Related papers: Explicit Solution Equation for Every Combinatorial…
Real-world decision-making systems are often subject to uncertainties that have to be resolved through observational data. Therefore, we are frequently confronted with combinatorial optimization problems of which the objective function is…
The authors proposed a general way to find particular solutions for overdetermined systems of PDEs previously, where the number of equations is greater than the number of unknown functions. In this paper, we propose an algorithm for finding…
Any satisfiability problem in conjunctive normal form can be solved in polynomial time by reducing it to a 3-sat formulation and transforming this to a Linear Complementarity problem (LCP) which is then solved as a linear program (LP). Any…
A fully tensorial theoretical framework for hypercomplex-valued neural networks is presented. The proposed approach enables neural network architectures to operate on data defined over arbitrary finite-dimensional algebras. The central…
The Matrix Torsion Problem (MTP) is: given a square matrix M with rational entries, decide whether two distinct powers of M are equal. It has been shown by Cassaigne and the author that the MTP reduces to the Matrix Power Problem (MPP) in…
We first investigate properties of M-tensor equations. In particular, we show that if the constant term of the equation is nonnegative, then finding a nonnegative solution of the equation can be done by finding a positive solution of a…
Determinant maximization provides an elegant generalization of problems in many areas, including convex geometry, statistics, machine learning, fair allocation of goods, and network design. In an instance of the determinant maximization…
This article introduces a pedagogical method for {\it solving combinatorial problems} that frequently involve structures that are unfamiliar or less familiar. Indeed, an indirect method has been proposed in order to evade any possible…
We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued, pairwise orthogonal vectors. Such decompositions do not generally exist, but we show that some symmetric tensor decomposition…
In this paper, we present two tensor network quantum-inspired algorithms to solve the knapsack and the shortest path problems, and enables to solve some of its variations. These methods provide an exact equation which returns the optimal…
A new unbiased Monte Carlo technique called Tensor Network Monte Carlo (TNMC) is introduced based on sampling all possible renormalizations (or course-grainings) of tensor networks, in this case matrix-product states. Tensor networks are a…
Combinatorial problems are formulated to find optimal designs within a fixed set of constraints. They are commonly found across diverse engineering and scientific domains. Understanding how to best use quantum computers for combinatorial…
Matrix Completion is the problem of recovering an unknown real-valued low-rank matrix from a subsample of its entries. Important recent results show that the problem can be solved efficiently under the assumption that the unknown matrix is…
Learning to optimize (L2O) has recently emerged as a promising approach to solving optimization problems by exploiting the strong prediction power of neural networks and offering lower runtime complexity than conventional solvers. While L2O…
Recently, neural heuristics based on deep reinforcement learning have exhibited promise in solving multi-objective combinatorial optimization problems (MOCOPs). However, they are still struggling to achieve high learning efficiency and…
We study the binary perceptron, a random constraint satisfaction problem that asks to find a Boolean vector in the intersection of independently chosen random halfspaces. A striking feature of this model is that at every positive constraint…
In physics, accurately modeling large-scale phenomena such as core-collapse supernovae, (CCSN), and neutron star mergers are computationally challenging and require solving large sets of partial and ordinary differential equations.…
The development of a satisfying and rigorous mathematical understanding of the performance of neural networks is a major challenge in artificial intelligence. Against this background, we study the expressive power of neural networks through…
We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the…
A numerical method for solving elliptic PDEs with variable coefficients on two-dimensional domains is presented. The method is based on high-order composite spectral approximations and is designed for problems with smooth solutions. The…