[Paper Review] A Quadratic Lower Bound for Homogeneous Algebraic Branching Programs

Yabe, Akihiro|arXiv (Cornell University)|Apr 1, 2015

Complexity and Algorithms in Graphs14 references10 citations

TL;DR

This paper introduces the bi-polynomial rank as a new rank-based lower bound technique for determinantal complexity, proving that it provides a lower bound on the size of arithmetic branching programs computing a polynomial. Over the reals, this yields a tight (d−1)²+1 lower bound for the permanent, improving upon the prior d²/2 bound and offering a novel optimization-based approach via concave minimization to attack the permanent vs. determinant problem.

ABSTRACT

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 variables x_1, x_2, ..., x_n over an underlying field. An ABP computes a polynomial in a natural way, as the sum of weights of all paths from s to t, where the weight of a path is the product of the weights of the edges in the path. An ABP is said to be homogeneous if the polynomial computed at every vertex is homogeneous. In this paper, we show that any homogeneous algebraic branching program which computes the polynomial x_1^n + x_2^n + ... + x_n^n has at least Omega(n^2) vertices (and edges). To the best of our knowledge, this seems to be the first non-trivial super-linear lower bound on the number of vertices for a general homogeneous ABP and slightly improves the known lower bound of Omega(n log n) on the number of edges in a general (possibly non-homogeneous) ABP, which follows from the classical results of Strassen (1973) and Baur--Strassen (1983). On the way, we also get an alternate and unified proof of an Omega(n log n) lower bound on the size of a homogeneous arithmetic circuit (follows from [Strassen, 1973] and [Baur-Strassen, 1983]), and an n/2 lower bound (n over reals) on the determinantal complexity of an explicit polynomial [Mignon-Ressayre, 2004], [Cai, Chen, Li, 2010], [Yabe, 2015]. These are currently the best lower bounds known for these problems for any explicit polynomial, and were originally proved nearly two decades apart using seemingly different proof techniques.

Motivation & Objective

• To develop a new rank-based lower bound technique for determinantal complexity.
• To improve the known quadratic lower bound of d²/2 for the permanent's determinantal complexity.
• To provide a computational framework using concave minimization to estimate bi-polynomial rank.
• To offer a new strategy for attacking the permanent vs. determinant problem in algebraic complexity theory.

Proposed method

• Define the bi-polynomial rank b-rank(p) of a homogeneous polynomial p ∈ K[x](2k) as the minimal n such that p = Σ_{i=1}^n fi gi with fi, gi ∈ K[x](k).
• Prove that determinantal complexity dc(p) ≥ (1/2^{2k−2}) × b-rank(p(2k)_x0) − 2(k−1)D^{k−1} for x0 ∈ Zeros(p).
• Formulate the computation of b-rank(p) as a rank minimization problem over an affine subspace Xp ⊆ Sym_n, intersected with positive semidefinite matrices.
• Use concave minimization theory: if the sum of the n−r smallest eigenvalues of all X ∈ Xp ∩ Psd_n is positive, then b-rank(p) > r/2.
• Propose a certificate via extreme points of a polyhedral convex set P containing Xp ∩ Psd_n to verify positivity of the smallest eigenvalue sum.
• Apply this to the permanent polynomial, showing that b-rank(perm(2k)_d,X_d) = Ω(d^{2k}) implies dc(permd) = Ω(d^{2k}).

Experimental results

Research questions

• RQ1 Can a higher-order generalization of the Hessian rank method yield improved lower bounds on determinantal complexity?
• RQ2 Does the bi-polynomial rank provide a meaningful lower bound for the determinantal complexity of the permanent?
• RQ3 Can concave minimization techniques be used to compute or lower-bound the bi-polynomial rank efficiently over the reals?
• RQ4 Is there a computational strategy that can establish Ω(d^4) lower bounds for dc(permd) via fourth derivatives of the permanent?
• RQ5 Can the bi-polynomial rank framework be used to construct a new optimization-theoretic flip strategy for the permanent vs. determinant problem?

Key findings

• The bi-polynomial rank provides a lower bound on determinantal complexity: dc(p) ≥ (1/2^{2k−2}) × b-rank(p(2k)_x0) − 2(k−1)D^{k−1}.
• Over the reals, the determinantal complexity of the d×d permanent satisfies dc(permd) ≥ (d−1)^2 + 1, improving the prior d²/2 bound.
• The bi-polynomial rank of a generic homogeneous polynomial of degree 2k is at least k! D^k / 2(2k)!, indicating potential for super-quadratic lower bounds.
• For k=1, the method recovers and sharpens the Hessian-based bound of Mignon and Ressayre.
• The approach enables a new strategy to attack the permanent vs. determinant problem by focusing on higher-order derivatives, such as the fourth derivative for Ω(d^4) bounds.
• A computational certificate via extreme points of a polyhedron ensures b-rank(p) > r/2 if the sum of the n−r smallest eigenvalues of all X ∈ Xp ∩ Psd_n is positive.