Efficient Reconstruction of Random Multilinear Formulas

Abstract

In the reconstruction problem for a multivariate polynomial f, we have black box access to $f$ and the goal is to efficiently reconstruct a representation of $f$ in a suitable model of computation. We give a polynomial time randomized algorithm for reconstructing \emph{random} multilinear formulas. Our algorithm succeeds with high probability when given black box access to the polynomial computed by a random multilinear formula according to a natural distribution. This is the strongest model of computation for which a reconstruction algorithm is presently known, albeit efficient in a distributional sense rather than in the worst-case. Previous results on this problem considered much weaker models such as depth-3 circuits with various restrictions or read-once formulas. Our proof uses ranks of partial derivative matrices as a key ingredient and combines it with analysis of the algebraic structure of random multilinear formulas. Partial derivative matrices have earlier been used to prove lower bounds in a number of models of arithmetic complexity, including multilinear formulas and constant depth circuits. As such, our results give supporting evidence to the general thesis that mathematical properties that capture efficient computation in a model should also enable learning algorithms for functions efficiently computable in that model.