6 research outputs found
Lower Bounds for Depth Three Arithmetic Circuits with Small Bottom Fanin
Shpilka and Wigderson (CCC 99) had posed the problem of proving exponential lower bounds for (nonhomogeneous) depth three arithmetic circuits with bounded bottom fanin over a field F of characteristic zero. We resolve this problem by proving a N^(Omega(d/t)) lower bound for (nonhomogeneous) depth three arithmetic circuits with bottom fanin at most t computing an explicit N-variate polynomial of degree d over F.
Meanwhile, Nisan and Wigderson (CC 97) had posed the problem of proving superpolynomial lower bounds for homogeneous depth five arithmetic circuits. Over fields of characteristic zero, we show a lower bound of N^(Omega(sqrt(d))) for homogeneous depth five circuits (resp. also for depth three circuits) with bottom fanin at most N^(u), for any fixed u < 1. This resolves the problem posed by Nisan and Wigderson only partially because of the added restriction on the bottom fanin (a general homogeneous depth five circuit has bottom fanin at most N)
On the complexity of partial derivatives
The method of partial derivatives is one of the most successful lower bound
methods for arithmetic circuits. It uses as a complexity measure the dimension
of the span of the partial derivatives of a polynomial. In this paper, we
consider this complexity measure as a computational problem: for an input
polynomial given as the sum of its nonzero monomials, what is the complexity of
computing the dimension of its space of partial derivatives? We show that this
problem is #P-hard and we ask whether it belongs to #P. We analyze the "trace
method", recently used in combinatorics and in algebraic complexity to lower
bound the rank of certain matrices. We show that this method provides a
polynomial-time computable lower bound on the dimension of the span of partial
derivatives, and from this method we derive closed-form lower bounds. We leave
as an open problem the existence of an approximation algorithm with reasonable
performance guarantees.A slightly shorter version of this paper was presented
at STACS'17. In this new version we have corrected a typo in Section 4.1, and
added a reference to Shitov's work on tensor rank
On the size of homogeneous and of depth four formulas with low individual degree
International audienceLet r ≥ 1 be an integer. Let us call a polynomial f (x_1,...,x_N) ∈ F[x] as a multi-r-ic polynomial if the degree of f with respect to any variable is at most r (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. We prove lower bounds for several subclasses of such circuits. Specifically, first define the formal degree of a node α with respect to a variable x_i inductively as follows. For a leaf α it is 1 if α is labelled with x_i and zero otherwise; for an internal node α labelled with × (respectively +) it is the sum of (respectively the maximum of) the formal degrees of the children with respect to x_i. We call an arithmetic circuit as a multi-r-ic circuit if the formal degree of the output node with respect to any variable is at most r. We prove lower bounds for various subclasses of multi-r-ic circuits