Related papers: Threshold rates for properties of random codes
An unknown vector f in R^n can be recovered from corrupted measurements y = Af + e where A^(m*n)(m>n) is the coding matrix if the unknown error vector e is sparse. We investigate the relationship of the fraction of errors and the recovering…
We consider the problem of erasure/list decoding using certain classes of simplified decoders. Specifically, we assume a class of erasure/list decoders, such that a codeword is in the list if its likelihood is larger than a threshold. This…
A graphon is said to have the $H$-property if a random undirected graph $G_n$ on $n$ nodes sampled from it has a node-wise disjoint cycle cover almost surely as $n\to\infty$. It has been shown in the earlier work that the $H$-property obeys…
We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in…
The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures finding optimal solutions by means…
The error exponent of the typical random code is defined as the asymptotic normalized expectation of the logarithm of the probability of error, as opposed to the traditional definition of the random coding exponent as the normalized…
A binary code with covering radius $R$ is a subset $C$ of the hypercube $Q_n=\{0,1\}^n$ such that every $x\in Q_n$ is within Hamming distance $R$ of some codeword $c\in C$, where $R$ is as small as possible. For a fixed coordinate…
The relationship between the quality of a string, as judged by a human reader, and its probability, $p(\boldsymbol{y})$ under a language model undergirds the development of better language models. For example, many popular algorithms for…
A matrix is given in ``shredded'' form if we are presented with the multiset of rows and the multiset of columns, but not told which row is which or which column is which. The matrix is reconstructible if it is uniquely determined by this…
In this work we give precise asymptotic expressions on the probability of the existence of fixed-size components at the threshold of connectivity for random geometric graphs.
The graph projection of a hypergraph is a simple graph with the same vertex set and with an edge between each pair of vertices that appear in a hyperedge. We consider the problem of reconstructing a random $d$-uniform hypergraph from its…
A code is locally recoverable when each symbol in one of its code words can be reconstructed as a function of $r$ other symbols. We use bundles of projective spaces over a line to construct locally recoverable codes with availability; that…
In this article we consider the graph alignment problem from the perspective of high-dimensional statistics: we aim to estimate an unknown permutation $\pi^*$ from the observation of two correlated random adjacency matrices $A_1$, $A_2$. We…
The problem of guessing a random string is revisited. A close relation between guessing and compression is first established. Then it is shown that if the sequence of distributions of the information spectrum satisfies the large deviation…
This paper considers '$\delta$-almost Reed-Muller codes', i.e., linear codes spanned by evaluations of all but a $\delta$ fraction of monomials of degree at most $d$. It is shown that for any $\delta > 0$ and any $\varepsilon>0$, there…
A Lagrange-dual (Gallager-style) lower bound is derived for the error exponent function of the typical random code (TRC) pertaining to the i.i.d. random coding ensemble and mismatched stochastic likelihood decoding. While the original…
Sampling is a fundamental problem in computer science and statistics. However, for a given task and stream, it is often not possible to choose good sampling probabilities in advance. We derive a general framework for adaptively changing the…
This article studies the achievable guarantees on the error rates of certain learning algorithms, with particular focus on refining logarithmic factors. Many of the results are based on a general technique for obtaining bounds on the error…
We consider limit probabilities of first order properties in random graphs with a given degree sequence. Under mild conditions on the degree sequence, we show that the closure set of limit probabilities is a finite union of closed…
We imagine an experiment on an unknown quantum mechanical system in which the system is prepared in various ways and a range of measurements are performed. For each measurement M and preparation rho the experimenter can determine, given…