Related papers: A Comprehensive Study on Ziv-Zakai Lower Bounds on…
The Korkine-Zolotareff (KZ) reduction is one of the often used reduction strategies for lattice decoding. In this paper, we first investigate some important properties of KZ reduced matrices. Specifically, we present a linear upper bound on…
Given finite-dimensional random vectors $Y$, $X$, and $Z$ that form a Markov chain in that order (i.e., $Y \to X \to Z$), we derive upper bounds on the excess minimum risk using generalized information divergence measures. Here, $Y$ is a…
This paper provides a general technique for lower bounding the Bayes risk of statistical estimation, applicable to arbitrary loss functions and arbitrary prior distributions. A lower bound on the Bayes risk not only serves as a lower bound…
Zero-noise extrapolation (ZNE) reduces noise-induced bias but can increase sampling variance through Richardson coefficients and shot splitting. We define a finite-shot help-harm boundary: the lower local mean-squared-error crossing where…
We consider the problem of learning a dictionary matrix from a number of observed signals, which are assumed to be generated via a linear model with a common underlying dictionary. In particular, we derive lower bounds on the minimum…
This paper extends the standard chaining technique to prove excess risk upper bounds for empirical risk minimization with random design settings even if the magnitude of the noise and the estimates is unbounded. The bound applies to many…
This paper gives upper and lower bounds on the minimum error probability of Bayesian $M$-ary hypothesis testing in terms of the Arimoto-R\'enyi conditional entropy of an arbitrary order $\alpha$. The improved tightness of these bounds over…
The performance of multi-image alignment, bringing different images into one coordinate system, is critical in many applications with varied signal-to-noise ratio (SNR) conditions. A great amount of effort is being invested into developing…
We examine codes, over the additive Gaussian noise channel, designed for reliable communication at some specific signal-to-noise ratio (SNR) and constrained by the permitted minimum mean-square error (MMSE) at lower SNRs. The maximum…
We consider the least singular value of a large random matrix with real or complex i.i.d. Gaussian entries shifted by a constant $z\in\mathbb{C}$. We prove an optimal lower tail estimate on this singular value in the critical regime where…
Mrs. Gerber's Lemma lower bounds the entropy at the output of a binary symmetric channel in terms of the entropy of the input process. In this paper, we lower bound the output entropy via a different measure of input uncertainty, pertaining…
Consider the minimum mean-square error (MMSE) of estimating an arbitrary random variable from its observation contaminated by Gaussian noise. The MMSE can be regarded as a function of the signal-to-noise ratio (SNR) as well as a functional…
We study large deviation upper bounds and mean-squared error (MSE) guarantees of a general framework of nonlinear stochastic gradient methods in the online setting, in the presence of heavy-tailed noise. Unlike existing works that rely on…
We consider unbiased estimation of a sparse nonrandom vector corrupted by additive white Gaussian noise. We show that while there are infinitely many unbiased estimators for this problem, none of them has uniformly minimum variance.…
In this paper, we give a short Bayesian proof of Talagrand's celebrated majorizing-measure theorem (MMT). While the upper-bound direction of MMT follows relatively directly from standard arguments, the lower-bound direction is widely…
In this note, we consider Szemer\'{e}di's theorem on $k$-term arithmetic progressions over finite fields $\mathbb{F}_p^n$, where the allowed set $S$ of common differences in these progressions is chosen randomly of fixed size. Combining a…
We study stochastic nonconvex optimization under heavy-tailed noise. In this setting, the stochastic gradients only have bounded $p$-th central moment ($p$-BCM) for some $p \in (1,2]$. Building on the foundational work of Arjevani et al.…
This paper investigates the minimum mean square error (MMSE) estimation of x, given the observation y = Hx+n, when x and n are independent and Gaussian Mixture (GM) distributed. The introduction of GM distributions, represents a…
We obtain new outer bounds on the capacity regions of the two-user multiple access channel with generalized feedback (MAC-GF) and the two-user interference channel with generalized feedback (IC-GF). These outer bounds are based on the idea…
Linear Least Squares is a very well known technique for parameter estimation, which is used even when sub-optimal, because of its very low computational requirements and the fact that exact knowledge of the noise statistics is not required.…