Revisit Sparse Polynomial Interpolation based on Randomized Kronecker Substitution

In this paper, a new reduction based interpolation algorithm for black-box multivariate polynomials over finite fields is given. The method is based on two main ingredients. A new Monte Carlo method is given to reduce black-box multivariate polynomial interpolation to black-box univariate polynomial interpolation over any ring. The reduction algorithm leads to multivariate interpolation algorithms with better or the same complexities most cases when combining with various univariate interpolation algorithms. We also propose a modified univariate Ben-or and Tiwarri algorithm over the finite field, which has better total complexity than the Lagrange interpolation algorithm. Combining our reduction method and the modified univariate Ben-or and Tiwarri algorithm, we give a Monte Carlo multivariate interpolation algorithm, which has better total complexity in most cases for sparse interpolation of black-box polynomial over finite fields

Sparse Polynomial Interpolation and Testing

Interpolation is the process of learning an unknown polynomial f from some set of its evaluations. We consider the interpolation of a sparse polynomial, i.e., where f is comprised of a small, bounded number of terms. Sparse interpolation dates back to work in the late 18th century by the French mathematician Gaspard de Prony, and was revitalized in the 1980s due to advancements by Ben-Or and Tiwari, Blahut, and Zippel, amongst others. Sparse interpolation has applications to learning theory, signal processing, error-correcting codes, and symbolic computation. Closely related to sparse interpolation are two decision problems. Sparse polynomial identity testing is the problem of testing whether a sparse polynomial f is zero from its evaluations. Sparsity testing is the problem of testing whether f is in fact sparse. We present effective probabilistic algebraic algorithms for the interpolation and testing of sparse polynomials. These algorithms assume black-box evaluation access, whereby the algorithm may specify the evaluation points. We measure algorithmic costs with respect to the number and types of queries to a black-box oracle. Building on previous work by Garg–Schost and Giesbrecht–Roche, we present two methods for the interpolation of a sparse polynomial modelled by a straight-line program (SLP): a sequence of arithmetic instructions. We present probabilistic algorithms for the sparse interpolation of an SLP, with cost softly-linear in the sparsity of the interpolant: its number of nonzero terms. As an application of these techniques, we give a multiplication algorithm for sparse polynomials, with cost that is sensitive to the size of the output. Multivariate interpolation reduces to univariate interpolation by way of Kronecker substitu- tion, which maps an n-variate polynomial f to a univariate image with degree exponential in n. We present an alternative method of randomized Kronecker substitutions, whereby one can more efficiently reconstruct a sparse interpolant f from multiple univariate images of considerably reduced degree. In error-correcting interpolation, we suppose that some bounded number of evaluations may be erroneous. We present an algorithm for error-correcting interpolation of polynomials that are sparse under the Chebyshev basis. In addition we give a method which reduces sparse Chebyshev-basis interpolation to monomial-basis interpolation. Lastly, we study the class of Boolean functions that admit a sparse Fourier representation. We give an analysis of Levin’s Sparse Fourier Transform algorithm for such functions. Moreover, we give a new algorithm for testing whether a Boolean function is Fourier-sparse. This method reduces sparsity testing to homomorphism testing, which in turn may be solved by the Blum–Luby–Rubinfeld linearity test

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Independence in Algebraic Complexity Theory

This thesis examines the concepts of linear and algebraic independence in algebraic complexity theory. Arithmetic circuits, computing multivariate polynomials over a field, form the framework of our complexity considerations. We are concerned with polynomial identity testing (PIT), the problem of deciding whether a given arithmetic circuit computes the zero polynomial. There are efficient randomized algorithms known for this problem, but as yet deterministic polynomial-time algorithms could be found only for restricted circuit classes. We are especially interested in blackbox algorithms, which do not inspect the given circuit, but solely evaluate it at some points. Known approaches to the PIT problem are based on the notions of linear independence and rank of vector subspaces of the polynomial ring. We generalize those methods to algebraic independence and transcendence degree of subalgebras of the polynomial ring. Thereby, we obtain efficient blackbox PIT algorithms for new circuit classes. The Jacobian criterion constitutes an efficient characterization for algebraic independence of polynomials. However, this criterion is valid only in characteristic zero. We deduce a novel Jacobian-like criterion for algebraic independence of polynomials over finite fields. We apply it to obtain another blackbox PIT algorithm and to improve the complexity of testing the algebraic independence of arithmetic circuits over finite fields.Die vorliegende Arbeit untersucht die Konzepte der linearen und algebraischen Unabhängigkeit innerhalb der algebraischen Komplexitätstheorie. Arithmetische Schaltkreise, die multivariate Polynome über einem Körper berechnen, bilden die Grundlage unserer Komplexitätsbetrachtungen. Wir befassen uns mit dem polynomial identity testing (PIT) Problem, bei dem entschieden werden soll ob ein gegebener Schaltkreis das Nullpolynom berechnet. Für dieses Problem sind effiziente randomisierte Algorithmen bekannt, aber deterministische Polynomialzeitalgorithmen konnten bisher nur für eingeschränkte Klassen von Schaltkreisen angegeben werden. Besonders von Interesse sind Blackbox-Algorithmen, welche den gegebenen Schaltkreis nicht inspizieren, sondern lediglich an Punkten auswerten. Bekannte Ansätze für das PIT Problem basieren auf den Begriffen der linearen Unabhängigkeit und des Rangs von Untervektorräumen des Polynomrings. Wir übertragen diese Methoden auf algebraische Unabhängigkeit und den Transzendenzgrad von Unteralgebren des Polynomrings. Dadurch erhalten wir effiziente Blackbox-PIT-Algorithmen für neue Klassen von Schaltkreisen. Eine effiziente Charakterisierung der algebraischen Unabhängigkeit von Polynomen ist durch das Jacobi-Kriterium gegeben. Dieses Kriterium ist jedoch nur in Charakteristik Null gültig. Wir leiten ein neues Jacobi-artiges Kriterium für die algebraische Unabhängigkeit von Polynomen über endlichen Körpern her. Dieses liefert einen weiteren Blackbox-PIT-Algorithmus und verbessert die Komplexität des Problems arithmetische Schaltkreise über endlichen Körpern auf algebraische Unabhängigkeit zu testen

Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials

International audienceSparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function

Algorithmic Contributions to the Theory of Regular Chains

Regular chains, introduced about twenty years ago, have emerged as one of the major tools for solving polynomial systems symbolically. In this thesis, we focus on different algorithmic aspects of the theory of regular chains, from theoretical questions to high- performance implementation issues. The inclusion test for saturated ideals is a fundamental problem in this theory. By studying the primitivity of regular chains, we show that a regular chain generates its saturated ideal if and only if it is primitive. As a result, a family of inclusion tests can be detected very efficiently. The algorithm to compute the regular GCDs of two polynomials modulo a regular chain is one of the key routines in the various triangular decomposition algorithms. By revisiting relations between subresultants and GCDs, we proposed a novel bottom-up algorithm for this task, which improves the previous algorithm in a significant manner and creates opportunities for parallel execution. This thesis also discusses the accelerations towards fast Fourier transform (FFT) over finite fields and FFT based subresultant chain constructions in the context of massively parallel GPU architectures, which speedup our algorithms by several orders of magnitude