Related papers: A complexity analysis of the F4 Gr\"obner basis al…
Algorithm- and data-dependent generalization bounds are required to explain the generalization behavior of modern machine learning algorithms. In this context, there exists information theoretic generalization bounds that involve (various…
In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides a rigorous framework for estimating the complexity of solving a system of polynomial equations via Groebner bases…
In this work, we extend modular techniques for computing Gr\"obner bases involving rational coefficients to (two-sided) ideals in free algebras. We show that the infinite nature of Gr\"obner bases in this setting renders the classical…
Degree heterogeneity and latent geometry, also referred to as popularity and similarity, are key explanatory components underlying the structure of real-world networks. The relationship between these components and the statistical…
This paper is the fourth and the last part of the series "Spherical higher order Fourier analysis over finite fields", aiming to develop the higher order Fourier analysis method along spheres over finite fields, and to solve the Geometric…
Quantifying the complexity of feed-forward neural networks (FFNNs) remains challenging due to their nonlinear, hierarchical structure and numerous parameters. We apply generalized degrees of freedom (GDF) to measure model complexity in…
The present article is a review of recent developments concerning the notion of F{\o}lner sequences both in operator theory and operator algebras. We also give a new direct proof that any essentially normal operator has an increasing…
We establish a lower bound concerning the computational complexity of Grover's algorithms on fractal networks. This bound provides general predictions for the quantum advantage gained for searching unstructured lists. It yields a…
Faugere's F5 algorithm is the fastest known algorithm to compute Groebner bases. It has a signature-based and an incremental structure that allow to apply the F5 criterion for deletion of unnecessary reductions. In this paper, we present an…
V.I. Arnold has recently defined the complexity of a sequence of $n$ zeros and ones with the help of the operator of finite differences. In this paper we describe the results obtained for almost most complicated sequences of elements of a…
We prove an effective version of the inverse theorem for the Gowers $U^3$-norm for functions supported on high-rank quadratic level sets in finite vector spaces. For configurations controlled by the $U^3$-norm (complexity-two…
Given a zero-dimensional ideal I in K[x1,...,xn] of degree D, the transformation of the ordering of its Groebner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving…
Given a computable sequence of natural numbers, it is a natural task to find a G\"odel number of a program that generates this sequence. It is easy to see that this problem is neither continuous nor computable. In algorithmic learning…
The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such…
Incremental gradient and incremental proximal methods are a fundamental class of optimization algorithms used for solving finite sum problems, broadly studied in the literature. Yet, without strong convexity, their convergence guarantees…
We give a criterion for a collection of polynomials to be a universal Gr\"{o}bner basis for an ideal in terms of the multidegree of the closure of the corresponding affine variety in $(\mathbb{P}^1)^N$. This criterion can be used to give…
We introduce a new complexity measure of a path of (problems, solutions) pairs in terms of the length of the path in the condition metric which we define in the article. The measure gives an upper bound for the number of Newton steps…
The Gr\"obner basis detection (GBD) is defined as follows: Given a set of polynomials, decide whether there exists -and if "yes" find- a term order such that the set of polynomials is a Gr\"obner basis. This problem was shown to be NP-hard…
A generalized criterion for signature-based algorithms to compute Gr\"obner bases is proposed in this paper. This criterion is named by "generalized criterion", because it can be specialized to almost all existing criteria for…
In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. Recently, a series of paper has been published for analysis of expansion complexity and for testing sequences in terms of this new…