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We consider the problem of estimating the joint distribution of $n$ independent random variables. Our approach is based on a family of candidate probabilities that we shall call a model and which is chosen to either contain the true…
Generative models aim to learn the distribution of datasets, such as images, so as to be able to generate samples that statistically resemble real data. However, learning the underlying probability distribution can be very challenging and…
In this paper, we study the distribution of the cokernels of random $p$-adic matrices with fixed zero entries. Let $X_n$ be a random $n \times n$ matrix over $\mathbb{Z}_p$ in which some entries are fixed to be zero and the other entries…
Christiano et al. (2022) define a *heuristic estimator* to be a hypothetical algorithm that estimates the values of mathematical expressions from arguments. In brief, a heuristic estimator $\mathbb{G}$ takes as input a mathematical…
A method to generate new classes of random matrix ensembles is proposed. Random matrices from these ensembles are Lax matrices of classically integrable systems with a certain distribution of momenta and coordinates. The existence of an…
We study the averaging-based distributed optimization solvers over random networks. We show a general result on the convergence of such schemes using weight-matrices that are row-stochastic almost surely and column-stochastic in expectation…
In this paper we show that the expected generalisation performance of a learning machine is determined by the distribution of risks or equivalently its logarithm -- a quantity we term the risk entropy -- and the fluctuations in a quantity…
We analyze complex networks under random matrix theory framework. Particularly, we show that $\Delta_3$ statistic, which gives information about the long range correlations among eigenvalues, provides a qualitative measure of randomness in…
We consider distributed estimation of the inverse covariance matrix, also called the concentration or precision matrix, in Gaussian graphical models. Traditional centralized estimation often requires global inference of the covariance…
The theorem of Furstenberg and Kesten provides a strong law of large numbers for the norm of a product of random matrices. This can be extended under various assumptions, covering nonnegative as well as invertible matrices, to a law of…
We extend to the matrix setting a recent result of Srivastava-Vershynin about estimating the covariance matrix of a random vector. The result can be in- terpreted as a quantified version of the law of large numbers for positive…
Motivated by questions in algebraic geometry, Yifeng Huang recently derived generating functions for counting mutually annihilating matrices and mutually annihilating nilpotent matrices over a finite field. We give a different derivation of…
We describe a novel algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm is based on Metropolis Monte Carlo sampling. The algorithm samples from a stretched Boltzmann…
The naive Nystrom extension forms a low-rank approximation to a positive-semidefinite matrix by uniformly randomly sampling from its columns. This paper provides the first relative-error bound on the spectral norm error incurred in this…
This paper describes a flexible framework for generalized low-rank tensor estimation problems that includes many important instances arising from applications in computational imaging, genomics, and network analysis. The proposed estimator…
We study a class of Markov chains that model the evolution of a quantum system subject to repeated measurements. Each Markov chain in this class is defined by a measure on the space of matrices. It is then given by a random product of…
We consider the problem of the estimation of a high-dimensional probability distribution from i.i.d. samples of the distribution using model classes of functions in tree-based tensor formats, a particular case of tensor networks associated…
Chaos reveals a fundamental paradox in the scientific understanding of Complex Systems. Although chaotic models may be mathematically deterministic, they are practically non-determinable due to the finite precision, which is inherent in all…
Herding is a deterministic algorithm used to generate data points that can be regarded as random samples satisfying input moment conditions. The algorithm is based on the complex behavior of a high-dimensional dynamical system and is…
The general limit distributions of the sum of random variables described by a finite matrix product ansatz are characterized. Using a mapping to a Hidden Markov Chain formalism, non-standard limit distributions are obtained, and related to…