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For a lattice/linear code, we define the Voronoi spherical cumulative density function (CDF) as the CDF of the $\ell _{2}$ -norm/Hamming weight of a random vector uniformly distributed over the Voronoi cell. Using the first moment method together with a simple application of Jensen’s inequality, we develop lower bounds on the expected Voronoi spherical CDF of a random lattice/linear code. Our bounds are valid for any finite dimension and are quite close to a ball-based lower bound. They immediately translate to new non-asymptotic upper bounds on the normalized second moment and the error probability of a random lattice over the additive white Gaussian noise channel, as well as new non-asymptotic upper bounds on the Hamming distortion and the error probability of a random linear code over the binary symmetric channel. In particular, we show that for most lattices in $\mathbb {R}^{n}$ the second moment is greater than that of a Euclidean ball with the same covolume only by a $\left ({{1+O\left ({{\frac {1}{n}}}\right)}}\right)$ multiplicative factor. Similarly, for most linear codes in $\mathbb {F}_{2}^{n}$ the expected Hamming distortion is greater than that of a corresponding Hamming ball only by an additive universal constant.

Recent work in machine learning community proposed multiple methods for performing lossy compression (quantization) of large matrices. This quantization is important for accelerating matrix multiplication (main component of large language models), which is often bottlenecked by the speed of loading these matrices from memory. Unlike classical vector quantization and rate-distortion theory, the goal of these new compression algorithms is to be able to approximate not the matrices themselves, but their matrix product. Specifically, given a pair of real matrices $A,B$ an encoder (compressor) is applied to each of them independently producing descriptions with $R$ bits per entry. These representations subsequently are used by the decoder to estimate matrix product $A^{\top } B$ . In this work, we provide a non-asymptotic lower bound on the mean squared error of this approximation (as a function of rate $R$ ) for the case of matrices $A,B$ with iid Gaussian entries. Algorithmically, we construct a universal quantizer based on nested lattices with an explicit guarantee of approximation error for any (non-random) pair of matrices $A$ , $B$ in terms of only Frobenius norms $\|\bar {A}\|_{F}, \|\bar {B}\|_{F}$ and $\|\bar {A}^{\top } \bar {B}\|_{F}$ , where $\bar {A},\bar {B}$ are versions of $A,B$ with zero-centered columns, respectively. For iid Gaussian matrices our quantizer achieves the lower bound and is, thus, asymptotically optimal. A practical low-complexity version of our quantizer achieves performance quite close to optimal. In addition, we derive rate-distortion function for matrix multiplication of iid Gaussian matrices, which exhibits an interesting phase-transition at $R\approx 0.906$ bit/entry, showing necessity of Johnson-Lindestrauss dimensionality reduction (sketching) in the low-rate regime.

The problem of statistical inference in its various forms has been the subject of decades-long extensive research. Most of the effort has been focused on characterizing the behavior as a function of the number of available samples, with far less attention given to the effect of memory limitations on performance. Recently, this latter topic has drawn much interest in the engineering and computer science literature. In this survey paper, we attempt to review the state-of-the-art of statistical inference under memory constraints in several canonical problems, including hypothesis testing, parameter estimation, and distribution property testing/estimation. We discuss the main results in this developing field, and by identifying recurrent themes, we extract some fundamental building blocks for algorithmic construction, as well as useful techniques for lower bound derivations.