Related papers: Computational Barriers to Estimation from Low-Degr…

The generalized phase retrieval problem over compact groups aims to recover a set of matrices -- representing an unknown signal -- from their associated Gram matrices. This framework generalizes the classical phase retrieval problem, which…

We consider the high-dimensional inference problem where the signal is a low-rank symmetric matrix which is corrupted by an additive Gaussian noise. Given a probabilistic model for the low-rank matrix, we compute the limit in the large…

This paper establishes a statistical versus computational trade-off for solving a basic high-dimensional machine learning problem via a basic convex relaxation method. Specifically, we consider the {\em Sparse Principal Component Analysis}…

Consider the setting where a $\rho$-sparse Rademacher vector is planted in a random $d$-dimensional subspace of $R^n$. A classical question is how to recover this planted vector given a random basis in this subspace. A recent result by…

Low-rank matrices play a fundamental role in modeling and computational methods for signal processing and machine learning. In many applications where low-rank matrices arise, these matrices cannot be fully sampled or directly observed, and…

In the undersampled phase retrieval problem, the goal is to recover an $N$-dimensional complex signal $\mathbf{x}$ from only $M<N$ noisy intensity measurements without phase information. This problem has drawn a lot of attention to reduce…

Many data-analysis problems involve large dense matrices that describe the covariance of stationary noise processes; the computational cost of inverting these matrices, or equivalently of solving linear systems that contain them, is often a…

In this paper, we study the problem of multivariate shuffled linear regression, where the correspondence between predictors and responses in a linear model is obfuscated by a latent permutation. Specifically, we investigate the model…

In many high-dimensional problems,polynomial-time algorithms fall short of achieving the statistical limits attainable without computational constraints. A powerful approach to probe the limits of polynomial-time algorithms is to study the…

There is a growing body of work on proving hardness results for average-case estimation problems by bounding the low-degree advantage (LDA) - a quantitative estimate of the closeness of low-degree moments - between a null distribution and a…

Subspace clustering is the problem of clustering data that lie close to a union of linear subspaces. In the abstract form of the problem, where no noise or other corruptions are present, the data are assumed to lie in general position…

The detection of weak and rare effects in large amounts of data arises in a number of modern data analysis problems. Known results show that in this situation the potential of statistical inference is severely limited by the large-scale…

Low-rank matrix recovery problems involving high-dimensional and heterogeneous data appear in applications throughout statistics and machine learning. The contribution of this paper is to establish the fundamental limits of recovery for a…

The discovering of low-dimensional manifolds in high-dimensional data is one of the main goals in manifold learning. We propose a new approach to identify the effective dimension (intrinsic dimension) of low-dimensional manifolds. The scale…

Suppose we are given an $n$-dimensional order-3 symmetric tensor $T \in (\mathbb{R}^n)^{\otimes 3}$ that is the sum of $r$ random rank-1 terms. The problem of recovering the rank-1 components is possible in principle when $r \lesssim n^2$…

In this article, we develop methods for estimating a low rank tensor from noisy observations on a subset of its entries to achieve both statistical and computational efficiencies. There have been a lot of recent interests in this problem of…

For subspace recovery, most existing low-rank representation (LRR) models performs in the original space in single-layer mode. As such, the deep hierarchical information cannot be learned, which may result in inaccurate recoveries for…

Unsupervised dimensionality reduction is one of the commonly used techniques in the field of high dimensional data recognition problems. The deep autoencoder network which constrains the weights to be non-negative, can learn a low…

We introduce the broad subclass of algebraic compressed sensing problems, where structured signals are modeled either explicitly or implicitly via polynomials. This includes, for instance, low-rank matrix and tensor recovery. We employ…

Clustering analysis is one of the most widely used statistical tools in many emerging areas such as microarray data analysis. For microarray and other high-dimensional data, the presence of many noise variables may mask underlying…