Related papers: An Approach to Circuit Lower Bounds via Bounded Wi…
We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various…
For $S \subseteq \{0,1\}^n$ a Boolean function $f \colon S \to \{-1,1\}$ is a polynomial threshold function (PTF) of degree $d$ and weight $W$ if there is a polynomial $p$ with integer coefficients of degree $d$ and with sum of absolute…
$f,g_1,...,g_m$ be elements of the polynomial ring $\mathbb{R}[x_1,...,x_n]$. The paper deals with the general problem of computing a lower bound for $f$ on the subset of $\mathbb{R}^n$ defined by the inequalities $g_i\ge 0$, $i=1,...,m$.…
Estimating numerically the spectral radius of a random walk on a nonamenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral…
We continue to study the notion of cancellation-free linear circuits. We show that every matrix can be computed by a cancellation- free circuit, and almost all of these are at most a constant factor larger than the optimum linear circuit…
In this paper, free boundary geodesic networks whose length realize the first min-max width of the length functional are investigated. This functional acts on the space of relative flat 1-dimensional cycles modulo 2 in a compact surface…
Considering the problem of risk-sensitive parameter estimation, we propose a fairly wide family of lower bounds on the exponential moments of the quadratic error, both in the Bayesian and the non--Bayesian regime. This family of bounds,…
Probabilistic circuits (PCs) are a prominent representation of probability distributions with tractable inference. While parameter learning in PCs is rigorously studied, structure learning is often more based on heuristics than on…
The Boolean circuit simplification problem involves finding a smaller circuit that computes the same function as a given Boolean circuit. This problem is closely related to several key areas with both theoretical and practical applications,…
At each iteration of a Block Coordinate Descent method one minimizes an approximation of the objective function with respect to a generally small set of variables subject to constraints in which these variables are involved. The…
In recent years it has become popular to study dynamic problems in a sensitivity setting: Instead of allowing for an arbitrary sequence of updates, the sensitivity model only allows to apply batch updates of small size to the original input…
The core of a cooperative game on a set of players $N$ is one of the most popular concept of solution. When cooperation is restricted (feasible coalitions form a subcollection $\cF$ of $2^N$), the core may become unbounded, which makes it…
We present a general method for obtaining strong bounds for discrete optimization problems that is based on a concept of branching duality. It can be applied when no useful integer programming model is available, and we illustrate this with…
Quantum circuits are time dependent diagrams describing the process of quantum computation. Usually, a quantum algorithm must be mapped into a quantum circuit. Optimal synthesis of quantum circuits is intractable and heuristic methods must…
We study the replacement paths problem in the $\mathsf{CONGEST}$ model of distributed computing. Given an $s$-$t$ shortest path $P$, the goal is to compute, for every edge $e$ in $P$, the shortest-path distance from $s$ to $t$ avoiding $e$.…
The construction of quantum computers is based on the synthesis of low-cost quantum circuits. The quantum circuit of any Boolean function expressed in a Positive Polarity Reed-Muller $PPRM$ expansion can be synthesized using…
We consider the problem of testing whether an unknown Boolean function $f$ is monotone versus $\epsilon$-far from every monotone function. The two main results of this paper are a new lower bound and a new algorithm for this well-studied…
Reducing circuit depth and identifying an optimal trade-off between circuit depth and width is crucial for successful quantum computation. In this context, midcircuit measurement and feedforward have been shown to significantly reduce the…
We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the…
A Boolean function $f$ on $n$ variables is said to be a bent function if the absolute value of all its Walsh coefficients is $2^{n/2}$. Our main result is a new asymptotic lower bound on the number of Boolean bent functions. It is based on…