All reports by Author Prahladh Harsha:

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TR18-081
| 20th April 2018
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Abhishek Bhrushundi, Prahladh Harsha, Pooya Hatami, Swastik Kopparty, Mrinal Kumar#### On Multilinear Forms: Bias, Correlation, and Tensor Rank

Revisions: 1

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TR18-075
| 23rd April 2018
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Yotam Dikstein, Irit Dinur, Yuval Filmus, Prahladh Harsha#### Boolean function analysis on high-dimensional expanders

Revisions: 1

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TR17-181
| 26th November 2017
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Irit Dinur, Yuval Filmus, Prahladh Harsha#### Agreement tests on graphs and hypergraphs

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TR17-180
| 26th November 2017
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Irit Dinur, Yuval Filmus, Prahladh Harsha#### Low degree almost Boolean functions are sparse juntas

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TR17-013
| 23rd January 2017
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Abhishek Bhrushundi, Prahladh Harsha, Srikanth Srinivasan#### On polynomial approximations over $\mathbb{Z}/2^k\mathbb{Z}$

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TR16-204
| 20th December 2016
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Prahladh Harsha, Srikanth Srinivasan#### Robust Multiplication-based Tests for Reed-Muller Codes

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TR16-160
| 26th October 2016
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Irit Dinur, Prahladh Harsha, Rakesh Venkat, Henry Yuen#### Multiplayer parallel repetition for expander games

Revisions: 1

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TR16-068
| 28th April 2016
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Prahladh Harsha, Srikanth Srinivasan#### On Polynomial Approximations to $\mathrm{AC}^0$

Revisions: 1

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TR15-199
| 7th December 2015
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Prahladh Harsha, Rahul Jain, Jaikumar Radhakrishnan#### Relaxed partition bound is quadratically tight for product distributions

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TR15-085
| 23rd May 2015
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Irit Dinur, Prahladh Harsha, Guy Kindler#### Polynomially Low Error PCPs with polyloglogn Queries via Modular Composition

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TR13-167
| 28th November 2013
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Venkatesan Guruswami, Prahladh Harsha, Johan Håstad, Srikanth Srinivasan, Girish Varma#### Super-polylogarithmic hypergraph coloring hardness via low-degree long codes

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TR09-144
| 24th December 2009
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Prahladh Harsha, Adam Klivans, Raghu Meka#### An Invariance Principle for Polytopes

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TR09-042
| 5th May 2009
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Irit Dinur, Prahladh Harsha#### Composition of low-error 2-query PCPs using decodable PCPs

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TR07-127
| 22nd November 2007
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Arie Matsliah, Eli Ben-Sasson, Prahladh Harsha, Oded Lachish#### Sound 3-query PCPPs are Long

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TR06-151
| 10th December 2006
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Prahladh Harsha, Rahul Jain, David McAllester, Jaikumar Radhakrishnan#### The communication complexity of correlation

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TR04-021
| 23rd March 2004
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Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, Salil Vadhan#### Robust PCPs of Proximity, Shorter PCPs and Applications to Coding

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TR03-006
| 23rd January 2003
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Eli Ben-Sasson, Prahladh Harsha, Sofya Raskhodnikova#### 3CNF Properties are Hard to Test

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TR03-004
| 24th December 2002
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Eli Ben-Sasson, Prahladh Harsha#### Lower Bounds for Bounded-Depth Frege Proofs via Buss-Pudlack Games

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TR00-061
| 14th August 2000
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Prahladh Harsha, Madhu Sudan#### Small PCPs with low query complexity

Abhishek Bhrushundi, Prahladh Harsha, Pooya Hatami, Swastik Kopparty, Mrinal Kumar

In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over $GF(2)= \{0,1\}$. We show the following results for multilinear forms and tensors.

1. Correlation bounds : We show that a random $d$-linear ... more >>>

Yotam Dikstein, Irit Dinur, Yuval Filmus, Prahladh Harsha

We initiate the study of Boolean function analysis on high-dimensional expanders. We describe an analog of the Fourier expansion and of the Fourier levels on simplicial complexes, and generalize the FKN theorem to high-dimensional expanders.

Our results demonstrate that a high-dimensional expanding complex X can sometimes serve as a sparse ... more >>>

Irit Dinur, Yuval Filmus, Prahladh Harsha

Agreement tests are a generalization of low degree tests that capture a local-to-global phenomenon, which forms the combinatorial backbone of most PCP constructions. In an agreement test, a function is given by an ensemble of local restrictions. The agreement test checks that the restrictions agree when they overlap, and the ... more >>>

Irit Dinur, Yuval Filmus, Prahladh Harsha

Nisan and Szegedy showed that low degree Boolean functions are juntas. Kindler and Safra showed that low degree functions which are *almost* Boolean are close to juntas. Their result holds with respect to $\mu_p$ for every *constant* $p$. When $p$ is allowed to be very small, new phenomena emerge. ... more >>>

Abhishek Bhrushundi, Prahladh Harsha, Srikanth Srinivasan

We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function F$:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be F$_k:\{0,1\}^n \rightarrow \{0,2^{k-1}\}$ by $F_k(x) = 2^{k-F(x)}$ (mod $2^k$). We consider the fractional agreement (which we refer to as $\gamma_{d,k}(F)$) of $F_k$ with ... more >>>

Prahladh Harsha, Srikanth Srinivasan

We consider the following multiplication-based tests to check if a given function $f: \mathbb{F}_q^n\to \mathbb{F}_q$ is the evaluation of a degree-$d$ polynomial over $\mathbb{F}_q$ for $q$ prime.

* $\mathrm{Test}_{e,k}$: Pick $P_1,\ldots,P_k$ independent random degree-$e$ polynomials and accept iff the function $fP_1\cdots P_k$ is the evaluation of a degree-$(d+ek)$ polynomial.

... more >>>Irit Dinur, Prahladh Harsha, Rakesh Venkat, Henry Yuen

We investigate the value of parallel repetition of one-round games with any number of players $k\ge 2$. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially ... more >>>

Prahladh Harsha, Srikanth Srinivasan

We make progress on some questions related to polynomial approximations of $\mathrm{AC}^0$. It is known, by works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. $6$th CCC 1991), that any $\mathrm{AC}^0$ circuit of size $s$ and depth $d$ has an $\varepsilon$-error probabilistic polynomial over the reals ... more >>>

Prahladh Harsha, Rahul Jain, Jaikumar Radhakrishnan

Let $f : \{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ be a 2-party function. For every product distribution $\mu$ on $\{0,1\}^n \times \{0,1\}^n$, we show that $${{CC}}^\mu_{0.49}(f) = O\left(\left(\log {{rprt}}_{1/4}(f) \cdot \log \log {{rprt}}_{1/4}(f)\right)^2\right),$$ where ${{CC}^\mu_\varepsilon(f)$ is the distributional communication complexity with error at most $\varepsilon$ under the distribution $\mu$ and ... more >>>

Irit Dinur, Prahladh Harsha, Guy Kindler

We show that every language in NP has a PCP verifier that tosses $O(\log n)$ random coins, has perfect completeness, and a soundness error of at most $1/poly(n)$, while making at most $O(poly\log\log n)$ queries into a proof over an alphabet of size at most $n^{1/poly\log\log n}$. Previous constructions that ... more >>>

Venkatesan Guruswami, Prahladh Harsha, Johan Håstad, Srikanth Srinivasan, Girish Varma

We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka, the “short code” of Barak et. al. [FOCS 2012]) and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate this code for inapproximability results.

In particular, we prove quasi-NP-hardness of the following problems on ... more >>>

Prahladh Harsha, Adam Klivans, Raghu Meka

Let $X$ be randomly chosen from $\{-1,1\}^n$, and let $Y$ be randomly

chosen from the standard spherical Gaussian on $\R^n$. For any (possibly unbounded) polytope $P$

formed by the intersection of $k$ halfspaces, we prove that

$$\left|\Pr\left[X \in P\right] - \Pr\left[Y \in P\right]\right| \leq \log^{8/5}k ...
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Irit Dinur, Prahladh Harsha

The main result of this paper is a simple, yet generic, composition theorem for low error two-query probabilistically checkable proofs (PCPs). Prior to this work, composition of PCPs was well-understood only in the constant error regime. Existing composition methods in the low error regime were non-modular (i.e., very much tailored ... more >>>

Arie Matsliah, Eli Ben-Sasson, Prahladh Harsha, Oded Lachish

We initiate the study of the tradeoff between the {\em length} of a

probabilistically checkable proof of proximity (PCPP) and the

maximal {\em soundness} that can be guaranteed by a $3$-query

verifier with oracle access to the proof. Our main observation is

that a verifier limited to querying a short ...
more >>>

Prahladh Harsha, Rahul Jain, David McAllester, Jaikumar Radhakrishnan

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 ...
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Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, Salil Vadhan

We continue the study of the trade-off between the length of PCPs

and their query complexity, establishing the following main results

(which refer to proofs of satisfiability of circuits of size $n$):

We present PCPs of length $\exp(\tildeO(\log\log n)^2)\cdot n$

that can be verified by making $o(\log\log n)$ Boolean queries.

more >>>

Eli Ben-Sasson, Prahladh Harsha, Sofya Raskhodnikova

For a boolean formula \phi on n variables, the associated property

P_\phi is the collection of n-bit strings that satisfy \phi. We prove

that there are 3CNF properties that require a linear number of queries,

even for adaptive tests. This contrasts with 2CNF properties

that are testable with O(\sqrt{n}) ...
more >>>

Eli Ben-Sasson, Prahladh Harsha

We present a simple proof of the bounded-depth Frege lower bounds of

Pitassi et. al. and Krajicek et. al. for the pigeonhole

principle. Our method uses the interpretation of proofs as two player

games given by Pudlak and Buss. Our lower bound is conceptually

simpler than previous ones, and relies ...
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Prahladh Harsha, Madhu Sudan

Most known constructions of probabilistically checkable proofs (PCPs)

either blow up the proof size by a large polynomial, or have a high

(though constant) query complexity. In this paper we give a

transformation with slightly-super-cubic blowup in proof size, with a

low query complexity. Specifically, the verifier probes the proof ...
more >>>