Related papers: Smoothed analysis in compressed sensing
The recovery of signals that are sparse not in a basis, but rather sparse with respect to an over-complete dictionary is one of the most flexible settings in the field of compressed sensing with numerous applications. As in the standard…
We introduce the notion of a random relaxed asymptotic contraction in the setting of random normed modules. The contraction condition employs two quasi-metrics that are built directly from the random operator: a lower quasi-metric which…
Sparse signal recovery based on nonconvex and nonsmooth optimization problems has significant applications and demonstrates superior performance in signal processing and machine learning. This work deals with a scale-invariant…
We study the singular values and Lyapunov exponents of non-stationary random matrix products subject to small, absolutely continuous, additive noise. Consider a fixed sequence of matrices of bounded norm. Independently perturb the matrices…
We focus on \emph{row sampling} based approximations for matrix algorithms, in particular matrix multipication, sparse matrix reconstruction, and \math{\ell_2} regression. For \math{\matA\in\R^{m\times d}} (\math{m} points in \math{d\ll m}…
Sensitivity analysis for the unconfoundedness assumption is crucial in observational studies. For this purpose, the marginal sensitivity model (MSM) gained popularity recently due to its good interpretability and mathematical properties.…
Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and…
Compressed Sensing aims to capture attributes of a sparse signal using very few measurements. Cand\`{e}s and Tao showed that sparse reconstruction is possible if the sensing matrix acts as a near isometry on all $\boldsymbol{k}$-sparse…
We propose a general random subspace framework for unconstrained nonconvex optimization problems that requires a weak probabilistic assumption on the subspace gradient, which we show to be satisfied by various random matrix ensembles, such…
Sorted L-One Penalized Estimator (SLOPE) is a relatively new convex optimization procedure for selecting predictors in large data bases. Contrary to LASSO, SLOPE has been proved to be asymptotically minimax in the context of sparse…
This paper addresses the topic of robust Bayesian compressed sensing over finite fields. For stationary and ergodic sources, it provides asymptotic (with the size of the vector to estimate) necessary and sufficient conditions on the number…
Tuning hyperparameters, such as the stepsize, presents a major challenge of training machine learning models. To address this challenge, numerous adaptive optimization algorithms have been developed that achieve near-optimal complexities,…
We study the lower tail behavior of the least singular value of an $n\times n$ random matrix $M_n := M+N_n$, where $M$ is a fixed complex matrix with operator norm at most $\exp(n^{c})$ and $N_n$ is a random matrix, each of whose entries is…
Compressed sensing (CS) shows that a signal having a sparse or compressible representation can be recovered from a small set of linear measurements. In classical CS theory, the sampling matrix and representation matrix are assumed to be…
We study statistical inference for small-noise-perturbed multiscale dynamical systems. We prove consistency, asymptotic normality, and convergence of all scaled moments of an appropriately-constructed maximum likelihood estimator (MLE) for…
Meaningful comparison between sets of observations often necessitates alignment or registration between them, and the resulting optimization problems range in complexity from those admitting simple closed-form solutions to those requiring…
Estimation problems with constrained parameter spaces arise in various settings. In many of these problems, the observations available to the statistician can be modelled as arising from the noisy realization of the image of a random linear…
We consider the problem of reconstructing a sparse signal $x^0\in\R^n$ from a limited number of linear measurements. Given $m$ randomly selected samples of $U x^0$, where $U$ is an orthonormal matrix, we show that $\ell_1$ minimization…
We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure $\mu$ on the space of infinite bit sequences is ML absolutely continuous if the non-ML-random bit sequences form a null set with…
Consider the task of estimating a 3-order $n \times n \times n$ tensor from noisy observations of randomly chosen entries in the sparse regime. We introduce a similarity based collaborative filtering algorithm for estimating a tensor from…