Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published June 2016 | Submitted + Published
Book Section - Chapter Open

Reconstruction of real depth-3 circuits with top fan-in 2

Sinha, Gaurav
Other:
Raz, Ran

Abstract

Reconstruction of arithmetic circuits has been heavily studied in the past few years and has connections to proving lower bounds and deterministic identity testing. In this paper we present a polynomial time randomized algorithm for reconstructing ΣΠΣ(2) circuits over F (char(F) = 0), i.e. depth-3 circuits with fan-in 2 at the top addition gate and having coefficients from a field of characteristic 0. The algorithm needs only a blackbox query access to the polynomial f ∈ F[x_1, ..., x_n] of degree d, computable by a ΣΠΣ(2) circuit C. In addition, we assume that the "simple rank" of this polynomial (essential number of variables after removing the gcd of the two multiplication gates) is bigger than a fixed constant. Our algorithm runs in time poly(n, d) and returns an equivalent ΣΠΣ(2) circuit (with high probability). The problem of reconstructing ΣΠΣ(2) circuits over finite fields was first proposed by Shpilka [24]. The generalization to ΣΠΣ(k) circuits, k = O(1) (over finite fields) was addressed by Karnin and Shpilka in [15]. The techniques in these previous involve iterating over all objects of certain kinds over the ambient field and thus the running time depends on the size of the field F. Their reconstruction algorithm uses lower bounds on the lengths of Linear Locally Decodable Codes with 2 queries. In our settings, such ideas immediately pose a problem and we need new ideas to handle the case of the characteristic 0 field F. Our main techniques are based on the use of Quantitative Sylvester Gallai Theorems from the work of Barak et al. [3] to find a small collection of "nice" subspaces to project onto. The heart of our paper lies in subtle applications of the Quantitative Sylvester Gallai theorems to prove why projections w.r.t. the "nice" subspaces can be "glued". We also use Brill's Equations from [8] to construct a small set of candidate linear forms (containing linear forms from both gates). Another important technique which comes very handy is the polynomial time randomized algorithm for factoring multivariate polynomials given by Kaltofen [14].

Additional Information

© 2016 Gaurav Sinha; licensed under Creative Commons License CC-BY. I am extremely thankful to Neeraj Kayal for introducing me to this problem. Sukhada Fadnavis, Neeraj Kayal and myself started working on the problem together during my summer internship at Microsoft Research India Labs in 2011. We solved the first important case together. I'm grateful to them for all helpful discussions, constant guidance and encouragement. I would like to thank Zeev Dvir for communicating the most recent rank bounds on δ − SGk configurations from [6] and his feedback on the work. This reduces the gap in the first problem we mentioned above. I would also like to thank Vinamra Agrawal, Pravesh Kothari and Piyush Srivastava for helpful discussions. I would also like to thank the anonymous reviewers for their suggestions. Lastly I would like to thank Microsoft Research for giving me the opportunity to intern at their Bangalore Labs with the Applied Mathematics Group.

Attached Files

Published - 33.pdf

Submitted - 1512.01256.pdf

Files

33.pdf
Files (2.0 MB)
Name Size Download all
md5:189cb6b9f66ebcabb8a048ee5c09e6f2
1.1 MB Preview Download
md5:59be8188d5e4746c73bafcf4aa5b48f9
937.9 kB Preview Download

Additional details

Created:
August 20, 2023
Modified:
January 14, 2024