AF: Small: Symbolic Computation with Certificates, Sparsity and Error Correction

North Carolina State University, Raleigh NC

Investigators

Abstract

This project continues the PI's productive work in certifying the results of symbolic computations and in polynomial computations. It aims for results in three areas: 1. Low-cost certification of the output of large-scale symbolic computations performed on high power servers, possibly in the computing cloud. The focus is on problems in high-dimensional linear algebra and non-linear optimization. Quickly checkable proofs can also expose bugs/faults in time- and space-consuming server computations. 2. Medical image processing needs robust interpolation from values sampled at points; The project will reconstruct functions as sparse sums of a few exponentials or orthogonal polynomials, using algorithms that account for noise and outliers (that is, catastrophic errors) while minimizing the number of sample points. 3. In fundamental problems in algebraic computational complexity, it takes a fresh look at polynomial multiplication and greatest common divisors. The PI continues to train undergraduate and Master students, and supervise Ph.D. students and a postdoctoral scholar, so that they develop advanced skills in designing algorithms and computer programs for doing mathematics. The students and postdoc will be encouraged to adopt advanced research communication skills by presenting posters and talks at international meetings, especially when their background is from under-represented groups in the discipline. The method to achieve the goal of low-cost certification is to develop certificates for the Frobenius form of sparse matrices and for the solvability of polynomial equations over the real numbers that are independent of the algorithms that produce the output. The approach is based on interactive proof protocols and crytographic hardness assumptions, but the project aims to reduce the use of cryptography in its protocols, so that the verifier can increase probability of correctness arbitrarily, even after the computation. The PI has used algebraic error correcting decoding to locate evaluation errors in sparse interpolation and dense rational vector recovery, the latter with early termination. The current focus is on sparseness in a Chebyshev basis, where list-decoders are missing, and on multivariate sparse interpolation using a generalization of Prony's Algorithm combined with Sakata's generalization of the Berlekamp/Massey Algorithm. Finally, the project studies the algebraic complexity of multiplying polynomials without the use of constants, computing the Sylvester resultant without divisions, and representing polynomials as the characteristic polynomial of matrices.