polynomial ridge functions under mild regularity conditions. Generalizing our analysis of this example, we formulate a necessary condition under which adversarial robustness can be achieved without significant degradation of the accuracy. This necessary condition is expressed in terms of a quantity that resembles the Poincaré constant of the data distribution.

stat.ML↗

arXiv2025

Variational Tail Bounds for Norms of Random Vectors and Matrices

We propose a variational tail bound for norms of random vectors and matrices under moment assumptions on their one-dimensional marginals. A simplified version of the bound that parametrizes the ``aggregating distribution'' using a certain pushforward of the Gaussian distribution is also provided. We apply the proposed method to reproduce some of the well-known bounds on norms of Gaussian random vectors, and also obtain dimension-free tail bounds for the Euclidean norm of random vectors with arbitrary moment profiles. Furthermore, we reproduce a dimension-free concentration inequality for sum of independent and identically distributed positive semidefinite matrices with sub-exponential marginals, and obtain a concentration inequality for the sample covariance matrix of sub-exponential random vectors. We also obtain a tail bound for the operator norm of a random matrix series whose random coefficients may have arbitrary moment profiles. Furthermore, we use coupling to formulate an abstraction of the proposed approach that applies more broadly. As a corollary, we derive a PAC-Bayesian-style bound in terms of a certain combination of the KL and R\'{e}nyi divergences between the prior and posterior distributions.

math.PR↗

arXiv2024

Max-Linear Regression by Convex Programming

We consider the multivariate max-linear regression problem where the model parameters $\boldsymbolβ_{1},\dotsc,\boldsymbolβ_{k}\in\mathbb{R}^{p}$ need to be estimated from $n$ independent samples of the (noisy) observations $y = \max_{1\leq j \leq k} \boldsymbolβ_{j}^{\mathsf{T}} \boldsymbol{x} + \mathrm{noise}$. The max-linear model vastly generalizes the conventional linear model, and it can approximate any convex function to an arbitrary accuracy when the number of linear models $k$ is large enough. However, the inherent nonlinearity of the max-linear model renders the estimation of the regression parameters computationally challenging. Particularly, no estimator based on convex programming is known in the literature. We formulate and analyze a scalable convex program given by anchored regression (AR) as the estimator for the max-linear regression problem. Under the standard Gaussian observation setting, we present a non-asymptotic performance guarantee showing that the convex program recovers the parameters with high probability. When the $k$ linear components are equally likely to achieve the maximum, our result shows a sufficient number of noise-free observations for exact recovery scales as {$k^{4}p$} up to a logarithmic factor. { This sample complexity coincides with that by alternating minimization (Ghosh et al., {2021}). Moreover, the same sample complexity applies when the observations are corrupted with arbitrary deterministic noise. We provide empirical results that show that our method performs as our theoretical result predicts, and is competitive with the alternating minimization algorithm particularly in presence of multiplicative Bernoulli noise. Furthermore, we also show empirically that a recursive application of AR can significantly improve the estimation accuracy.}

stat.ML↗

arXiv2022

Instance-dependent uniform tail bounds for empirical processes

We formulate a uniform tail bound for empirical processes indexed by a class of functions, in terms of the individual deviations of the functions rather than the worst-case deviation in the considered class. The tail bound is established by introducing an initial ``deflation'' step to the standard generic chaining argument. The resulting tail bound is the sum of the complexity of the ``deflated function class'' in terms of a generalization of Talagrand's $\gamma$ functional, and the deviation of the function instance, both of which are formulated based on the natural seminorm induced by the corresponding Cram\'{e}r functions. Leveraging another less demanding natural seminorm, we also show similar bounds, though with implicit dependence on the sample size, in the more general case where finite exponential moments cannot be assumed. We also provide approximations of the tail bounds in terms of the more prevalent Orlicz norms or their ``incomplete'' versions under suitable moment conditions.

math.PR↗

arXiv2021

Decentralized Feature-Distributed Optimization for Generalized Linear Models

We consider the "all-for-one" decentralized learning problem for generalized linear models. The features of each sample are partitioned among several collaborating agents in a connected network, but only one agent observes the response variables. To solve the regularized empirical risk minimization in this distributed setting, we apply the Chambolle--Pock primal--dual algorithm to an equivalent saddle-point formulation of the problem. The primal and dual iterations are either in closed-form or reduce to coordinate-wise minimization of scalar convex functions. We establish convergence rates for the empirical risk minimization under two different assumptions on the loss function (Lipschitz and square root Lipschitz), and show how they depend on the characteristics of the design matrix and the Laplacian of the network.

math.OC↗

arXiv2021

Low-Rank Matrix Estimation From Rank-One Projections by Unlifted Convex Optimization

We study an estimator with a convex formulation for recovery of low-rank matrices from rank-one projections. Using initial estimates of the factors of the target $d_1\times d_2$ matrix of rank-$r$, the estimator admits a practical subgradient method operating in a space of dimension $r(d_1+d_2)$. This property makes the estimator significantly more scalable than the convex estimators based on lifting and semidefinite programming. Furthermore, we present a streamlined analysis for exact recovery under the real Gaussian measurement model, as well as the partially derandomized measurement model by using the spherical $t$-design. We show that under both models the estimator succeeds, with high probability, if the number of measurements exceeds $r^2 (d_1+d_2)$ up to some logarithmic factors. This sample complexity improves on the existing results for nonconvex iterative algorithms.

math.ST↗

arXiv2020

Nearly Optimal Robust Mean Estimation via Empirical Characteristic Function

We propose an estimator for the mean of random variables in separable real Banach spaces using the empirical characteristic function. Assuming that the covariance operator of the random variable is bounded in a precise sense, we show that the proposed estimator achieves the optimal sub-Gaussian rate, except for a faster decaying mean-dependent term. Under a mild condition, an iterative refinement procedure can essentially eliminate the mean-dependent term and provide a shift-equivariant estimate. Our results particularly suggests that a certain Gaussian width that appears in the best known rate in the literature might not be necessary. Furthermore, using the boundedness of the characteristic functions, we also show that, except possibly at high signal-to-noise ratios, the proposed estimator is gracefully robust against adversarial "contamination". Our analysis is overall concise and transparent, thanks to the tractability of the characteristic functions.

math.ST↗

arXiv2020

Phase Retrieval of Low-Rank Matrices by Anchored Regression

We study the low-rank phase retrieval problem, where we try to recover a $d_1\times d_2$ low-rank matrix from a series of phaseless linear measurements. This is a fourth-order inverse problem, as we are trying to recover factors of matrix that have been put through a quadratic nonlinearity after being multiplied together. We propose a solution to this problem using the recently introduced technique of anchored regression. This approach uses two different types of convex relaxations: we replace the quadratic equality constraints for the phaseless measurements by a search over a polytope, and enforce the rank constraint through nuclear norm regularization. The result is a convex program that works in the space of $d_1 \times d_2$ matrices. We analyze two specific scenarios. In the first, the target matrix is rank-$1$, and the observations are structured to correspond to a phaseless blind deconvolution. In the second, the target matrix has general rank, and we observe the magnitudes of the inner products against a series of independent Gaussian random matrices. In each of these problems, we show that the anchored regression returns an accurate estimate from a near-optimal number of measurements given that we have access to an anchor matrix of sufficient quality. We also show how to create such an anchor in the phaseless blind deconvolution problem, again from an optimal number of measurements, and present a partial result in this direction for the general rank problem.

cs.IT↗

arXiv2019

Convex Programming for Estimation in Nonlinear Recurrent Models

We propose a formulation for nonlinear recurrent models that includes simple parametric models of recurrent neural networks as a special case. The proposed formulation leads to a natural estimator in the form of a convex program. We provide a sample complexity for this estimator in the case of stable dynamics, where the nonlinear recursion has a certain contraction property, and under certain regularity conditions on the input distribution. We evaluate the performance of the estimator by simulation on synthetic data. These numerical experiments also suggest the extent at which the imposed theoretical assumptions may be relaxed.

stat.ML↗

arXiv2019

Estimation from Non-Linear Observations via Convex Programming with Application to Bilinear Regression

We propose a computationally efficient estimator, formulated as a convex program, for a broad class of non-linear regression problems that involve difference of convex (DC) non-linearities. The proposed method can be viewed as a significant extension of the "anchored regression" method formulated and analyzed in [10] for regression with convex non-linearities. Our main assumption, in addition to other mild statistical and computational assumptions, is availability of a certain approximation oracle for the average of the gradients of the observation functions at a ground truth. Under this assumption and using a PAC-Bayesian analysis we show that the proposed estimator produces an accurate estimate with high probability. As a concrete example, we study the proposed framework in the bilinear regression problem with Gaussian factors and quantify a sufficient sample complexity for exact recovery. Furthermore, we describe a computationally tractable scheme that provably produces the required approximation oracle in the considered bilinear regression problem.

stat.ML↗

arXiv2018

Solving Equations of Random Convex Functions via Anchored Regression

We consider the question of estimating a solution to a system of equations that involve convex nonlinearities, a problem that is common in machine learning and signal processing. Because of these nonlinearities, conventional estimators based on empirical risk minimization generally involve solving a non-convex optimization program. We propose anchored regression, a new approach based on convex programming that amounts to maximizing a linear functional (perhaps augmented by a regularizer) over a convex set. The proposed convex program is formulated in the natural space of the problem, and avoids the introduction of auxiliary variables, making it computationally favorable. Working in the native space also provides great flexibility as structural priors (e.g., sparsity) can be seamlessly incorporated. For our analysis, we model the equations as being drawn from a fixed set according to a probability law. Our main results provide guarantees on the accuracy of the estimator in terms of the number of equations we are solving, the amount of noise present, a measure of statistical complexity of the random equations, and the geometry of the regularizer at the true solution. We also provide recipes for constructing the anchor vector (that determines the linear functional to maximize) directly from the observed data.

cs.LG↗

arXiv2017

Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation

We propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with "lifting" and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techniques for phase retrieval. We relax the quadratic equations for phaseless measurements to inequality constraints each of which representing a symmetric "slab". Through a simple convex program, our proposed estimator finds an extreme point of the intersection of these slabs that is best aligned with a given anchor vector. We characterize geometric conditions that certify success of the proposed estimator. Furthermore, using classic results in statistical learning theory, we show that for random measurements the geometric certificates hold with high probability at an optimal sample complexity. Phase transition of our estimator is evaluated through simulations. Our numerical experiments also suggest that the proposed method can solve phase retrieval problems with coded diffraction measurements as well.

cs.IT↗

arXiv2017

Algebraic Connectivity Under Site Percolation in Finite Weighted Graphs

We study the behavior of algebraic connectivity in a weighted graph that is subject to site percolation, random deletion of the vertices. Using a refined concentration inequality for random matrices we show in our main theorem that the (augmented) Laplacian of the percolated graph concentrates around its expectation. This concentration bound then provides a lower bound on the algebraic connectivity of the percolated graph. As a special case for $(n,d,λ)$-graphs (i.e., $d$-regular graphs on $n$ vertices with non-trivial eigenvalues less than $λ$ in magnitude) our result shows that, with high probability, the graph remains connected under a homogeneous site percolation with survival probability $p\ge 1-C_{1}n^{-C_{2}/d}$ with $C_{1}$ and $C_{2}$ depending only on $λ/d$.

math.PR↗

arXiv2016

Near-Optimal Estimation of Simultaneously Sparse and Low-Rank Matrices from Nested Linear Measurements

In this paper we consider the problem of estimating simultaneously low-rank and row-wise sparse matrices from nested linear measurements where the linear operator consists of the product of a linear operator $\mathcal{W}$ and a matrix $\mathbf{\varPsi}$. Leveraging the nested structure of the measurement operator, we propose a computationally efficient two-stage algorithm for estimating the simultaneously structured target matrix. Assuming that $\mathcal{W}$ is a restricted isometry for low-rank matrices and $\mathbf{\varPsi}$ is a restricted isometry for row-wise sparse matrices, we establish an accuracy guarantee that holds uniformly for all sufficiently low-rank and row-wise sparse matrices with high probability. Furthermore, using standard tools from information theory, we establish a minimax lower bound for estimation of simultaneously low-rank and row-wise sparse matrices from linear measurements that need not be nested. The accuracy bounds established for the algorithm, that also serve as a minimax upper bound, differ from the derived minimax lower bound merely by a polylogarithmic factor of the dimensions. Therefore, the proposed algorithm is nearly minimax optimal. We also discuss some applications of the proposed observation model and evaluate our algorithm through numerical simulation.

math.ST↗

arXiv2016

Learning Model-Based Sparsity via Projected Gradient Descent

Several convex formulation methods have been proposed previously for statistical estimation with structured sparsity as the prior. These methods often require a carefully tuned regularization parameter, often a cumbersome or heuristic exercise. Furthermore, the estimate that these methods produce might not belong to the desired sparsity model, albeit accurately approximating the true parameter. Therefore, greedy-type algorithms could often be more desirable in estimating structured-sparse parameters. So far, these greedy methods have mostly focused on linear statistical models. In this paper we study the projected gradient descent with non-convex structured-sparse parameter model as the constraint set. Should the cost function have a Stable Model-Restricted Hessian the algorithm produces an approximation for the desired minimizer. As an example we elaborate on application of the main results to estimation in Generalized Linear Model.

stat.ML↗

arXiv2015

Sketching for Simultaneously Sparse and Low-Rank Covariance Matrices

We introduce a technique for estimating a structured covariance matrix from observations of a random vector which have been sketched. Each observed random vector $\boldsymbol{x}_t$ is reduced to a single number by taking its inner product against one of a number of pre-selected vector $\boldsymbol{a}_\ell$. These observations are used to form estimates of linear observations of the covariance matrix $\boldsymbol{\varSigma}$, which is assumed to be simultaneously sparse and low-rank. We show that if the sketching vectors $\boldsymbol{a}_\ell$ have a special structure, then we can use straightforward two-stage algorithm that exploits this structure. We show that the estimate is accurate when the number of sketches is proportional to the maximum of the rank times the number of significant rows/columns of $\boldsymbol{\varSigma}$. Moreover, our algorithm takes direct advantage of the low-rank structure of $\boldsymbol{\varSigma}$ by only manipulating matrices that are far smaller than the original covariance matrix.

cs.IT↗

arXiv2015

Efficient Compressive Phase Retrieval with Constrained Sensing Vectors

We propose a robust and efficient approach to the problem of compressive phase retrieval in which the goal is to reconstruct a sparse vector from the magnitude of a number of its linear measurements. The proposed framework relies on constrained sensing vectors and a two-stage reconstruction method that consists of two standard convex programs that are solved sequentially. In recent years, various methods are proposed for compressive phase retrieval, but they have suboptimal sample complexity or lack robustness guarantees. The main obstacle has been that there is no straightforward convex relaxations for the type of structure in the target. Given a set of underdetermined measurements, there is a standard framework for recovering a sparse matrix, and a standard framework for recovering a low-rank matrix. However, a general, efficient method for recovering a jointly sparse and low-rank matrix has remained elusive. Deviating from the models with generic measurements, in this paper we show that if the sensing vectors are chosen at random from an incoherent subspace, then the low-rank and sparse structures of the target signal can be effectively decoupled. We show that a recovery algorithm that consists of a low-rank recovery stage followed by a sparse recovery stage will produce an accurate estimate of the target when the number of measurements is $\mathsf{O}(k\,\log\frac{d}{k})$, where $k$ and $d$ denote the sparsity level and the dimension of the input signal. We also evaluate the algorithm through numerical simulation.

cs.IT↗

arXiv2014

Lifting for Blind Deconvolution in Random Mask Imaging: Identifiability and Convex Relaxation

In this paper we analyze the blind deconvolution of an image and an unknown blur in a coded imaging system. The measurements consist of subsampled convolution of an unknown blurring kernel with multiple random binary modulations (coded masks) of the image. To perform the deconvolution, we consider a standard lifting of the image and the blurring kernel that transforms the measurements into a set of linear equations of the matrix formed by their outer product. Any rank-one solution to this system of equation provides a valid pair of an image and a blur. We first express the necessary and sufficient conditions for the uniqueness of a rank-one solution under some additional assumptions (uniform subsampling and no limit on the number of coded masks). These conditions are special case of a previously established result regarding identifiability in the matrix completion problem. We also characterize a low-dimensional subspace model for the blur kernel that is sufficient to guarantee identifiability, including the interesting instance of "bandpass"` blur kernels. Next, assuming the bandpass model for the blur kernel, we show that the image and the blur kernel can be found using nuclear norm minimization. Our main results show that recovery is achieved (with high probability) when the number of masks is on the order of $\mu\log^{2}L\,\log\frac{Le}{\mu}\,\log\log\left(N+1\right)$ where $\mu$ is the \emph{coherence} of the blur, $L$ is the dimension of the image, and $N$ is the number of measured samples per mask.

cs.IT↗