Related papers: The analytic computability of the Shannon transfor…
The notion of Schnorr randomness refers to computable reals or computable functions. We propose a version of Schnorr randomness for subcomputable classes and characterize it in different ways: by Martin L\"of tests, martingales or measure…
The open problem of calculating the limiting spectrum (or its Shannon transform) of increasingly large random Hermitian finite-band matrices is described. In general, these matrices include a finite number of non-zero diagonals around their…
This short course offers a new perspective on randomized algorithms for matrix computations. It explores the distinct ways in which probability can be used to design algorithms for numerical linear algebra. Each design template is…
The conventional rounding error analysis provides worst-case bounds with an associated failure probability and ignores the statistical property of the rounding errors. In this paper, we develop a new statistical rounding error analysis for…
A ternary ring of operators (TRO) in finite dimensions is a diagonal sum of spaces of rectangular matrices. TRO as operator space corresponds to quantum channels that are diagonal sums of partial traces, which we call TRO channels. TRO…
We define a class of "algebraic" random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner…
In this letter, we introduce the computational-limited (comp-limited) signals, a communication capacity regime in which the signal time computational complexity overhead is the key constraint -- rather than power or bandwidth -- to the…
The primary objective of quantum Shannon theory is to evaluate the capacity of quantum channels. In spite of the existence of rigorous coding theorems that quantify the transmission of information through quantum channels, superadditivity…
The generalization of Shannon's theory to include messages with given autocorrelations is presented. The analytical calculation of the channel capacity is based on the transfer matrix method of the effective 1D Hamiltonian. This bridge…
In 2010, Silva, Kschischang and K\"otter studied certain classes of finite field matrix channels in order to model random linear network coding where exactly $t$ random errors are introduced. In this paper we consider a generalisation of…
In the problem of channel resolvability, where a given output probability distribution via a channel is approximated by transforming the uniform random numbers, characterizing the asymptotically minimum rate of the size of the random…
Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all…
Quantum channels can be mathematically represented as completely positive trace-preserving maps that act on a density matrix. A general quantum channel can be written as a convex sum of `extremal' channels. We show that for an $N$-level…
We present a method for estimating conditionally Gaussian random vectors with random covariance matrices, which uses techniques from the field of machine learning. Such models are typical in communication systems, where the covariance…
The alternate row and column scaling algorithm applied to a positive $n\times n$ matrix $A$ converges to a doubly stochastic matrix $S(A)$, sometimes called the \emph{Sinkhorn limit} of $A$. For every positive integer $n$, a two parameter…
Analytic combinatorics in several variables is a branch of mathematics that deals with deriving the asymptotic behavior of combinatorial quantities by analyzing multivariate generating functions. We study information-theoretic questions…
The development of randomized algorithms for numerical linear algebra, e.g. for computing approximate QR and SVD factorizations, has recently become an intense area of research. This paper studies one of the most frequently discussed…
We consider two classes of computations which admit taking linear combinations of execution runs: probabilistic sampling and generalized animation. We argue that the task of program learning should be more tractable for these architectures…
The process of alternately row scaling and column scaling a positive $n \times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$ matrix $S(A)$, called the \emph{Sinkhorn limit} of $A$. Exact formulae for the Sinkhorn…
We determine the minimal experimental resources that ensure a unique solution in the estimation of trace-preserving quantum channels with both direct and convex optimization methods. A convenient parametrization of the constrained set is…