10 research outputs found
Bounds on an exponential sum arising in Boolean circuit complexity
We study exponential sums of the form S = 2-n ∑x∈{0,1}n em (h(x))eq (p(x)), where m, q ∈ Z+ are relatively prime, p is a polynomial with coefficients in Zq, and h(x) = a(x1 +⋯+ xn) for some 1 ≤ a \u3c m. We prove an upper bound of the form 2-Ω(n) on S . This generalizes a result of J. Bourgain, who establishes this bound in the case where q is odd. This bound has consequences in Boolean circuit complexity. © Académie des sciences. Published by Elsevier SAS. All rights reserved
Incomplete Quadratic Exponential Sums in Several Variables
We consider incomplete exponential sums in several variables of the form
S(f,n,m) = \frac{1}{2^n} \sum_{x_1 \in \{-1,1\}} ... \sum_{x_n \in \{-1,1\}}
x_1 ... x_n e^{2\pi i f(x)/p}, where m>1 is odd and f is a polynomial of degree
d with coefficients in Z/mZ. We investigate the conjecture, originating in a
problem in computational complexity, that for each fixed d and m the maximum
norm of S(f,n,m) converges exponentially fast to 0 as n grows to infinity. The
conjecture is known to hold in the case when m=3 and d=2, but existing methods
for studying incomplete exponential sums appear to be insufficient to resolve
the question for an arbitrary odd modulus m, even when d=2. In the present
paper we develop three separate techniques for studying the problem in the case
of quadratic f, each of which establishes a different special case of the
conjecture. We show that a bound of the required sort holds for almost all
quadratic polynomials, a stronger form of the conjecture holds for all
quadratic polynomials with no more than 10 variables, and for arbitrarily many
variables the conjecture is true for a class of quadratic polynomials having a
special form.Comment: 31 pages (minor corrections from original draft, references to new
results in the subject, publication information
Uniqueness of Optimal Mod 3 Circuits for Parity
We prove that the quadratic polynomials modulo
with the largest correlation with parity are unique up to
permutation of variables and constant factors. As a consequence of
our result, we completely characterize the smallest
MAJ~ circuits that compute parity, where a
MAJ~ circuit is one that has a
majority gate as output, a middle layer of MOD gates and a
bottom layer of AND gates of fan-in . We
also prove that the sub-optimal circuits exhibit a stepped behavior:
any sub-optimal circuits of this class that compute parity
must have size at least a factor of times the
optimal size. This verifies, for the special case of ,
two conjectures made
by Due~{n}ez, Miller, Roy and Straubing (Journal of Number Theory, 2006) for general MAJ~ circuits for any odd . The correlation
and circuit bounds are obtained by studying the associated
exponential sums, based on some of the techniques developed
by Green (JCSS, 2004). We regard this as a step towards
obtaining tighter bounds both for the quadratic
case as well as for
higher degrees
Limits on Representing Boolean Functions by Linear Combinations of Simple Functions: Thresholds, ReLUs, and Low-Degree Polynomials
We consider the problem of representing Boolean functions exactly by "sparse"
linear combinations (over ) of functions from some "simple" class
. In particular, given we are interested in finding
low-complexity functions lacking sparse representations. When is the
set of PARITY functions or the set of conjunctions, this sort of problem has a
well-understood answer, the problem becomes interesting when is
"overcomplete" and the set of functions is not linearly independent. We focus
on the cases where is the set of linear threshold functions, the set
of rectified linear units (ReLUs), and the set of low-degree polynomials over a
finite field, all of which are well-studied in different contexts.
We provide generic tools for proving lower bounds on representations of this
kind. Applying these, we give several new lower bounds for "semi-explicit"
Boolean functions. For example, we show there are functions in nondeterministic
quasi-polynomial time that require super-polynomial size:
Depth-two neural networks with sign activation function, a special
case of depth-two threshold circuit lower bounds.
Depth-two neural networks with ReLU activation function.
-linear combinations of -degree
-polynomials, for every prime (related to problems regarding
Higher-Order "Uncertainty Principles"). We also obtain a function in
requiring linear combinations.
-linear combinations of circuits of
polynomial size (further generalizing the recent lower bounds of Murray and the
author).
(The above is a shortened abstract. For the full abstract, see the paper.
The correlation between parity and quadratic polynomials mod 3
We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2 Ω(n). This is the first result of this type for general mod3 subcircuits with ANDs of fan-in greater than 1. This yields an exponential improvement over a long-standing result of Smolensky, answering a question recently posed by Alon and Beigel. The proof uses a novel inductive estimate of the relevant exponential sums introduced by Cai, Green and Thierauf. The exponential sum and correlation bounds presented here are tight. © 2004 Elsevier Inc. All rights reserved
The Correlation Between Parity and Quadratic Polynomials
We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2 . This is the first result of this type for general mod 3 subcircuits with ANDs of fan-in greater than 1. This yields an exponential improvement over a long-standing result of Smolensky, answering a question recently posed by Alon and Beigel. The proof uses a novel inductive estimate of the relevant exponential sums introduced by Cai, Green and Thierauf. The exponential sum and correlation bounds presented here are tight