Log-concavity and lower bounds for arithmetic circuits

One question that we investigate in this paper is, how can we build log-concave polynomials using sparse polynomials as building blocks? More precisely, let $f = \sum\_{i = 0}^d a\_i X^i \in \mathbb{R}^+[X]$ be a polynomial satisfying the log-concavity condition a\_i^2 \textgreater{} \tau a\_{i-1}a\_{i+1} for every $i \in \{1,\ldots,d-1\},$ where \tau \textgreater{} 0. Whenever $f$ can be written under the form $f = \sum\_{i = 1}^k \prod\_{j = 1}^m f\_{i,j}$ where the polynomials $f\_{i,j}$ have at most $t$ monomials, it is clear that $d \leq k t^m$. Assuming that the $f\_{i,j}$ have only non-negative coefficients, we improve this degree bound to $d = \mathcal O(k m^{2/3} t^{2m/3} {\rm log^{2/3}}(kt))$ if \tau \textgreater{} 1, and to $d \leq kmt$ if $\tau = d^{2d}$. This investigation has a complexity-theoretic motivation: we show that a suitable strengthening of the above results would imply a separation of the algebraic complexity classes VP and VNP. As they currently stand, these results are strong enough to provide a new example of a family of polynomials in VNP which cannot be computed by monotone arithmetic circuits of polynomial size

Superpolynomial lower bounds for general homogeneous depth 4 arithmetic circuits

In this paper, we prove superpolynomial lower bounds for the class of homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP of degree $n$ in $n^2$ variables such that any homogeneous depth 4 arithmetic circuit computing it must have size $n^{\Omega(\log \log n)}$. Our results extend the works of Nisan-Wigderson [NW95] (which showed superpolynomial lower bounds for homogeneous depth 3 circuits), Gupta-Kamath-Kayal-Saptharishi and Kayal-Saha-Saptharishi [GKKS13, KSS13] (which showed superpolynomial lower bounds for homogeneous depth 4 circuits with bounded bottom fan-in), Kumar-Saraf [KS13a] (which showed superpolynomial lower bounds for homogeneous depth 4 circuits with bounded top fan-in) and Raz-Yehudayoff and Fournier-Limaye-Malod-Srinivasan [RY08, FLMS13] (which showed superpolynomial lower bounds for multilinear depth 4 circuits). Several of these results in fact showed exponential lower bounds. The main ingredient in our proof is a new complexity measure of {\it bounded support} shifted partial derivatives. This measure allows us to prove exponential lower bounds for homogeneous depth 4 circuits where all the monomials computed at the bottom layer have {\it bounded support} (but possibly unbounded degree/fan-in), strengthening the results of Gupta et al and Kayal et al [GKKS13, KSS13]. This new lower bound combined with a careful "random restriction" procedure (that transforms general depth 4 homogeneous circuits to depth 4 circuits with bounded support) gives us our final result

A Measure of Space for Computing over the Reals

We propose a new complexity measure of space for the BSS model of computation. We define LOGSPACE\_W and PSPACE\_W complexity classes over the reals. We prove that LOGSPACE\_W is included in NC^2\_R and in P\_W, i.e. is small enough for being relevant. We prove that the Real Circuit Decision Problem is P\_R-complete under LOGSPACE\_W reductions, i.e. that LOGSPACE\_W is large enough for containing natural algorithms. We also prove that PSPACE\_W is included in PAR\_R

Arithmetic Circuit Lower Bounds via MaxRank

We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results : As our main result, we prove that any homogeneous depth-3 circuit for computing the product of $d$ matrices of dimension $n \times n$ requires $\Omega(n^{d-1}/2^d)$ size. This improves the lower bounds by Nisan and Wigderson(1995) when $d=\omega(1)$. There is an explicit polynomial on $n$ variables and degree at most $\frac{n}{2}$ for which any depth-3 circuit $C$ of product dimension at most $\frac{n}{10}$ (dimension of the space of affine forms feeding into each product gate) requires size $2^{\Omega(n)}$. This generalizes the lower bounds against diagonal circuits proved by Saxena(2007). Diagonal circuits are of product dimension 1. We prove a $n^{\Omega(\log n)}$ lower bound on the size of product-sparse formulas. By definition, any multilinear formula is a product-sparse formula. Thus, our result extends the known super-polynomial lower bounds on the size of multilinear formulas by Raz(2006). We prove a $2^{\Omega(n)}$ lower bound on the size of partitioned arithmetic branching programs. This result extends the known exponential lower bound on the size of ordered arithmetic branching programs given by Jansen(2008).Comment: 22 page

Adversary lower bounds for nonadaptive quantum algorithms

International audienceWe present general methods for proving lower bounds on the query complexity of nonadaptive quantum algorithms. Our results are based on the adversary method of Ambainis

On Σ A Σ A Σ Circuits: The Role of Middle Σ Fan-in, Homogeneity and Bottom Degree.

We study polynomials computed by depth five Σ ∧ Σ ∧ Σ arithmetic circuits where ‘Σ’ and ‘∧’ represent gates that compute sum and power of their inputs respectively. Such circuits compute polynomials of the form Pt i=1 Q αi i , where Qi = Pri j=1 ` dij ij where `ij are linear forms and ri, αi, t > 0. These circuits are a natural generalization of the well known class of Σ ∧ Σ circuits and received significant attention recently. We prove an exponential lower bound for the monomial x1 · · · xn against depth five Σ ∧ Σ [≤n] ∧ [≥21] Σ and Σ ∧ Σ [≤2 √n/1000] ∧ [≥ √n] Σ arithmetic circuits where the bottom Σ gate is homogeneous. Our results show that the fan-in of the middle Σ gates, the degree of the bottom powering gates and the homogeneity at the bottom Σ gates play a crucial role in the computational power of Σ ∧ Σ ∧ Σ circuits

Valiant's model: from exponential sums to exponential products

12 pagesWe study the power of big products for computing multivariate polynomials in a Valiant-like framework. More precisely, we define a new class \vpip as the set of families of polynomials that are exponential products of easily computable polynomials. We investigate the consequences of the hypothesis that these big products are themselves easily computable. For instance, this hypothesis would imply that the nonuniform versions of P and NP coincide. Our main result relates this hypothesis to Blum, Shub and Smale's algebraic version of P versus NP. Let $K$ be a field of characteristic 0. Roughly speaking, we show that in order to separate \p_K from \np_K using a problem from a fairly large class of ``simple'' problems, one should first be able to show that exponential products are not easily computable. The class of ``simple'' problems under consideration is the class of NP problems in the structure $(K,+,-,=)$, in which multiplication is not allowed

Reachability problems for PAMs

Piecewise affine maps (PAMs) are frequently used as a reference model to show the openness of the reachability questions in other systems. The reachability problem for one-dimentional PAM is still open even if we define it with only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on p-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM's orbits, reachability problems and representation of numbers in a rational base system. Finally we show a particular instance where the uniform distribution of the original orbit may not remain uniform or even dense after making regular shifts and taking a fractional part in that sequence.Comment: 16 page

Improved bounds for reduction to depth 4 and depth 3

Koiran showed that if a $n$-variate polynomial of degree $d$ (with $d=n^{O(1)}$) is computed by a circuit of size $s$, then it is also computed by a homogeneous circuit of depth four and of size $2^{O(\sqrt{d}\log(d)\log(s))}$. Using this result, Gupta, Kamath, Kayal and Saptharishi gave a $\exp(O(\sqrt{d\log(d)\log(n)\log(s)}))$ upper bound for the size of the smallest depth three circuit computing a $n$-variate polynomial of degree $d=n^{O(1)}$ given by a circuit of size $s$. We improve here Koiran's bound. Indeed, we show that if we reduce an arithmetic circuit to depth four, then the size becomes $\exp(O(\sqrt{d\log(ds)\log(n)}))$. Mimicking Gupta, Kamath, Kayal and Saptharishi's proof, it also implies the same upper bound for depth three circuits. This new bound is not far from optimal in the sense that Gupta, Kamath, Kayal and Saptharishi also showed a $2^{\Omega(\sqrt{d})}$ lower bound for the size of homogeneous depth four circuits such that gates at the bottom have fan-in at most $\sqrt{d}$. Finally, we show that this last lower bound also holds if the fan-in is at least $\sqrt{d}$

On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields

Recently, Gupta et.al. [GKKS2013] proved that over Q any $n^{O(1)}$-variate and $n$-degree polynomial in VP can also be computed by a depth three $\Sigma\Pi\Sigma$ circuit of size $2^{O(\sqrt{n}\log^{3/2}n)}$. Over fixed-size finite fields, Grigoriev and Karpinski proved that any $\Sigma\Pi\Sigma$ circuit that computes $Det_n$ (or $Perm_n$) must be of size $2^{\Omega(n)}$ [GK1998]. In this paper, we prove that over fixed-size finite fields, any $\Sigma\Pi\Sigma$ circuit for computing the iterated matrix multiplication polynomial of $n$ generic matrices of size $n\times n$, must be of size $2^{\Omega(n\log n)}$. The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the $n^{O(1)}$-variate and $n$-degree polynomials in VP by depth 3 circuits of size $2^{o(n\log n)}$. The result [GK1998] can only rule out such a possibility for depth 3 circuits of size $2^{o(n)}$. We also give an example of an explicit polynomial ($NW_{n,\epsilon}(X)$) in VNP (not known to be in VP), for which any $\Sigma\Pi\Sigma$ circuit computing it (over fixed-size fields) must be of size $2^{\Omega(n\log n)}$. The polynomial we consider is constructed from the combinatorial design. An interesting feature of this result is that we get the first examples of two polynomials (one in VP and one in VNP) such that they have provably stronger circuit size lower bounds than Permanent in a reasonably strong model of computation. Next, we prove that any depth 4 $\Sigma\Pi^{[O(\sqrt{n})]}\Sigma\Pi^{[\sqrt{n}]}$ circuit computing $NW_{n,\epsilon}(X)$ (over any field) must be of size $2^{\Omega(\sqrt{n}\log n)}$. To the best of our knowledge, the polynomial $NW_{n,\epsilon}(X)$ is the first example of an explicit polynomial in VNP such that it requires $2^{\Omega(\sqrt{n}\log n)}$ size depth four circuits, but no known matching upper bound