1,354 research outputs found
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
List decoding Reed-Muller codes over small fields
The list decoding problem for a code asks for the maximal radius up to which
any ball of that radius contains only a constant number of codewords. The list
decoding radius is not well understood even for well studied codes, like
Reed-Solomon or Reed-Muller codes.
Fix a finite field . The Reed-Muller code
is defined by -variate degree-
polynomials over . In this work, we study the list decoding radius
of Reed-Muller codes over a constant prime field ,
constant degree and large . We show that the list decoding radius is
equal to the minimal distance of the code.
That is, if we denote by the normalized minimal distance of
, then the number of codewords in any ball of
radius is bounded by independent
of . This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008],
who among other results proved it in the special case of
; and extends the work of Gopalan [FOCS 2010] who
proved the conjecture in the case of .
We also analyse the number of codewords in balls of radius exceeding the
minimal distance of the code. For , we show that the number of
codewords of in a ball of radius is bounded by , where
is independent of . The dependence on is tight.
This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who
proved similar bounds over .
The proof relies on several new ingredients: an extension of the
Frieze-Kannan weak regularity to general function spaces, higher-order Fourier
analysis, and an extension of the Schwartz-Zippel lemma to compositions of
polynomials.Comment: fixed a bug in the proof of claim 5.6 (now lemma 5.5
Low Density Lattice Codes
Low density lattice codes (LDLC) are novel lattice codes that can be decoded
efficiently and approach the capacity of the additive white Gaussian noise
(AWGN) channel. In LDLC a codeword x is generated directly at the n-dimensional
Euclidean space as a linear transformation of a corresponding integer message
vector b, i.e., x = Gb, where H, the inverse of G, is restricted to be sparse.
The fact that H is sparse is utilized to develop a linear-time iterative
decoding scheme which attains, as demonstrated by simulations, good error
performance within ~0.5dB from capacity at block length of n = 100,000 symbols.
The paper also discusses convergence results and implementation considerations.Comment: 24 pages, 4 figures. Submitted for publication in IEEE transactions
on Information Theor
- …