Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Numerically Stable Polynomially Coded Computing   We study the numerical stability of polynomial based encoding methods, which
has emerged to be a powerful class of techniques for providing straggler and
fault tolerance in the area of coded computing. Our contributions are as
follows: 1) We construct new codes for matrix multiplication that achieve the
same fault/straggler tolerance as the previously constructed MatDot Codes and
Polynomial Codes. Unlike previous codes that use polynomials expanded in a
monomial basis, our codes uses a basis of orthogonal polynomials. 2) We show
that the condition number of every $m \times m$ sub-matrix of an $m \times n, n
\geq m$ Chebyshev-Vandermonde matrix, evaluated on the $n$-point Chebyshev
grid, grows as $O(n^{2(n-m)})$ for $n > m$. An implication of this result is
that, when Chebyshev-Vandermonde matrices are used for coded computing, for a
fixed number of redundant nodes $s=n-m,$ the condition number grows at most
polynomially in the number of nodes $n$. 3) By specializing our orthogonal
polynomial based constructions to Chebyshev polynomials, and using our
condition number bound for Chebyshev-Vandermonde matrices, we construct new
numerically stable techniques for coded matrix multiplication. We empirically
demonstrate that our constructions have significantly lower numerical errors
compared to previous approaches which involve inversion of Vandermonde
matrices. We generalize our constructions to explore the trade-off between
computation/communication and fault-tolerance. 4) We propose a numerically
stable specialization of Lagrange coded computing. Motivated by our condition
number bound, our approach involves the choice of evaluation points and a
suitable decoding procedure that involves inversion of an appropriate
Chebyshev-Vandermonde matrix. Our approach is demonstrated empirically to have
lower numerical errors as compared to standard methods.

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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: eess
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The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: q-bio
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: physics