Related papers: On the computation of the M{\"o}bius transform
Feature attribution aims to explain the reasoning behind a black-box model's prediction by identifying the impact of each feature on the prediction. Recent work has extended feature attribution to interactions between multiple features.…
Efficient algorithms for computing linear convolutions based on the fast Fourier transform are developed. A hybrid approach is described that combines the conventional practice of explicit dealiasing (explicitly padding the input data with…
We develop a new algorithm to compute a basis for $M_k(\Gamma_0(N))$, the space of weight $k$ holomorphic modular forms on $\Gamma_0(N)$, in the case when the graded algebra of modular forms over $\Gamma_0(N)$ is generated at weight two.…
In this paper, we create a systematic and automatic procedure for transforming the integer factorization problem into the problem of solving a system of Boolean equations. Surprisingly, the resulting system of Boolean equations takes on a…
A celebrated theorem of Klein implies that any hypergeometric differential equation with algebraic solutions is a pull-back of one of the few standard hypergeometric equations with algebraic solutions. The most interesting cases are…
Boolean calculus has been studied extensively in the past in the context of switching circuits, error-correcting codes etc. This work generalizes several approaches to defining a differential calculus for Boolean functions. A unified theory…
Non-Abelian geometric phases form the foundation of fault-tolerant holonomic quantum computation. An "all-geometric" approach leveraging these phases enables robust unitary operations in condensed matter systems. Photonics, with rich…
We consider a generalization of polynomial programs: algebraic programs, which are optimization or feasibility problems with algebraic objectives or constraints. Algebraic functions are defined as zeros of multivariate polynomials. They are…
This paper develops upper and lower bounds for the probability of Boolean functions by treating multiple occurrences of variables as independent and assigning them new individual probabilities. We call this approach dissociation and give an…
In our setting enumeration amounts to generate all solutions of a problem instance without duplicates. We address the problem of enumerating the models of B-formulae. A B-formula is a propositional formula whose connectives are taken from a…
A polynomial transformation of the real plane $\Bbb R^2$ is a mapping $\Bbb R^2\to\Bbb R^2$ given by two polynomials of two variables. Such a transformation is called quadratic if the degrees of its polynomials are not greater than two. In…
Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…
We propose to interpret machine learning functions as physical observables, opening up the possibility to apply "standard" statistical-mechanical methods to outputs from neural networks. This includes histogram reweighting and finite-size…
We develop Boolean-valued domain theory and show how the lambda-calculus can be interpreted in using domain-valued random variables. We focus on the reflexive domain construction rather than the language and its semantics. The notion of…
The Matrix Bochner Problem aims to classify weight matrices $W$ such that the algebra $\mathcal D(W)$, of all differential operators that have a sequence of matrix-valued orthogonal polynomials for $W$ as eigenfunctions, contains a…
This thesis is intended to provide an account of the theory and applications of Operational Methods that allow the "translation" of the theory of special functions and polynomials into a "different" mathematical language. The language we…
This paper traces a straight line from classical M\"obius inversion to Hopf-algebraic perturbative renormalisation. This line, which is logical but not entirely historical, consists of just a few main abstraction steps, and some…
In the present paper we describe new heuristic technique, which can be applied to the optimization of pseudo-Boolean functions including Black-Box functions. This technique is based on a simple procedure which consists in transition from…
A space of entire functions of several complex variables rapidly decreasing on ${\mathbb R}^n$ and such that their growth along $i{\mathbb R}^n$ is majorized with the help of a family of weight functions is considered in this paper. For…
In this paper we develop a new approach to the design of direct numerical methods for multidimensional problems of the calculus of variations. The approach is based on a transformation of the problem with the use of a new class of…