Stochastic approximation algorithms are iterative procedures which are used to approximate a target value in an environment where the target is unknown and direct observations are corrupted by noise. These algorithms are useful, for instance, for root-finding and function minimization when the target function or model is not directly known. Originally introduced in a 1951 paper by Robbins and Monro, the field of Stochastic approximation has grown enormously and has come to influence application domains from adaptive signal processing to artificial intelligence. As an example, the Stochastic Gradient Descent algorithm which is ubiquitous in various subdomains of Machine Learning is based on stochastic approximation theory. In this paper, we give a formal proof (in the Coq proof assistant) of a general convergence theorem due to Aryeh Dvoretzky [Dvoretzky, 1956] (proven in 1956) which implies the convergence of important classical methods such as the Robbins-Monro and the Kiefer-Wolfowitz algorithms. In the process, we build a comprehensive Coq library of measure-theoretic probability theory and stochastic processes.

Reinforcement learning algorithms solve sequential decision-making problems in probabilistic environments by optimizing for long-term reward. The desire to use reinforcement learning in safety-critical settings inspires a recent line of work on formally constrained reinforcement learning; however, these methods place the implementation of the learning algorithm in their Trusted Computing Base. The crucial correctness property of these implementations is a guarantee that the learning algorithm converges to an optimal policy.

Multi-bit screening is an extension of binary screening, in which every pixel in continuous-tone image can be rendered to one among multiple absorptance levels. Many multi-bit screen algorithms face the problem of contouring artifacts due to sudden changes in the majority absorptance level between gray levels. In this paper, we have extended the direct binary search to the multi-bit case where at every pixel the algorithm chooses the best drop absorptance level to create a visually pleasing halftone pattern without any user defined guidance. This is repeated throughout the entire range of gray levels to create a high quality multi-bit screen.

We illustrate how employing Graphics Processing Units (GPU) can speed-up intensive image processing operations. In particular, we demonstrate the use of the NVIDIA CUDA architecture to implement a color digital binary halftoning algorithm based on Direct Binary Search (DBS). Halftoning a color image is more computationally expensive than the single color case as there is a need to minimize dot interaction between different color planes as well. We propose processing all color planes in parallel. In addition we employ processing several non-overlapping neighborhoods in parallel, by utilizing the GPU's parallel architecture, to further improve the computational efficiency. This parallel approach allows us to use a large neighborhood and filter size, to achieve the highest halftone quality, while having minimal impact on performance.

Programmable Graphics Processing Unit (GPU) has emerged as a powerful parallel processing architecture for various applications requiring a large amount of CPU cycles. In this paper, we study the feasibility for using this architecture for image halftoning, in particular implementing computationally intensive neighborhood halftoning algorithms such as error diffusion and Direct Binary Search (DBS). We show that it is possible to deliver very high performance even for high speed printers.

We study the problem of the computation of the radical of an ideal of polynomials with coefficients over fields of arbitrary characteristic. We show how to use Seidenberg's condition P to solve this problem in the case of positive characteristic.

\Ve discuss the use of a single generic linear combination of multiplication matrices, and its reordered Schur factorization, to find the roots of a system of multivariate polynomial equations.The principal contribution of this paper is to show how to reduce thr multivariate problem to a univariate problem, even in the raw of multiple roots, in a numerically stable wav,

After giving a proposition which reduces the problem of computing the integral closure of a general noetherian ring to the three problems: Compute a universal denominator d (element in the conductor). Compute radical of the ideal generated by d. Compute ideal quotients. We show that for the common case of affine domains, i.e. domains which are finitely generated over fields, of characteristic zero, we can use an effective localization in order to perform most of the computation in one dimensional rings where it can be done with linear algebra.

This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coefficients are inexact or imperfectly known. We first give an algorithm for computing univariate GCD's which gives exact results for interesting nearby problems, and give efficient algorithms for computing precisely how nearby. We generalize this to multivariate GCD computation. Next, we adapt Lazard's u-resultant algorithm for the solution of overdetermined systems of polynomial equations to the inexact-coefficient case. We also briefly discuss an application of the modied Lazard's method to the location of singular points on approximately known projections of algebraic curves.

Article Free Access Share on How to make AXIOM into a scratchpad Authors: Richard D. Jenks IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NYView Profile , Barry M. Trager IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NYView Profile Authors Info & Claims ISSAC '94: Proceedings of the international symposium on Symbolic and algebraic computationAugust 1994 Pages 32–40https://doi.org/10.1145/190347.190357Published:01 August 1994Publication History 3citation264DownloadsMetricsTotal Citations3Total Downloads264Last 12 Months6Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

This paper explains how Scratchpad solves the problem of presenting a categorical view of factorization in unique factorization domains, i.e. a view which can be propagated by functors such as SparseUnivariatePolynomial or Fraction. This is not easy, as the constructive version of the classical concept of UniqueFactorizationDomain cannot be so propagated. The solution adopted is based largely on Seidenberg's conditions (F) and (P), but there are several additional points that have to be borne in mind to produce reasonably efficient algorithms in the required generality. The consequence of the algorithms and interfaces presented in this paper is that Scratchpad can factorize in any extension of the integers or finite fields by any combination of polynomial, fraction and algebraic extensions: a capability far more general than any other computer algebra system possesses.

Algorithms are developed that adopt a novel implicit representation for multivariate polynomials and rational functions with rational coefficients, that of black boxes for their evaluation. It is shown that within this evaluation-box representation, the polynomial greatest common divisor and factorization problems as well as the problem of extracting the numerator and denominator of a rational function can be solved in random polynomial time in the usual parameters. Since the resulting evaluation programs for the goal polynomials can be converted efficiently to sparse format, solutions to sparse problems such as the sparse ration interpolation problem follow as a consequence.>

It is proved that, under the usual restrictions, the denominator of the integral of a purely logarithmic function is the expected one, that is, all factors of the denominator of the integrand have their multiplicity decreased by one. Furthermore, it is determined which new logarithms may appear in the integration.

Previously-known algorithms for polynomial square-free decomposition rely on greatest common divisor (gcd) computations over the same coefficient domain where the decomposition is to be performed. In particular, gcd of the given polynomial and its first derivative (with respect to some variable) is obtained to begin with. Application of modular homomorphism and p-adic construction (multivariate case) or the Chinese remainder algorithm (univariate case) results in new square-free decomposition algorithms which, generally speaking, take less time than a single gcd between the given polynomial and its first derivative. The key idea is to obtain one or several “correct” homomorphic images of the desired square-free decomposition first. This provides information as to how many different square-free factors there are, their multiplicities and their homomorphic images. Since the multiplicities are known, only the square-free factors need be constructed. Thus, these new algorithms are relatively insensitive to the multiplicities of the square-free factors.