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Deterministic identity testing for sum of readonce oblivious arithmetic branching programs
 In 30th Conference on Computational Complexity, CCC 2015
"... A readonce oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding bl ..."
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A readonce oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasipolynomial time complexity nO(logn). In both the cases, our time complexity is double exponential in the number of ROABPs. ROABPs are a generalization of setmultilinear depth3 circuits. The prior results for the sum of constantly many setmultilinear depth3 circuits were only slightly better than bruteforce, i.e. exponentialtime. Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and lowsupport rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension (or partial derivatives).
Dimension Expanders via Rank Condensers
, 2014
"... An emerging theory of “linearalgebraic pseudorandomness” aims to understand the linearalgebraic analogs of fundamental Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In this work, we study and highlight the interrelationships between several such al ..."
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An emerging theory of “linearalgebraic pseudorandomness” aims to understand the linearalgebraic analogs of fundamental Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In this work, we study and highlight the interrelationships between several such algebraic objects such as subspace designs, dimension expanders, seeded rank condensers, twosource rank condensers, and rankmetric codes. In particular, with the recent construction of nearoptimal subspace designs by Guruswami and Kopparty [GK13] as a starting point, we construct good (seeded) rank condensers (both lossless and lossy versions), which are a small collection of linear maps Fn → Ft for t n such that for every subset of Fn of small rank, its rank is preserved (up to a constant factor in the lossy case) by at least one of the maps. We then compose a tensoring operation with our lossy rank condenser to construct constantdegree dimension expanders over polynomially large fields. That is, we give O(1) explicit linear maps Ai: Fn→ Fn such that for any subspace V ⊆ Fn of dimension at most n/2, dim(∑i Ai(V))> (1+Ω(1))dim(V). Previous constructions of such constantdegree dimension expanders were based on Kazhdan’s property T (for the case when F has characteristic zero) or monotone expanders (for every field F); in either case