Citation
Sinha, Gaurav (2016) Blackbox Reconstruction of Depth Three Circuits with Top Fanin Two. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z92N507D. http://resolver.caltech.edu/CaltechTHESIS:06082016032155301
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 thesis we present a polynomial time randomized algorithm for reconstructing ΣΠΣ(2) circuits over characteristic zero fields F i.e. depth−3 circuits with fanin 2 at the top addition gate and having coefficients from a field of characteristic zero.
The algorithm needs only a blackbox query access to the polynomial f ∈ F[x1,...,xn] 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 g.c.d. of the two multiplication gates) is bigger than a fixed constant. Our algorithm runs in time polynomial in n and d and with high probability returns an equivalent ΣΠΣ(2) circuit.
The problem of reconstructing ΣΠΣ(2) circuits over finite fields was first proposed by Shpilka [27]. The generalization to ΣΠΣ(k) circuits, k = O(1) (over finite fields) was addressed by Karnin and Shpilka in [18]. 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 setting, such ideas immediately pose a problem and we need new techniques.
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 this work 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 [9] 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 [17].
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Reconstruction, Interpolation, Arithmetic circuits, Blackbox  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  25 May 2016  
NonCaltech Author Email:  sinhagaur88 (AT) gmail.com  
Record Number:  CaltechTHESIS:06082016032155301  
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:06082016032155301  
DOI:  10.7907/Z92N507D  
Related URLs: 
 
ORCID: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9861  
Collection:  CaltechTHESIS  
Deposited By:  Gaurav Sinha  
Deposited On:  22 Jun 2016 17:55  
Last Modified:  18 May 2017 19:44 
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