Cleaning-up data for sparse model synthesis

Abstract

EXTENDED ABSTRACTThe discipline of symbolic computation contributes to mathematical model synthesist in several ways. One is the pioneering creation of interpolation algorithms that can account for sparsity in the resulting multi-dimensional models, for example, by Zippel (12], Ben-Or and Tiwari (1), and in their recent numerical counterparts by Giesbrecht-Labahn-Lee (5) and Kaltofen-Yang-Zhi (9).The theme of our talk is the discovery of sparsity in interpolation algorithms, while at the same time allowing for erroneous input data. As shown in Figure 1, not removing erroneous input points can result in wrong, of course, but also dense outputs. Thus one may use the sparsity constraint to correct for errors, although this is easier said than done: we shall deal with the difficult case where the underlying •This research wBS supported in part by the National Science Foundation under Grant CCF-1115772. tModels are artifacts made by humans, not discovered. When constraining to sparse models, on e has a sequence of less and less sparse models that flt the data better and better. Pennission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored Abstracting with credit is pemlitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific pennission and/or a fee. Request permissions from Permissions@acm.org. SNC'/4, 1 140 120 100 80 60 40 20 10 -20F igure 1: Quadratic fit without and with outlier sparsity structure, for example the non-zero terms or nonzero entries in the model, are not known on input. Therefore we must compute two lists of discrete quantities: the supports of the output, that is, the sparsity structure, and the location of erroneous input scalars. The corresponding scalar coefficients can be considered, over the real and complex numbers, as continuous quantities. In our symbolic-numeric algorithms we allow imprecision in the input scalars, besides large outlier errors. Sparsity in many situations turns out to be a stabilizing constraint for numerical computation with floating point scalars, but outlier errors are quite destructive when not properly removed.Sparse interpolation of polynomials or rational functions is the process of computing a sparse multivariate rational function j=l m=l a;,bmEK,a;:;t:O,bm#O, (1) from values /t = (f /g)(f.1,£, ... , f.n ,t), where the (unknown) terms of the non-zero monomials are denoted by xil; = d; t d; n d 1! em 1 em n Th bl Xi ' • • • Xn'an X m = Xi ' • • • Xn ' • e pro em essentially constitutes sparse model recovery. We consider