Sparse multivariate function recovery with a high error rate in the evaluations

Abstract

In [Kaltofen and Yang, Proc. ISSAC 2013] we have generalized algebraic error-correcting decoding to multivariate sparse rational function interpolation from evaluations that can be numerically inaccurate and where several evaluations can have severe errors ("outliers"). Here we present a different algorithm that can interpolate a sparse multivariate rational function from evaluations where the error rate is 1/q for any q > 2, which our ISSAC 2013 algorithm could not handle. When implemented as a numerical algorithm we can, for instance, reconstruct a fraction of trinomials of degree 15 in 50 variables with non-outlier evaluations of relative noise as large as 10 −7 and where as much as 1/4 of the 14717 evaluations are outliers with relative error as small as 0.01 (large outliers are easily located by our method).For the algorithm with exact arithmetic and exact values at non-erroneous points, we provide a proof that for random evaluations one can avoid quadratic oversampling. Our argument already applies to our original 2007 sparse rational function interpolation algorithm [Kaltofen, Yang and Zhi, Proc. SNC 2007], where we have experimentally observed that for T unknown non-zero coefficients in a sparse candidate ansatz one only needs T + O(1) evaluations rather than the proven O(T 2 ) (cf. Candès and Tao sparse sensing). Here we finally can give the probabilistic analysis for this fact.

Error-Correcting Sparse Interpolation in the Chebyshev Basis

“…In [8] such a numerical method is formulated. Algorithms for error correction in the multivariate setting are given in [19].…”

“…Suppose now Λ(z) = τ κ=1 (z − ω ηκ ) where τ < τ . Since Λ linearly generates all {c κ /2 ω iηκ } i≥1−2t for 1 ≤ κ ≤ τ , where c κ is the coefficient corresponding to the exponent ηκ in (19), and by definition linearly generates the sequence (19) for i ≥ 1 − 2t, Λ must be a linear generator for { τ κ=τ +1 c κ /2 ω iηκ } i≥1−2t . The latter sequence is linearly generated by τ κ=τ +1 (z − ω ηκ ) and by a Vandermonde matrix argument the minimal generator Λ [2] (z) must be a polynomial factor = 1.…”

Cleaning-up data for sparse model synthesis

“…Our SPINO (sparse polynomial interpolation with noise and outliers) algorithm [4] solves the univariate sparse polynomial approximate fitting problem with outlier removal. Our sparse multivariate rational function interpolation algorithms [7,8] are based on dense univariate algorithms and can also approximate noisy points and remove outliers. From a dense univariate algorithm we can compute a scalar shift σ for the variable x = y + σ to obtain a sparse model in y [3].…”

Numerical linear system solving with parametric entries by error correction

“…If the scalar coefficients in the polynomial coefficients ai,j(u), bi(u) are floating point numbers, evaluations at u = ξ may drop the numeric rank below full rank, but that condition is dependent on a threshold of the condition number of the evaluated matrix A(ξ ). By interpreting the solution of the evaluated numerically low rank system A(ξ )x [ ] = b(ξ ) as an error in the data of the reconstruction problem, we can deploy techniques from algebraic error correcting codes [3,12] and their numeric counterparts [5,6,11] for recovering the solution from those error-free x [ ] where the numeric rank of A(ξ ) remained full. The Welch/Berlekamp decoder for Reed/Solomon codes can compute a reduced rational function solution of the algebraic interpolation-with-errors problem without locating the error.…”

“…If m = n = 1 and A = I1 =[1] (which implieŝ E1 = 0) we recover the polynomial b(u) from d f +dg +2Ê2+1 values, where ≤Ê of the values can be erroneous. The algorithm then is Welch/Berlekamp decoding of an algebraic Reed/Solomon error correcting code[3,6,11,12]. If m = n and A = In we recover the rational function vectorf (u)/g(u) = b(u) ∈ K(u) n from d f + dg +Ê1 + 2Ê2 + 1 values b(ξ ) ∈ K,where ≤Ê1 evaluations are roots of g, indicated by b(ξ ) = ∞ n leading to the equationΨ(ξ ) = 0 in(5), and ≤Ê2 evaluations are erroneous in one or more components ofb [ ] = b(ξ) (cf [3]…”