13 research outputs found
Multi-k-ic Depth Three Circuit Lower Bound
In a multi-k-ic depth three circuit every variable appears in at most k of the linear polynomials in every product gate of the circuit. This model is a natural generalization of multilinear depth three circuits that allows the formal degree of the circuit to exceed the number of underlying variables (as the formal degree of a multi-k-ic depth three circuit can be kn where n is the number of variables). The problem of proving lower bounds for depth three circuits with high formal degree has gained in importance following a work by Gupta, Kamath, Kayal and Saptharishi [7] on depth reduction to high formal degree depth three circuits. In this work, we show an exponential lower bound for multi-k-ic depth three circuits for any arbitrary constant k
Multi-k-ic depth three circuit lower bound
Abstract In a multi-k-ic depth three circuit every variable appears in at most k of the linear polynomials in every product gate of the circuit. This model is a natural generalization of multilinear depth three circuits that allows the formal degree of the circuit to exceed the number of underlying variables (as the formal degree of a multi-k-ic depth three circuit can be kn where n is the number of variables). The problem of proving lower bounds for depth three circuits with high formal degree has gained in importance following a work by Gupta, Kamath, Kayal and Saptharishi [GKKS13a] on depth reduction to high formal degree depth three circuits. In this work, we show an exponential lower bound for multi-k-ic depth three circuits for any arbitrary constant k
On Computing Multilinear Polynomials Using Multi-r-ic Depth Four Circuits
International audienceIn this paper, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which polynomial computed at every node has a bound on the individual degree of r (referred to as multi-r-ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better and better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing a multilinear polynomial on n^O(1) variables and degree d = o(n), must have size at least (n/r^1.1)^{\sqrt{d/r}} when r is o(d) and is strictly less than n^1/1.1. This bound however deteriorates with increasing r. It is a natural question to ask if we can prove a bound that does not deteriorate with increasing r or a bound that holds for a larger regime of r. We here prove a lower bound which does not deteriorate with r , however for a specific instance of d = d (n) but for a wider range of r. Formally, we show that there exists an explicit polynomial on n^{O(1)} variables and degree Θ(log^2(n)) such that any depth four circuit of bounded individual degree r < n^0.2 must have size at least exp(Ω (log^2 n)). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017)
An Almost Cubic Lower Bound for Depth Three Arithmetic Circuits
We show an almost cubic lower bound on the size of any depth three arithmetic circuit computing an explicit multilinear polynomial in n variables over any field. This improves upon the previously known quadratic lower bound by Shpilka and Wigderson [CCC, 1999]
Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four
Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists
an explicit -variate and degree polynomial such
that if any depth four circuit of bounded formal degree which computes
a polynomial of bounded individual degree , that is functionally
equivalent to , then must have size .
The motivation for their work comes from Boolean Circuit Complexity. Based on
a characterization for circuits by Yao [FOCS, 1985] and Beigel and
Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that
functions in can also be computed by algebraic
circuits (i.e., circuits of the form -- sums
of powers of polynomials) of size. Thus they argued that a
"functional" lower bound for an explicit
polynomial against circuits would imply a
lower bound for the "corresponding Boolean function" of against non-uniform
. In their work, they ask if their lower bound be extended to
circuits.
In this paper, for large integers and such that , we show that any circuit of
bounded individual degree at most that
functionally computes Iterated Matrix Multiplication polynomial
() over must have size . Since Iterated
Matrix Multiplication over is functionally in
, improvement of the afore mentioned lower bound to hold for
quasipolynomially large values of individual degree would imply a fine-grained
separation of from
Sum of Products of Read-Once Formulas
We study limitations of polynomials computed by depth two circuits built over read-once formulas (ROFs). In particular,
1. We prove an exponential lower bound for the sum of ROFs computing the 2n-variate polynomial in VP defined by Raz and Yehudayoff [CC,2009].
2. We obtain an exponential lower bound on the size of arithmetic circuits computing sum of products of restricted ROFs of unbounded depth computing the permanent of an n by n matrix. The restriction is on the number of variables with + gates as a parent in a proper sub formula of the ROF to be bounded by sqrt(n). Additionally, we restrict the product fan in to be bounded by a sub linear function. This proves an exponential lower bound for a subclass of possibly non-multilinear formulas of unbounded depth computing the permanent polynomial.
3. We also show an exponential lower bound for the above model against a polynomial in VP.
4. Finally we observe that the techniques developed yield an exponential lower bound on the size of sums of products of syntactically multilinear arithmetic circuits computing a product of variable disjoint linear forms where the bottom sum gate and product gates at the second level have fan in bounded by a sub linear function.
Our proof techniques are built on the measure developed by Kumar et al.[ICALP 2013] and are based on a non-trivial analysis of ROFs under random partitions. Further, our results exhibit strengths and provide more insight into the lower bound techniques introduced by Raz [STOC 2004]
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
Multi-k-ic Depth Three Circuit Lower Bound
In a multi-k-ic depth three circuit every variable appears in at most k of the linear polynomials in every product gate of the circuit. This model is a natural generalization of multilinear depth three circuits that allows the formal degree of the circuit to exceed the number of underlying variables (as the formal degree of a multi-k-ic depth three circuit can be kn where n is the number of variables). The problem of proving lower bounds for depth three circuits with high formal degree has gained in importance following a work by Gupta et al. (2013) on depth reduction to high formal degree depth three circuits. In this work, we show an exponential lower bound for multi-k-ic depth three circuits for any arbitrary constant k