Related papers: Optimal bounds for single-source Kolmogorov extrac…

In this paper we give a definition for quantum Kolmogorov complexity. In the classical setting, the Kolmogorov complexity of a string is the length of the shortest program that can produce this string as its output. It is a measure of the…

A recolouring sequence, between $k$-colourings $\alpha$ and $\beta$ of a graph $G$, transforms $\alpha$ into $\beta$ by recolouring one vertex at a time, such that after each recolouring step we again have a proper $k$-colouring of $G$. The…

Although information content is invariant up to an additive constant, the range of possible additive constants applicable to programming languages is so large that in practice it plays a major role in the actual evaluation of K(s), the…

Rank 1 inhomogeneous random graphs are a natural generalization of Erd\H{o}s R\'enyi random graphs. In this generalization each node is given a weight. Then the probability that an edge is present depends on the product of the weights of…

We study the rate of convergence of the empirical spectral distribution of products of independent non-Hermitian random matrices to the power of the Circular Law. The distance to the deterministic limit distribution will be measured in…

Maximum satisfiability is a canonical NP-hard optimization problem that appears empirically hard for random instances. Let us say that a Conjunctive normal form (CNF) formula consisting of $k$-clauses is $p$-satisfiable if there exists a…

We consider the maximum coding rate achievable by uniformly-random codes for the deletion channel. We prove an upper bound that's within 0.1 of the best known lower bounds for all values of the deletion probability $d,$ and much closer for…

A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum $\lambda_{\min}$ of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower…

Entropy rate of sequential data-streams naturally quantifies the complexity of the generative process. Thus entropy rate fluctuations could be used as a tool to recognize dynamical perturbations in signal sources, and could potentially be…

We suggest necessary conditions of soficness of multidimensional shifts formulated in termsof resource-bounded Kolmogorov complexity. Using this technique we provide examples ofeffective and non-sofic shifts on $\mathbb{Z}^2$ with very low…

Let $\{s_{ij}:i,j=1,2,...\}$ consist of i.i.d. random variables in $\mathbb{C}$ with $\mathsf{E}s_{11}=0$, $\mathsf{E}|s_{11}|^2=1$. For each positive integer $N$, let $\mathbf{s}_k={\mathbf{s}}_k(N)=(s_{1k},s_{2k},...,s_{Nk})^T$, $1\leq…

We revisit, beyond the uniform case, some aspects of the convergence of the cumulative shape of the RSK Young diagrams associated with random words, obtaining rates of convergence in Kolmogorov's distance. Since the length of the top row of…

Let $\Sigma$ be a countable alphabet. For $r\geq 1$, an infinite sequence $s$ with characters from $\Sigma$ is called $r$-quasi-regular, if for each $\sigma\in\Sigma$ the ratio of the longest to shortest interval between consecutive…

This paper investigates the problem of variable-length lossy source coding allowing a positive excess distortion probability and an overflow probability of codeword lengths. Novel one-shot achievability and converse bounds of the optimal…

We focus on the problem of minimizing a convex function $f$ over a convex set $S$ given $T$ queries to a stochastic first order oracle. We argue that the complexity of convex minimization is only determined by the rate of growth of the…

The set of all error--correcting block codes over a fixed alphabet with $q$ letters determines a recursively enumerable set of rational points in the unit square with coordinates $(R,\delta)$:= (relative transmission rate, relative minimal…

We obtain sharp estimates on the growth rate of stable commutator length on random (geodesic) words, and on random walks, in hyperbolic groups and groups acting nondegenerately on hyperbolic spaces. In either case, we show that with high…

This paper extends a previous bound of order $O(n^{-1})$ of the authors (arXiv:1405.7820[math.PR]), for the rate of convergence in Kolmogorov distance of the expected spectral distribution of a Wigner random matrix ensemble to the…

We determine, up to a multiplicative constant, the optimal number of random edges that need to be added to a $k$-graph $H$ with minimum vertex degree $\Omega(n^{k-1})$ to ensure an $F$-factor with high probability, for any $F$ that belongs…

Random reshuffling with momentum (RRM) corresponds to the SGD optimizer with momentum option enabled, as found in many machine learning libraries like PyTorch and TensorFlow. Despite its widespread use, the convergence properties of RRM do…