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Finding a Hadamard matrix (H-matrix) among the set of all binary matrices of corresponding order is a hard problem, which potentially can be solved by quantum computing. We propose a method to formulate the Hamiltonian of finding H-matrix…
To determine if two lists of numbers are the same set, we sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms arise in graph isomorphism…
Simplification of expressions in computer algebra systems often involves a step known as "canonicalisation", which reduces equivalent expressions to the same form. However, such forms may not be natural from the perspective of a…
Canonical correlation analysis is a statistical technique that is used to find relations between two sets of variables. An important extension in pattern analysis is to consider more than two sets of variables. This problem can be expressed…
We give a procedure that can be used to automatically satisfy invariants of a certain shape. These invariants may be written with the operations intersection, composition and converse over binary relations, and equality over these…
We investigate quantum algorithms for classification, a fundamental problem in machine learning, with provable guarantees. Given $n$ $d$-dimensional data points, the state-of-the-art (and optimal) classical algorithm for training…
This paper develops an invariant--geometric interpretation of the canonization problem for simple undirected weighted graphs based on the {discrete moving frame method} for finite groups. We consider the action of the {pair group}…
In this paper we describe a parallel Gaussian elimination algorithm for matrices with entries in a finite field. Unlike previous approaches, our algorithm subdivides a very large input matrix into smaller submatrices by subdividing both…
Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute…
We propose a method for low-rank semidefinite programming in application to the semidefinite relaxation of unconstrained binary quadratic problems. The method improves an existing solution of the semidefinite programming relaxation to…
Let $R$ be a real closed field. We consider basic semi-algebraic sets defined by $n$-variate equations/inequalities of $s$ symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by $2d < n$. Such a…
In this paper, an exact algorithm in polynomial time is developed to solve unrestricted binary quadratic programs. The computational complexity is $O\left( n^{\frac{15}{2}}\right) $, although very conservative, it is sufficient to prove…
Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of…
Matrix completion is a classical problem in data science wherein one attempts to reconstruct a low-rank matrix while only observing some subset of the entries. Previous authors have phrased this problem as a nuclear norm minimization…
Classification with a large number of classes is a key problem in machine learning and corresponds to many real-world applications like tagging of images or textual documents in social networks. If one-vs-all methods usually reach top…
A semi-algebraic set is a subset of $\mathbb{R}^n$ defined by a finite collection of polynomial equations and inequalities. In this paper, we investigate the problem of determining whether two points in such a set belong to the same…
We propose a quantum algorithm to solve systems of nonlinear algebraic equations. In the ideal case the complexity of the algorithm is linear in the number of variables $n$, which means our algorithm's complexity is less than $O(n^{3})$ of…
We introduce a novel technique to analyse unambiguous B\"uchi automata quantitatively, and apply this to the model checking problem. It is based on linear-algebra arguments that originate from the analysis of matrix semigroups with constant…
Classification is a central task in deep learning algorithms. Usually, images are first captured and then processed by a sequence of operations, of which the artificial neuron represents one of the fundamental units. This paradigm requires…
Combinatorial Exploration is a new domain-agnostic algorithmic framework to automatically and rigorously study the structure of combinatorial objects and derive their counting sequences and generating functions. We describe how it works and…