Daniel Minahan, Ilya Volkovich

In this paper we study the identity testing problem of \emph{arithmetic read-once formulas} (ROF) and some related models. A read-once formula is formula (a circuit whose underlying graph is a tree) in which the

operations are $\set{+,\times}$ and such that every input variable labels at most one leaf. We obtain ...
more >>>

morris yau

In "An Almost Cubic Lower Bound for $\sum\prod\sum$ circuits in VP", [BLS16] present an infinite family of polynomials, $\{P_n\}_{n \in \mathbb{Z}^+}$, with $P_n$

on $N = \Theta(n polylog(n))$

variables with degree $N$ being in VP such that every

$\sum\prod\sum$ circuit computing $P_n$ is of size $\Omega\big(\frac{N^3}{2^{\sqrt{\log N}}}\big)$.

We ...
more >>>

Mrinal Kumar

An algebraic branching program (ABP) is a directed acyclic graph, with a start vertex $s$, and end vertex $t$ and each edge having a weight which is an affine form in $\F[x_1, x_2, \ldots, x_n]$. An ABP computes a polynomial in a natural way, as the sum of weights of ... more >>>

Michael Forbes, Amir Shpilka

In this paper we study the complexity of constructing a hitting set for $\overline{VP}$, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given ... more >>>

Pavel Hrubes

We show that strong-enough lower bounds on monotone arithmetic circuits or the non-negative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial $f\in {\mathbb {R}}[x_1,\dots, x_n]$ of degree $d$ has an arithmetic circuit of size $s$ then $(x_1+\dots+x_n+1)^d+\epsilon ... more >>>

Oded Goldreich

We consider arithmetic circuits with arbitrary large (multi-linear) gates for computing multi-linear functions. An adequate complexity measure for such circuits is the maximum between the arity of the gates and their number.

This model and the corresponding complexity measure were introduced by Goldreich and Wigderson (ECCC, TR13-043, 2013), ...
more >>>

Nutan Limaye, Srikanth Srinivasan, Sébastien Tavenas

An Algebraic Circuit for a polynomial $P\in F[x_1,\ldots,x_N]$ is a computational model for constructing the polynomial $P$ using only additions and multiplications. It is a \emph{syntactic} model of computation, as opposed to the Boolean Circuit model, and hence lower bounds for this model are widely expected to be easier to ... more >>>

Vishwas Bhargava, Ankit Garg, Neeraj Kayal, Chandan Saha

Consider a homogeneous degree $d$ polynomial $f = T_1 + \cdots + T_s$, $T_i = g_i(\ell_{i,1}, \ldots, \ell_{i, m})$ where $g_i$'s are homogeneous $m$-variate degree $d$ polynomials and $\ell_{i,j}$'s are linear polynomials in $n$ variables. We design a (randomized) learning algorithm that given black-box access to $f$, computes black-boxes for ... more >>>

Vishwas Bhargava, Shubhangi Saraf, Ilya Volkovich

We develop a new technique for analyzing linear independence of multivariate polynomials. One of our main technical contributions is a \emph{Small Witness for Linear Independence} (SWLI) lemma which states the following.

If the polynomials $f_1,f_2, \ldots, f_k \in \F[X]$ over $X=\{x_1, \ldots, x_n\}$ are $\F$-linearly independent then there exists ...
more >>>

Prasad Chaugule, Nutan Limaye

In this paper we prove the following two results.

* We show that for any $C \in {mVF, mVP, mVNP}$, $C = \overline{C}$. Here, $mVF, mVP$, and $mVNP$ are monotone variants of $VF, VP$, and $VNP$, respectively. For an algebraic complexity class $C$, $\overline{C}$ denotes the closure of $C$. ...
more >>>