1,142 research outputs found
The List-Decoding Size of Fourier-Sparse Boolean Functions
A function defined on the Boolean hypercube is k-Fourier-sparse if it has at most k nonzero Fourier coefficients. For a function f: F_2^n -> R and parameters k and d, we prove a strong upper bound on the number of k-Fourier-sparse Boolean functions that disagree with f on at most d inputs. Our bound implies that the number of uniform and independent random samples needed for learning the class of k-Fourier-sparse Boolean functions on n variables exactly is at most O(n * k * log(k)).
As an application, we prove an upper bound on the query complexity of testing Booleanity of Fourier-sparse functions. Our bound is tight up to a logarithmic factor and quadratically improves on a result due to Gur and Tamuz [Chicago J. Theor. Comput. Sci.,2013]
The Restricted Isometry Property of Subsampled Fourier Matrices
A matrix satisfies the restricted isometry
property of order with constant if it preserves the
norm of all -sparse vectors up to a factor of . We prove
that a matrix obtained by randomly sampling rows from an Fourier matrix satisfies the restricted
isometry property of order with a fixed with high
probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math.,
2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).Comment: 16 page
Two new results about quantum exact learning
We present two new results about exact learning by quantum computers. First,
we show how to exactly learn a -Fourier-sparse -bit Boolean function from
uniform quantum examples for that function. This
improves over the bound of uniformly random classical
examples (Haviv and Regev, CCC'15). Our main tool is an improvement of Chang's
lemma for the special case of sparse functions. Second, we show that if a
concept class can be exactly learned using quantum membership
queries, then it can also be learned using classical membership queries. This improves the
previous-best simulation result (Servedio and Gortler, SICOMP'04) by a -factor.Comment: v3: 21 pages. Small corrections and clarification
Finding Skewed Subcubes Under a Distribution
Say that we are given samples from a distribution ? over an n-dimensional space. We expect or desire ? to behave like a product distribution (or a k-wise independent distribution over its marginals for small k). We propose the problem of enumerating/list-decoding all large subcubes where the distribution ? deviates markedly from what we expect; we refer to such subcubes as skewed subcubes. Skewed subcubes are certificates of dependencies between small subsets of variables in ?. We motivate this problem by showing that it arises naturally in the context of algorithmic fairness and anomaly detection.
In this work we focus on the special but important case where the space is the Boolean hypercube, and the expected marginals are uniform. We show that the obvious definition of skewed subcubes can lead to intractable list sizes, and propose a better definition of a minimal skewed subcube, which are subcubes whose skew cannot be attributed to a larger subcube that contains it. Our main technical contribution is a list-size bound for this definition and an algorithm to efficiently find all such subcubes. Both the bound and the algorithm rely on Fourier-analytic techniques, especially the powerful hypercontractive inequality.
On the lower bounds side, we show that finding skewed subcubes is as hard as the sparse noisy parity problem, and hence our algorithms cannot be improved on substantially without a breakthrough on this problem which is believed to be intractable. Motivated by this, we study alternate models allowing query access to ? where finding skewed subcubes might be easier
Some Applications of Coding Theory in Computational Complexity
Error-correcting codes and related combinatorial constructs play an important
role in several recent (and old) results in computational complexity theory. In
this paper we survey results on locally-testable and locally-decodable
error-correcting codes, and their applications to complexity theory and to
cryptography.
Locally decodable codes are error-correcting codes with sub-linear time
error-correcting algorithms. They are related to private information retrieval
(a type of cryptographic protocol), and they are used in average-case
complexity and to construct ``hard-core predicates'' for one-way permutations.
Locally testable codes are error-correcting codes with sub-linear time
error-detection algorithms, and they are the combinatorial core of
probabilistically checkable proofs
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