1,174 research outputs found
Deterministic Black-Box Identity Testing -Ordered Algebraic Branching Programs
In this paper we study algebraic branching programs (ABPs) with restrictions
on the order and the number of reads of variables in the program. Given a
permutation of variables, for a -ordered ABP (-OABP), for
any directed path from source to sink, a variable can appear at most once
on , and the order in which variables appear on must respect . An
ABP is said to be of read , if any variable appears at most times in
. Our main result pertains to the identity testing problem. Over any field
and in the black-box model, i.e. given only query access to the polynomial,
we have the following result: read -OABP computable polynomials can be
tested in \DTIME[2^{O(r\log r \cdot \log^2 n \log\log n)}].
Our next set of results investigates the computational limitations of OABPs.
It is shown that any OABP computing the determinant or permanent requires size
and read . We give a multilinear polynomial
in variables over some specifically selected field , such that
any OABP computing must read some variable at least times. We show
that the elementary symmetric polynomial of degree in variables can be
computed by a size read OABP, but not by a read OABP, for
any . Finally, we give an example of a polynomial and two
variables orders , such that can be computed by a read-once
-OABP, but where any -OABP computing must read some variable at
least $2^n
Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas
In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation
between black-box PIT and lower bounds we obtain lower bounds for these models.
For depth-3 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 2*delta/3))). This implies a lower bound of exp(~Omega(n^(1/2))) for depth-3 multilinear formulas, for some explicit polynomial.
For depth-4 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 4*delta/3)). This implies a lower bound of exp(~Omega(n^(1/4))) for depth-4 multilinear formulas, for some explicit polynomial.
A regular formula consists of alternating layers of +,* gates, where all gates at layer i have the same fan-in. We give a
hitting set of size (roughly) exp(n^(1-delta)), for regular depth-d multilinear formulas of size exp(n^delta), where delta = O(1/sqrt(5)^d)). This result implies a lower bound of roughly exp(~Omega(n^(1/sqrt(5)^d))) for such formulas.
We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of
a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known.
Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a
read-once algebraic branching program (from depth-3 formulas we go straight to read-once algebraic branching programs)
Lower Bounds for Multilinear Order-Restricted ABPs
Proving super-polynomial lower bounds on the size of syntactic multilinear Algebraic Branching Programs (smABPs) computing an explicit polynomial is a challenging problem in Algebraic Complexity Theory. The order in which variables in {x_1,...,x_n} appear along any source to sink path in an smABP can be viewed as a permutation in S_n. In this article, we consider the following special classes of smABPs where the order of occurrence of variables along a source to sink path is restricted:
1) Strict circular-interval ABPs: For every sub-program the index set of variables occurring in it is contained in some circular interval of {1,..., n}.
2) L-ordered ABPs: There is a set of L permutations (orders) of variables such that every source to sink path in the smABP reads variables in one of these L orders, where L 0.
We prove exponential (i.e., 2^{Omega(n^delta)}, delta>0) lower bounds on the size of above models computing an explicit multilinear 2n-variate polynomial in VP.
As a main ingredient in our lower bounds, we show that any polynomial that can be computed by an smABP of size S, can be written as a sum of O(S) many multilinear polynomials where each summand is a product of two polynomials in at most 2n/3 variables, computable by smABPs. As a corollary, we show that any size S syntactic multilinear ABP can be transformed into a size S^{O(sqrt{n})} depth four syntactic multilinear Sigma Pi Sigma Pi circuit where the bottom Sigma gates compute polynomials on at most O(sqrt{n}) variables.
Finally, we compare the above models with other standard models for computing multilinear polynomials
Black-Box Identity Testing of Noncommutative Rational Formulas in Deterministic Quasipolynomial Time
Rational Identity Testing (RIT) is the decision problem of determining
whether or not a noncommutative rational formula computes zero in the free skew
field. It admits a deterministic polynomial-time white-box algorithm [Garg,
Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018);
Hamada and Hirai (2021)], and a randomized polynomial-time algorithm [Derksen
and Makam (2017)] in the black-box setting, via singularity testing of linear
matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in
the white-box setting follows from the result of Derksen and Makam (2017).
Designing an efficient deterministic black-box algorithm for RIT and
understanding the parallel complexity of RIT are major open problems in this
area. Despite being open since the work of Garg, Gurvits, Oliveira, and
Wigderson (2016), these questions have seen limited progress. In fact, the only
known result in this direction is the construction of a quasipolynomial-size
hitting set for rational formulas of only inversion height two [Arvind,
Chatterjee, Mukhopadhyay (2022)].
In this paper, we significantly improve the black-box complexity of this
problem and obtain the first quasipolynomial-size hitting set for all rational
formulas of polynomial size. Our construction also yields the first
deterministic quasi-NC upper bound for RIT in the white-box setting.Comment: A white-box quasi-NC RIT algorithm has been adde
Black-Box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial Time
Hrube\v{s} and Wigderson (2015) initiated the complexity-theoretic study of
noncommutative formulas with inverse gates. They introduced the Rational
Identity Testing (RIT) problem which is to decide whether a noncommutative
rational formula computes zero in the free skew field. In the white-box
setting, deterministic polynomial-time algorithms are known for this problem
following the works of Garg, Gurvits, Oliveira, and Wigderson (2016) and
Ivanyos, Qiao, and Subrahmanyam (2018).
A central open problem in this area is to design efficient deterministic
black-box identity testing algorithm for rational formulas. In this paper, we
solve this problem for the first nested inverse case. More precisely, we obtain
a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative
rational formulas of inversion height two via a hitting set construction.
Several new technical ideas are involved in the hitting set construction,
including key concepts from matrix coefficient realization theory
(Vol\v{c}i\v{c}, 2018) and properties of cyclic division algebra (Lam, 2001).
En route to the proof, an important step is to embed the hitting set of Forbes
and Shpilka for noncommutative formulas (2013) inside a cyclic division algebra
of small index
Deterministic Identity Testing Paradigms for Bounded Top-Fanin Depth-4 Circuits
Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-4 reduction (Agrawal & Vinay, FOCS\u2708) has made PIT for depth-4 circuits, an enticing pursuit. The largely open special-cases of sum-product-of-sum-of-univariates (?^[k] ? ? ?) and sum-product-of-constant-degree-polynomials (?^[k] ? ? ?^[?]), for constants k, ?, have been a source of many great ideas in the last two decades. For eg. depth-3 ideas (Dvir & Shpilka, STOC\u2705; Kayal & Saxena, CCC\u2706; Saxena & Seshadhri, FOCS\u2710, STOC\u2711); depth-4 ideas (Beecken, Mittmann & Saxena, ICALP\u2711; Saha,Saxena & Saptharishi, Comput.Compl.\u2713; Forbes, FOCS\u2715; Kumar & Saraf, CCC\u2716); geometric Sylvester-Gallai ideas (Kayal & Saraf, FOCS\u2709; Shpilka, STOC\u2719; Peleg & Shpilka, CCC\u2720, STOC\u2721). We solve two of the basic underlying open problems in this work.
We give the first polynomial-time PIT for ?^[k] ? ? ?. Further, we give the first quasipolynomial time blackbox PIT for both ?^[k] ? ? ? and ?^[k] ? ? ?^[?]. No subexponential time algorithm was known prior to this work (even if k = ? = 3). A key technical ingredient in all the three algorithms is how the logarithmic derivative, and its power-series, modify the top ?-gate to ?
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