arXiv subjects

Abhishek Bhrushundi

Publications and source records attributed to Abhishek Bhrushundi.

3 recordsLinked to original sources

arXiv2019

Torus polynomials: an algebraic approach to ACC lower bounds

We propose an algebraic approach to proving circuit lower bounds for ACC0 by defining and studying the notion of torus polynomials. We show how currently known polynomial-based approximation results for AC0 and ACC0 can be reformulated in this framework, implying that ACC0 can be approximated by low-degree torus polynomials. Furthermore, as a step towards proving ACC0 lower bounds for the majority function via our approach, we show that MAJORITY cannot be approximated by low-degree symmetric torus polynomials. We also pose several open problems related to our framework.

cs.CC↗

arXiv2018

On Multilinear Forms: Bias, Correlation, and Tensor Rank

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 form has exponentially low correlation with low-degree polynomials. More precisely, for $d \ll 2^{o(k)}$, we show that a random $d$-linear form $f(X_1,X_2, \dots, X_d) : \left(GF(2)^{k}\right)^d \rightarrow GF(2)$ has correlation $2^{-k(1-o(1))}$ with any polynomial of degree at most $d/10$. This result is proved by giving near-optimal bounds on the bias of random $d$-linear form, which is in turn proved by giving near-optimal bounds on the probability that a random rank-$t$ $d$-linear form is identically zero. 2. Tensor-rank vs Bias : We show that if a $d$-dimensional tensor has small rank, then the bias of the associated $d$-linear form is large. More precisely, given any $d$-dimensional tensor $$T :\underbrace{[k]\times \ldots [k]}_{\text{$d$ times}}\to GF(2)$$ of rank at most $t$, the bias of the associated $d$-linear form $$f_T(X_1,\ldots,X_d) := \sum_{(i_1,\dots,i_d) \in [k]^d} T(i_1,i_2,\ldots, i_d) X_{1,i_1}\cdot X_{1,i_2}\cdots X_{d,i_d}$$ is at least $\left(1-\frac1{2^{d-1}}\right)^t$. The above bias vs tensor-rank connection suggests a natural approach to proving nontrivial tensor-rank lower bounds for $d=3$. In particular, we use this approach to prove that the finite field multiplication tensor has tensor rank at least $3.52 k$ matching the best known lower bound for any explicit tensor in three dimensions over $GF(2)$.

cs.CC↗

arXiv2017

On polynomial approximations over $\mathbb{Z}/2^k\mathbb{Z}$

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 $F_k(x) = 2^{k-F(x)} \pmod {2^k}$. We consider the fractional agreement (which we refer to as $γ_{d,k}(F)$) of $F_k$ with degree $d$ polynomials from $\mathbb{Z}/2^k\mathbb{Z}[x_1,\ldots,x_n]$. Our results are the following: - Increasing $k$ can help: We observe that as $k$ increases, $γ_{d,k}(F)$ cannot decrease. We give two kinds of examples where $γ_{d,k}(F)$ actually increases. The first is an infinite family of functions $F$ such that $γ_{2d,2}(F) - γ_{3d-1,1}(F) \geq Ω(1)$. The second is an infinite family of functions $F$ such that $γ_{d,1}(F)\leq\frac{1}{2}+o(1)$ -- as small as possible -- but $γ_{d,3}(F) \geq \frac{1}{2}+Ω(1)$. - Increasing $k$ doesn't always help: Adapting a proof of Green [Comput. Complexity, 9(1):16-38, 2000], we show that irrespective of the value of $k$, the Majority function $\mathrm{Maj}_n$ satisfies $γ_{d,k}(\mathrm{Maj}_n) \leq \frac{1}{2}+\frac{O(d)}{\sqrt{n}}$. In other words, polynomials over $\mathbb{Z}/2^k\mathbb{Z}$ for large $k$ do not approximate the majority function any better than polynomials over $\mathbb{Z}/2\mathbb{Z}$. We observe that the model we study subsumes the model of non-classical polynomials in the sense that proving bounds in our model implies bounds on the agreement of non-classical polynomials with Boolean functions. In particular, our results answer questions raised by Bhowmick and Lovett [In Proc. 30th Computational Complexity Conf., pages 72-87, 2015] that ask whether non-classical polynomials approximate Boolean functions better than classical polynomials of the same degree.

cs.CC↗