We prove strengthened lower bounds for constant-depth set-multilinear formulas. More precisely, we show that over any field, there is an explicit polynomial $f$ in VNP defined over $n^2$ variables, and of degree $n$, such that any product-depth $Δ$ set-multilinear formula computing $f$ has size at least $n^{Ω\left( n^{1/Δ}/Δ\right)}$. The hard polynomial $f$ comes from the class of Nisan-Wigderson (NW) design-based polynomials.
Our lower bounds improve upon the recent work of Limaye, Srinivasan and Tavenas (STOC 2022), where a lower bound of the form $(\log n)^{Ω(Δn^{1/Δ})}$ was shown for the size of product-depth $Δ$ set-multilinear formulas computing the iterated matrix multiplication (IMM) polynomial of the same degree and over the same number of variables as $f$. Moreover, our lower bounds are novel for any $Δ\geq 2$.
For general set-multilinear formulas, a lower bound of the form $ n^{Ω(\log n)}$ was already obtained by Raz (J. ACM 2009) for the more general model of multilinear formulas. The techniques of LST give a different route to set-multilinear formula lower bounds, and allow them to obtain a lower bound of the form $(\log n)^{Ω(\log n)}$ for the size of general set-multilinear formulas computing the IMM polynomial. Our proof techniques are another variation on those of LST, and enable us to show an improved lower bound (matching that of Raz) of the form $n^{Ω(\log n)}$, albeit for the same polynomial $f$ in VNP (the NW polynomial). As observed by LST, if the same $n^{Ω(\log n)}$ size lower bounds for unbounded-depth set-multilinear formulas could be obtained for the IMM polynomial, then using the self-reducibility of IMM and using hardness escalation results, this would imply super-polynomial lower bounds for general algebraic formulas.
In this paper, we study the computational complexity of the commutative determinant polynomial computed by a class of set-multilinear circuits which we call regular set-multilinear circuits. Regular set-multilinear circuits are commutative circuits with a restriction on the order in which they can compute polynomials. A regular circuit can be seen as the commutative analogue of the ordered circuit defined by Hrubes,Wigderson and Yehudayoff [HWY10]. We show that if the commutative determinant polynomial has small representation in the sum of constantly many regular set-multilinear circuits, the…
We prove a lower bound of $Ω(n^2/\log^2 n)$ on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial $f(x_1, \ldots, x_n)$. Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([RSY08]), who proved a lower bound of $Ω(n^{4/3}/\log^2 n)$ for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized version of Galvin's problem in extremal set theory.
We study the size blow-up that is necessary to convert an algebraic circuit of product-depth $Δ+1$ to one of product-depth $Δ$ in the multilinear setting. We show that for every positive $Δ= Δ(n) = o(\log n/\log \log n),$ there is an explicit multilinear polynomial $P^{(Δ)}$ on $n$ variables that can be computed by a multilinear formula of product-depth $Δ+1$ and size $O(n)$, but not by any multilinear circuit of product-depth $Δ$ and size less than $\exp(n^{Ω(1/Δ)})$. This result is tight up to the constant implicit in the double exponent for all $Δ= o(\log n/\log \log n).$ This strengthens a…
We show that any $n$-variate polynomial computable by a syntactically multilinear circuit of size $\operatorname{poly}(n)$ can be computed by a depth-$4$ syntactically multilinear ($ΣΠΣΠ$) circuit of size at most $\exp\left({O\left(\sqrt{n\log n}\right)}\right)$. For degree $d = ω(n/\log n)$, this improves upon the upper bound of $\exp\left({O(\sqrt{d}\log n)}\right)$ obtained by Tavenas~\cite{T15} for general circuits, and is known to be asymptotically optimal in the exponent when $d < n^ε$ for a small enough constant $ε$. Our upper bound matches the lower bound of $\exp\left({Ω\left(\sqrt{n\…
In this paper, we study the algebraic formula complexity of multiplying $d$ many $2\times 2$ matrices, denoted $\mathrm{IMM}_{d}$, and show that the well-known divide-and-conquer algorithm cannot be significantly improved at any depth, as long as the formulas are multilinear. Formally, for each depth $Δ\leq \log d$, we show that any product-depth $Δ$ multilinear formula for $\mathrm{IMM}_d$ must have size $\exp(Ω(Δd^{1/Δ})).$ It also follows from this that any multilinear circuit of product-depth $Δ$ for the same polynomial of the above form must have a size of $\exp(Ω(d^{1/Δ})).$ In particula…
Algebraic Branching Programs(ABPs) are standard models for computing polynomials. Syntactic multilinear ABPs (smABPs) are restrictions of ABPs where every variable is allowed to occur at most once in every path from the start to the terminal node. Proving lower bounds against syntactic multilinear ABPs remains a challenging open question in Algebraic Complexity Theory. The current best known bound is only quadratic [Alon-Kumar-Volk, ECCC 2017]. In this article we develop a new approach upper bounding the rank of the partial derivative matrix of syntactic multlinear ABPs: Convert the ABP to a s…
We give new and efficient black-box reconstruction algorithms for some classes of depth-$3$ arithmetic circuits. As a consequence, we obtain the first efficient algorithm for computing the tensor rank and for finding the optimal tensor decomposition as a sum of rank-one tensors when then input is a constant-rank tensor. More specifically, we provide efficient learning algorithms that run in randomized polynomial time over general fields and in deterministic polynomial time over the reals and the complex numbers for the following classes: (1) Set-multilinear depth-$3$ circuits of constant top f…
We prove super-polynomial lower bounds on the size of propositional proof systems operating with constant-depth algebraic circuits over fields of zero characteristic. Specifically, we show that the subset-sum variant $\sum_{i,j,k,\ell\in[n]} z_{ijk\ell}x_ix_j x_k x_\ell - β=0$, for Boolean variables, does not have polynomial-size IPS refutations where the refutations are multilinear and written as constant-depth circuits. Andrews and Forbes (STOC'22) established recently a constant-depth IPS lower bound, but their hard instance does not have itself small constant-depth circuits, while our inst…
Weakening Assumptions for Deterministic Subexponential Time Non-Singular Matrix Completion
在（Kabanets，Impagliazzo，2004）中，给出了如何在确定性次指数时间内确定电路多项式恒等式测试问题（CPIT），假设算术电路的一些显式多线性多项式族的硬度。本文考虑了CPIT的一种特殊情况，即低次非奇异矩阵完备化（NSMC）。对于这个子类的问题，它示出了如何获得相同的确定性的时间范围，使用较弱的假设，在确定性的复杂性。硬度随机性权衡也将显示在相反的方向，在努力取得进展的Valiant的VP与VNP的问题。为了分离VP和VNP，已知证明m × m持久式的行列式复杂度是$m^{ω(\log m)}$就足够了。本文证明了，对于一个适当的显式性概念，具有行列式复杂度m^{ω（\log m）} $ is equivalent to the existence of an efficiently computable generator $ G_n $ for multilinear NSMC with seed length $ O（n^{1/ \sqrt{\log n} }）$. The latter is a combinatorial object that provides an efficient deterministic black-box algorithm for NSMC. ``Multilinear NSMC'' indicates that $ G…
We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS 2020], [Kayal-Nair-Saha, STACS 2016]. The recent breakthrough work of Limaye, Srinivasan and Tavenas [FOCS 2021] proved these lower bounds by proving lower bounds for low-depth set-multilinear circuits. An interesting aspect of our proof is that it does not require conversion of a circuit to a set-multilinear circuit, nor does it involve a random restrictio…
An arithmetic read-once formula (ROF) is a formula (circuit of fan-out 1) over $+,\times$ where each variable labels at most one leaf. Every multilinear polynomial can be expressed as the sum of ROFs. In this work, we prove, for certain multilinear polynomials, a tight lower bound on the number of summands in such an expression.
An arithmetic read-once formula (ROF) is a formula (circuit of fan-out 1) over $+, \times$ where each variable labels at most one leaf. Every multilinear polynomial can be expressed as the sum of ROFs. In this work, we prove, for certain multilinear polynomials, a tight lower bound on the number of summands in such an expression.
The complexity class PPA consists of NP-search problems which are reducible to the parity principle in undirected graphs. It contains a wide variety of interesting problems from graph theory, combinatorics, algebra and number theory, but only a few of these are known to be complete in the class. Before this work, the known complete problems were all discretizations or combinatorial analogues of topological fixed point theorems. Here we prove the PPA-completeness of two problems of radically different style. They are PPA-Circuit CNSS and PPA-Circuit Chevalley, related respectively to the Combin…
This text is a development of a preprint of Ankit Gupta. We present an approach for devising a deterministic polynomial time blackbox identity testing (PIT) algorithm for depth-$4$ circuits with bounded top fanin. This approach is similar to Kayal-Shubhangi approach for depth-$3$ circuits. Kayal and Shubhangi based their algorithm on Sylvester-Gallai-type theorem about linear polynomials. We show how it is possible to generalize this approach to depth-$4$ circuits. However we failed to implement this plan completely. We succeeded to construct a polynomial time deterministic algorithm for depth…
The best known size lower bounds against unrestricted circuits have remained around $3n$ for several decades. Moreover, the only known technique for proving lower bounds in this model, gate elimination, is inherently limited to proving lower bounds of less than $5n$. In this work, we propose a non-gate-elimination approach for obtaining circuit lower bounds, via certain depth-three lower bounds. We prove that every (unbounded-depth) circuit of size $s$ can be expressed as an OR of $2^{s/3.9}$ $16$-CNFs. For DeMorgan formulas, the best known size lower bounds have been stuck at around $n^{3-o(1…
We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [LMP16] and Lagarde, Limaye and Srinivasan [LLS17]) and give the following constructions: (1) An explicit hitting set of quasipolynomial size for UPT circuits, (2) An explicit hitting set of quasipolynomial size for FewPT circuits (circuits with constantly many parse tree shapes), (3) An explicit hitting set of polynomial size for UPT circuits (of known parse tree shape), when a para…
What is the power of constant-depth circuits with $MOD_m$ gates, that can count modulo $m$? Can they efficiently compute MAJORITY and other symmetric functions? When $m$ is a constant prime power, the answer is well understood: Razborov and Smolensky proved in the 1980s that MAJORITY and $MOD_m$ require super-polynomial-size $MOD_q$ circuits, where $q$ is any prime power not dividing $m$. However, relatively little is known about the power of $MOD_m$ circuits for non-prime-power $m$. For example, it is still open whether every problem in $EXP$ can be computed by depth-$3$ circuits of polynomia…
Proving super-polynomial size lower bounds for syntactic multilinear Algebraic Branching Programs(smABPs) computing an explicit polynomial is a challenging problem in Algebraic Complexity Theory. The order in which variables in $\{x_1,\ldots,x_n\}$ appear along source to sink paths in any smABP can be viewed as a permutation in $S_n$. In this article, we consider the following special classes of smABPs where the order of occurrence of variables along a source to sink path is restricted: Strict circular-interval ABPs: For every subprogram the index set of variables occurring in it is contained …
In this paper we give an Immerman's Theorem for real-valued computation. We define circuits operating over real numbers and show that families of such circuits of polynomial size and constant depth decide exactly those sets of vectors of reals that can be defined in first-order logic on R-structures in the sense of Cucker and Meer. Our characterization holds both non-uniformily as well as for many natural uniformity conditions.
We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-$3$ and depth-$4$ arithmetic circuits for functional computation. We prove the following results : 1. Exponential lower bounds homogeneous depth-$3$ arithmetic circuits for a polynomial in $VNP$. 2. Exponential lower bounds for homogeneous depth-$4$ arithmetic circuits …
Let $C$ be a depth-3 arithmetic circuit of size at most $s$, computing a polynomial $ f \in \mathbb{F}[x_1,\ldots, x_n] $ (where $\mathbb{F}$ = $\mathbb{Q}$ or $\mathbb{C}$) and the fan-in of the product gates of $C$ is bounded by $d$. We give a deterministic polynomial identity testing algorithm to check whether $f\equiv 0$ or not in time $ 2^d \text{ poly}(n,s) $.
We show that over the field of complex numbers, \emph{every} homogeneous polynomial of degree $d$ can be approximated (in the border complexity sense) by a depth-$3$ arithmetic circuit of top fan-in at most $d+1$. This is quite surprising since there exist homogeneous polynomials $P$ on $n$ variables of degree $2$, such that any depth-$3$ arithmetic circuit computing $P$ must have top fan-in at least $Ω(n)$. As an application, we get a new tradeoff between the top fan-in and formal degree in an approximate analog of the celebrated depth reduction result of Gupta, Kamath, Kayal and Saptharishi …
The paper discusses the gate complexity and the depth of reversible circuits consisting of NOT, CNOT and 2-CNOT gates in the case, when the number of additional inputs is limited. We study Shannon's gate complexity function $L(n, q)$ and depth function $D(n, q)$ for a reversible circuit implementing a Boolean transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$ with $8n < q \lesssim n2^{n-o(n)}$ additional inputs. The general upper bounds $L(n,q) \lesssim 2^n + 8n2^n \mathop / (\log_2 (q-4n) - \log_2 n - 2)$ and $D(n,q) \lesssim 2^{n+1}(2,5 + \log_2 n - \log_2 (\log_2 (q - 4n) - \log_2 n -…
In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes $\textrm{NC}^1$, $\textrm{SAC}^1$ and $\textrm{AC}^1$ as well as their arithmetic counterparts $\#\textrm{NC}^1$, $\#\textrm{SAC}^1$ and $\#\textrm{AC}^1$. We build on Immerman's characterization of constant-depth polynomial-size circuits by formulas of first-order logic, i.e., $\textrm{AC}^0 = \textrm{FO}$, and augment the logical language with an operator for defining relations in an inductive way. Considering sli…
We study the following computational problem: for which values of $k$, the majority of $n$ bits $\text{MAJ}_n$ can be computed with a depth two formula whose each gate computes a majority function of at most $k$ bits? The corresponding computational model is denoted by $\text{MAJ}_k \circ \text{MAJ}_k$. We observe that the minimum value of $k$ for which there exists a $\text{MAJ}_k \circ \text{MAJ}_k$ circuit that has high correlation with the majority of $n$ bits is equal to $Θ(n^{1/2})$. We then show that for a randomized $\text{MAJ}_k \circ \text{MAJ}_k$ circuit computing the majority of $n…
