Error Correction in Fast Matrix Multiplication and Inverse

Abstract

We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs and the erroneous output. Their running time is softly linear in the number of nonzero entries in these matrices when the number of errors is sufficiently small, and they also incorporate fast matrix multiplication so that the cost scales well when the number of errors is large. These algorithms build on the recent result of Gasieniec, Levcopoulos, Lingas, Pagh, and Tokuyama [2017] on correcting matrix products, as well as existing work on verification algorithms, sparse low-rank linear algebra, and sparse polynomial interpolation. 14 for i Ð 1, 2, . . . , r do 15 Set pJ i , eqth entry of E to c for each term cx e of f i 16 J Ð FindNonzeroRowspV Þ Ñ pC´EqV´ApBV q, ǫ 1 q 17 if #J ą r{2 then 18 k Ð 2k 19 if k ě 2n#J then return C´AB 20 foreach i P J do 21 Clear entries from row i of E added on this iteration 22 return E 17 for i Ð 1, 2, . . . , r do 18 Set pJ i , eqth entry of E to c for each term cx e of f i 19 J Ð FindNonzeroRowspV Þ Ñ V´pB`EqpAV q, ǫ 1 q 20 if #J ą r{2 then 21 k Ð 2k 22 if k ě 2n#J then return A´1´B 23 foreach i P J do 24 Clear entries from row i of E added on this iteration 25 return E

LU Factorization with Errors

“…The cleaning algorithm to be used needs to satisfy the following properties: 1. it has to be a block algorithm, gathering most operations into matrix multiplications where error cleaning can be efficiently performed by means of sparse interpolation, as in [30,32]; 2. it has to be recursive in order to make an efficient usage of fast matrix multiplication. 3. each block operation must be between operands that are submatrices of either the input matrix A or the approximate L and U factors.…”

“…Now, computing an inverse, as well as correcting it, is more expensive than using an LU factorization: for the computation itself, the inverse is more expensive by a constant factor of 3 (assuming classic matrix arithmetic), and for InverseEC the complexity of [32,Theorem 8.3] requires the fast selection of linearly independent rows using [7], which might be prohibitive in practice.…”

“…We explain the process to correct errors in U 23 (an equivalent process works in the transpose for L The idea is to be able to correct the next parts of an approximate U, namely U 23 , without recomputing it. Algorithm 2 below does this following the approach of [32]: for k mn total errors within c n erroneous columns, less than c/2 columns can have more than s = 2k/c errors. Therefore, we start with a candidate for k, then try to correct s errors per erroneous column.…”

“…Fault-tolerant computer algebra Some recent progress on fault tolerant algorithms has been made for Chinese remaindering [21,28,3], system solving [5,25], matrix multiplication and inversion [30,20,32], and function recovery [8,26].…”

Proof-of-Work Certificates that Can Be Efficiently Computed in the Cloud (Invited Talk)

“…Thus the question of the complexity of problem specific unbounded error correction also arises. This path again was first taken for matrix multiplication [35] and was recently extended to the matrix inverse [51].…”