We consider optimization using random projections as a statistical estimation problem, where the squared distance between the predictions from the estimator and the true solution is the error metric. In approximately solving a large-scale least squares problem using Gaussian sketches, we show that the sketched solution has a conditional Gaussian distribution with the true solution as its mean.

Estimating a binary vector from noisy linear measurements is a prototypical problem for MIMO systems. A popular algorithm, called the box-relaxation decoder, estimates the target signal by solving a least squares problem with convex constraints. This article shows that the performance of the algorithm, measured by the number of incorrectly-decoded bits, has a limiting Poisson law. This occurs when the sampling ratio and noise variance, two key parameters of the problem, follow certain scalings as the system dimension grows.

Given side information that an Ising tree-structured graphical model is homogeneous and has no external field, we derive the exact asymptotics of learning its structure from independently drawn samples. Our results, which leverage the use of probabilistic tools from the theory of strong large deviations, refine the large deviation (error exponents) results of Tan et al. (2011) and strictly improve those of Bresler and Karzand (2020). In addition, we extend our results to the scenario in which the samples are observed in random noise.

This work examines a novel question: how much randomness is needed to achieve local differential privacy (LDP)? A motivating scenario is providing multiple levels of privacy to multiple analysts, either for distribution or for heavy hitter estimation, using the same (randomized) output. We call this setting successive refinement of privacy, as it provides hierarchical access to the raw data with different privacy levels.

This work studies the robust subspace tracking (ST) problem. Robust ST can be simply understood as a (slow) time-varying subspace extension of robust PCA. It assumes that the true data lies in a low-dimensional subspace that is either fixed or changes slowly with time. The goal is to track the changing subspaces over time in the presence of additive sparse outliers and to do this quickly (with a short delay). We introduce a “fast” mini-batch robust ST solution that is provably correct under mild assumptions.

We consider rank-one symmetric tensor estimation when the tensor is corrupted by gaussian noise and the spike forming the tensor is a structured signal coming from a generalized linear model. The latter is a mathematically tractable model of a non-trivial hidden lower-dimensional latent structure in a signal. We work in a large dimensional regime with fixed ratio of signal-to-latent space dimensions.

This article studies a high-dimensional inference problem involving the matrix tensor product of random matrices. This problem generalizes a number of contemporary data science problems including the spiked matrix models used in sparse principal component analysis and covariance estimation and the stochastic block model used in network analysis.

We introduce a new general modeling approach for multivariate discrete event data with categorical interacting marks, which we refer to as marked Bernoulli processes. In the proposed model, the probability of an event of a specific category to occur in a location may be influenced by past events at this and other locations. We do not restrict interactions to be positive or decaying over time as it is commonly adopted, allowing us to capture an arbitrary shape of influence from historical events, locations, and events of different categories.

We study the minimax estimation of α-divergences between discrete distributions for integer α ≥ 1, which include the Kullback-Leibler divergence and the χ2-divergences as special examples. Dropping the usual theoretical tricks to acquire independence, we construct the first minimax rate-optimal estimator which does not require any Poissonization, sample splitting, or explicit construction of approximating polynomials.

We consider a finite-armed structured bandit problem in which mean rewards of different arms are known functions of a common hidden parameter 8*. Since we do not place any restrictions on these functions, the problem setting subsumes several previously studied frameworks that assume linear or invertible reward functions. We propose a novel approach to gradually estimate the hidden 8* and use the estimate together with the mean reward functions to substantially reduce exploration of sub-optimal arms.