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We consider the problem of packing congruent circles with the maximum radius in a unit square as a mathematical optimization problem. Due to the presence of non-overlapping constraints, this problem is a notoriously difficult nonconvex…
We consider uniform random permutations of length $n$ conditioned to have no cycle longer than $n^\beta$ with $0<\beta<1$, in the limit of large $n$. Since in unconstrained uniform random permutations most of the indices are in cycles of…
The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of…
Symmetry is one of the most central concepts in physics, and it is no surprise that it has also been widely adopted as an inductive bias for machine-learning models applied to the physical sciences. This is especially true for models…
In model selection problems for machine learning, the desire for a well-performing model with meaningful structure is typically expressed through a regularized optimization problem. In many scenarios, however, the meaningful structure is…
When analyzing parametric statistical models, a useful approach consists in modeling geometrically the parameter space. However, even for very simple and commonly used hierarchical models like statistical mixtures or stochastic deep neural…
Discretizations of infinite-dimensional variational inequalities lead to linear and nonlinear complementarity problems with many degrees of freedom. To solve these problems in a parallel computing environment, we propose two active-set…
We use the fact that certain cosets of the stabilizer of points are pairwise conjugate in a symmetric group $S_n$ in order to construct recurrence relations for enumerating certain subsets of $S_n$. Occasionally one can find `closed form'…
Cyclic block coordinate methods are a fundamental class of optimization methods widely used in practice and implemented as part of standard software packages for statistical learning. Nevertheless, their convergence is generally not well…
The property of cyclicity of a linear operator, or equivalently the property of simplicity of its spectrum, is an important spectral characteristic that appears in many problems of functional analysis and applications to mathematical…
This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of…
We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof…
We solve the Symmetrized Principal Minor Assignment Problem, that is we show how to determine if for a given vector $v\in \mathbb{C}^{n}$ there is an $n\times n$ matrix that has all $i\times i$ principal minors equal to $v_{i}$. We use a…
The isomorphism problem means to decide if two given finite-dimensional simple algebras over the same centre are isomorphic and, if so, to construct an isomorphism between them. A solution to this problem has applications in computational…
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…
Variational inequalities are an important tool, which includes minimization, saddles, games, fixed-point problems. Modern large-scale and computationally expensive practical applications make distributed methods for solving these problems…
When writing a constraint program, we have to choose which variables should be the decision variables, and how to represent the constraints on these variables. In many cases, there is considerable choice for the decision variables.…
We study the problem of compression for the purpose of similarity identification, where similarity is measured by the mean square Euclidean distance between vectors. While the asymptotical fundamental limits of the problem - the minimal…
We provide a solution to the problem of simultaneous $diagonalization$ $via$ $congruence$ of a given set of $m$ complex symmetric $n\times n$ matrices $\{A_{1},\ldots,A_{m}\}$, by showing that it can be reduced to a possibly…
The variational quantum-classical algorithms are the most promising approach for achieving quantum advantage on near-term quantum simulators. Among these methods, the variational quantum eigensolver has attracted a lot of attention in…