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Electronic Colloquium on Computational Complexity

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Reports tagged with Correlation:
TR06-097 | 9th August 2006
Emanuele Viola

New correlation bounds for GF(2) polynomials using Gowers uniformity

We study the correlation between low-degree GF(2) polynomials p and explicit functions. Our main results are the following:

(I) We prove that the Mod_m unction on n bits has correlation at most exp(-Omega(n/4^d)) with any GF(2) polynomial of degree d, for any fixed odd integer m. This improves on the ... more >>>

TR06-107 | 26th August 2006
Arkadev Chattopadhyay

An improved bound on correlation between polynomials over Z_m and MOD_q

Revisions: 1

Let m,q > 1 be two integers that are co-prime and A be any subset of Z_m. Let P be any multi-linear polynomial of degree d in n variables over Z_m. We show that the MOD_q boolean function on n variables has correlation at most exp(-\Omega(n/(m2^{m-1})^d)) with the boolean function ... more >>>

TR06-151 | 10th December 2006
Prahladh Harsha, Rahul Jain, David McAllester, Jaikumar Radhakrishnan

The communication complexity of correlation

We examine the communication required for generating random variables
remotely. One party Alice will be given a distribution D, and she
has to send a message to Bob, who is then required to generate a
value with distribution exactly D. Alice and Bob are allowed
to share random bits generated ... more >>>

TR07-050 | 25th May 2007
Arkadev Chattopadhyay

Discrepancy and the power of bottom fan-in in depth-three circuits

We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty 'Number on the Forehead' model. Our method is based on the notion of voting polynomial degree of functions and extends the Degree-Discrepancy Lemma in the recent work of Sherstov (STOC'07). ... more >>>

TR11-039 | 19th March 2011
Frederic Green, Daniel Kreymer, Emanuele Viola

In Brute-Force Search of Correlation Bounds for Polynomials

We report on some initial results of a brute-force search for determining the maximum correlation between degree-$d$ polynomials modulo $p$ and the $n$-bit mod $q$ function. For various settings of the parameters $n,d,p,$ and $q$, our results indicate that symmetric polynomials yield the maximum correlation. This contrasts with the previously-analyzed ... more >>>

TR12-108 | 4th September 2012
Arkadev Chattopadhyay, Rahul Santhanam

Lower Bounds on Interactive Compressibility by Constant-Depth Circuits

We formulate a new connection between instance compressibility \cite{Harnik-Naor10}), where the compressor uses circuits from a class $\C$, and correlation with
circuits in $\C$. We use this connection to prove the first lower bounds
on general probabilistic multi-round instance compression. We show that there
is no
probabilistic multi-round ... more >>>

TR12-134 | 22nd October 2012
Alexander Razborov, Emanuele Viola

Real Advantage

Revisions: 1

We highlight the challenge of proving correlation bounds
between boolean functions and integer-valued polynomials,
where any non-boolean output counts against correlation.

We prove that integer-valued polynomials of degree $\frac 12
\log_2 \log_2 n$ have zero correlation with parity. Such a
result is false for modular and threshold polynomials.
Its proof ... more >>>

TR12-160 | 20th November 2012
Frederic Green, Daniel Kreymer, Emanuele Viola

Block-symmetric polynomials correlate with parity better than symmetric

We show that degree-$d$ block-symmetric polynomials in
$n$ variables modulo any odd $p$ correlate with parity
exponentially better than degree-$d$ symmetric
polynomials, if $n \ge c d^2 \log d$ and $d \in [0.995
\cdot p^t - 1,p^t)$ for some $t \ge 1$. For these
infinitely many degrees, our result ... more >>>

TR13-119 | 2nd September 2013
Emanuele Viola

Challenges in computational lower bounds

We draw two incomplete, biased maps of challenges in
computational complexity lower bounds. Our aim is to put
these challenges in perspective, and to present some
connections which do not seem widely known.

more >>>

TR14-175 | 15th December 2014
Abhishek Bhowmick, Shachar Lovett

Nonclassical polynomials as a barrier to polynomial lower bounds

The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness, constructions of Ramsey graphs and locally decodable codes. Still, most of the known lower bounds become trivial for polynomials of ... more >>>

TR15-122 | 29th July 2015
Shiteng Chen, Periklis Papakonstantinou

Correlation lower bounds from correlation upper bounds

We show that for any coprime $m,r$ there is a circuit of the form $\text{MOD}_m\circ \text{AND}_{d(n)}$ whose correlation with $\text{MOD}_r$ is at least $2^{-O\left( \frac{n}{d(n)} \right) }$. This is the first correlation lower bound for arbitrary $m,r$, whereas previously lower bounds were known for prime $m$. Our motivation is the ... more >>>

TR16-158 | 9th October 2016
Alexander Kulikov, Vladimir Podolskii

Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates

We study the following computational problem: for which values of $k$, the majority of $n$ bits $\text{MAJ}_n$ can be computed with a depth two formula whose each gate computes a majority function of at most $k$ bits? The corresponding computational model is denoted by $\text{MAJ}_k \circ \text{MAJ}_k$. We observe that ... more >>>

TR18-192 | 12th November 2018
Alexander Golovnev, Alexander Kulikov

Circuit Depth Reductions

Revisions: 2

The best known circuit lower bounds against unrestricted circuits remained around $3n$ for several decades. Moreover, the only known technique for proving lower bounds in this model, gate elimination, is inherently limited to proving lower bounds of less than $5n$. In this work, we suggest a first non-gate-elimination approach for ... more >>>

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