68 research outputs found
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)
Sums of products of polynomials in few variables : lower bounds and polynomial identity testing
We study the complexity of representing polynomials as a sum of products of
polynomials in few variables. More precisely, we study representations of the
form such that each is
an arbitrary polynomial that depends on at most variables. We prove the
following results.
1. Over fields of characteristic zero, for every constant such that , we give an explicit family of polynomials , where
is of degree in variables, such that any
representation of the above type for with requires . This strengthens a recent result of Kayal and Saha
[KS14a] which showed similar lower bounds for the model of sums of products of
linear forms in few variables. It is known that any asymptotic improvement in
the exponent of the lower bounds (even for ) would separate VP
and VNP[KS14a].
2. We obtain a deterministic subexponential time blackbox polynomial identity
testing (PIT) algorithm for circuits computed by the above model when and
the individual degree of each variable in are at most and
for any constant . We get quasipolynomial running
time when . The PIT algorithm is obtained by combining our
lower bounds with the hardness-randomness tradeoffs developed in [DSY09, KI04].
To the best of our knowledge, this is the first nontrivial PIT algorithm for
this model (even for the case ), and the first nontrivial PIT algorithm
obtained from lower bounds for small depth circuits
Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs
Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs). In this work, we give an exponential lower bound of exp(n/k^{O(k)}) on the width of any read-k oblivious ABP computing some explicit multilinear polynomial f that is computed by a polynomial size depth-3 circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time 2^{~O(n^{1-1/2^{k-1}})} and needs white box access only to know the order in which the variables appear in the ABP
Shallow Circuits with High-Powered Inputs
A polynomial identity testing algorithm must determine whether an input
polynomial (given for instance by an arithmetic circuit) is identically equal
to 0. In this paper, we show that a deterministic black-box identity testing
algorithm for (high-degree) univariate polynomials would imply a lower bound on
the arithmetic complexity of the permanent. The lower bounds that are known to
follow from derandomization of (low-degree) multivariate identity testing are
weaker. To obtain our lower bound it would be sufficient to derandomize
identity testing for polynomials of a very specific norm: sums of products of
sparse polynomials with sparse coefficients. This observation leads to new
versions of the Shub-Smale tau-conjecture on integer roots of univariate
polynomials. In particular, we show that a lower bound for the permanent would
follow if one could give a good enough bound on the number of real roots of
sums of products of sparse polynomials (Descartes' rule of signs gives such a
bound for sparse polynomials and products thereof). In this third version of
our paper we show that the same lower bound would follow even if one could only
prove a slightly superpolynomial upper bound on the number of real roots. This
is a consequence of a new result on reduction to depth 4 for arithmetic
circuits which we establish in a companion paper. We also show that an even
weaker bound on the number of real roots would suffice to obtain a lower bound
on the size of depth 4 circuits computing the permanent.Comment: A few typos correcte
Succinct Hitting Sets and Barriers to Proving Lower Bounds for Algebraic Circuits
We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich (1997) for Boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike in the Boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding whether algebraic circuits are expressive enough to support “cryptography” secure against algebraic circuits.
Following a similar result of Williams (2016) in the Boolean setting, we show that the existence of an algebraic natural proofs barrier is equivalent to the existence of succinct derandomization of the polynomial identity testing problem, that is, to the existence of a hitting set for the class of poly(N)-degree poly(N)-size circuits which consists of coefficient vectors of polynomials of polylog(N) degree with polylog(N)-size circuits. Further, we give an explicit universal construction showing that if such a succinct hitting set exists, then our universal construction suffices.
Further, we assess the existing literature constructing hitting sets for restricted classes of algebraic circuits and observe that none of them are succinct as given. Yet, we show how to modify some of these constructions to obtain succinct hitting sets. This constitutes the first evidence supporting the existence of an algebraic natural proofs barrier.
Our framework is similar to the Geometric Complexity Theory (GCT) program of Mulmuley and Sohoni (2001), except that here we emphasize constructiveness of the proofs while the GCT program emphasizes symmetry. Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits.
A conference version of this paper appeared in the Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC 2017)
Hitting Sets for Orbits of Circuit Classes and Polynomial Families
The orbit of an n-variate polynomial f(?) over a field ? is the set {f(A?+?) : A ? GL(n,?) and ? ? ??}. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of:
1) Low-individual-degree polynomials computable by commutative ROABPs. This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials.
2) Multilinear polynomials computable by constant-width ROABPs. This implies a quasi-polynomial time hitting set for the orbits of the family {IMM_{3,d}}_{d ? ?}, which is complete for arithmetic formulas.
3) Polynomials computable by constant-depth, constant-occur formulas. This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in, and also polynomial-time hitting sets for the orbits of the power symmetric and the sum-product polynomials.
4) Polynomials computable by occur-once formulas
Proof complexity lower bounds from algebraic circuit complexity
We give upper and lower bounds on the power of subsystems of the Ideal Proof
System (IPS), the algebraic proof system recently proposed by Grochow and
Pitassi, where the circuits comprising the proof come from various restricted
algebraic circuit classes. This mimics an established research direction in the
boolean setting for subsystems of Extended Frege proofs, where proof-lines are
circuits from restricted boolean circuit classes. Except one, all of the
subsystems considered in this paper can simulate the well-studied
Nullstellensatz proof system, and prior to this work there were no known lower
bounds when measuring proof size by the algebraic complexity of the polynomials
(except with respect to degree, or to sparsity).
We give two general methods of converting certain algebraic lower bounds into
proof complexity ones. Our methods require stronger notions of lower bounds,
which lower bound a polynomial as well as an entire family of polynomials it
defines. Our techniques are reminiscent of existing methods for converting
boolean circuit lower bounds into related proof complexity results, such as
feasible interpolation. We obtain the relevant types of lower bounds for a
variety of classes (sparse polynomials, depth-3 powering formulas, read-once
oblivious algebraic branching programs, and multilinear formulas), and infer
the relevant proof complexity results. We complement our lower bounds by giving
short refutations of the previously-studied subset-sum axiom using IPS
subsystems, allowing us to conclude strict separations between some of these
subsystems
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