89 research outputs found
Transformées rapides sur les corps finis de caractéristique deux
We describe new fast algorithms for evaluation and interpolation on the "novel" polynomial basis over finite fields of characteristic two introduced by Lin, Chung and Han (FOCS 2014). Fast algorithms are also described for converting between their basis and the monomial basis, as well as for converting to and from the Newton basis associated with the evaluation points of the evaluation and interpolation algorithms. Combining algorithms yields a new truncated additive fast Fourier transform (FFT) and inverse truncated additive FFT which improve upon some previous algorithms when the field possesses an appropriate tower of subfields
Efficient deterministic approximate counting for low-degree polynomial threshold functions
We give a deterministic algorithm for approximately counting satisfying
assignments of a degree- polynomial threshold function (PTF). Given a
degree- input polynomial over and a parameter
, our algorithm approximates to within an additive in time . (Any sort of efficient multiplicative approximation is
impossible even for randomized algorithms assuming .) Note that the
running time of our algorithm (as a function of , the number of
coefficients of a degree- PTF) is a \emph{fixed} polynomial. The fastest
previous algorithm for this problem (due to Kane), based on constructions of
unconditional pseudorandom generators for degree- PTFs, runs in time
for all .
The key novel contributions of this work are: A new multivariate central
limit theorem, proved using tools from Malliavin calculus and Stein's Method.
This new CLT shows that any collection of Gaussian polynomials with small
eigenvalues must have a joint distribution which is very close to a
multidimensional Gaussian distribution. A new decomposition of low-degree
multilinear polynomials over Gaussian inputs. Roughly speaking we show that (up
to some small error) any such polynomial can be decomposed into a bounded
number of multilinear polynomials all of which have extremely small
eigenvalues. We use these new ingredients to give a deterministic algorithm for
a Gaussian-space version of the approximate counting problem, and then employ
standard techniques for working with low-degree PTFs (invariance principles and
regularity lemmas) to reduce the original approximate counting problem over the
Boolean hypercube to the Gaussian version
Approximate degree in classical and quantum computing
In this book, the authors survey what is known about a particularly natural notion of approximation by polynomials, capturing pointwise approximation over the real numbers.FG-2022-18482 - Alfred P. Sloan Foundation; CNS-2046425 - National Science Foundation; CCF-1947889 - National Science FoundationAccepted manuscrip
Practical Verified Computation with Streaming Interactive Proofs
As the cloud computing paradigm has gained prominence, the need for verifiable computation has grown urgent. Protocols for verifiable computation enable a weak client to outsource difficult computations to a powerful, but untrusted, server. These protocols provide the client with a (probabilistic) guarantee that the server performed the requested computations correctly, without requiring the client to perform the computations herself.Engineering and Applied Science
Algorithms Seminar, 2001-2002
These seminar notes constitute the proceedings of a seminar devoted to the analysis of algorithms and related topics. The subjects covered include combinatorics, symbolic computation, asymptotic analysis, number theory, as well as the analysis of algorithms, data structures, and network protocols
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