Efficient reconstruction of random multilinear formulas

Ankit Gupta*, Neeraj Kayal, Satya Lokam

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 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.

Original languageEnglish
Title of host publicationProceedings - 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011
Number of pages10
StatePublished - 2011
Externally publishedYes
Event2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011 - Palm Springs, CA, United States
Duration: 22 Oct 201125 Oct 2011

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428


Conference2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011
Country/TerritoryUnited States
CityPalm Springs, CA


  • arithmetic circuits
  • learning
  • multilinear formulas
  • reconstruction


Dive into the research topics of 'Efficient reconstruction of random multilinear formulas'. Together they form a unique fingerprint.

Cite this