3,933 research outputs found
Polynomial Threshold Functions, AC^0 Functions and Spectral Norms
The class of polynomial-threshold functions is studied using harmonic analysis, and the results are used to derive lower bounds related to AC^0 functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). The main result is that polynomial-threshold functions can be characterized by means of their spectral representation. In particular, it is proved that a Boolean function whose L_1 spectral norm is bounded by a polynomial in n is a polynomial-threshold function, and that a Boolean function whose L_β^(-1) spectral norm is not bounded by a polynomial in n is not a polynomial-threshold function. Some results for AC^0 functions are derived
A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits
We give a nontrivial algorithm for the satisfiability problem for cn-wire
threshold circuits of depth two which is better than exhaustive search by a
factor 2^{sn} where s= 1/c^{O(c^2)}. We believe that this is the first
nontrivial satisfiability algorithm for cn-wire threshold circuits of depth
two. The independently interesting problem of the feasibility of sparse 0-1
integer linear programs is a special case. To our knowledge, our algorithm is
the first to achieve constant savings even for the special case of Integer
Linear Programming. The key idea is to reduce the satisfiability problem to the
Vector Domination Problem, the problem of checking whether there are two
vectors in a given collection of vectors such that one dominates the other
component-wise.
We also provide a satisfiability algorithm with constant savings for depth
two circuits with symmetric gates where the total weighted fan-in is at most
cn.
One of our motivations is proving strong lower bounds for TC^0 circuits,
exploiting the connection (established by Williams) between satisfiability
algorithms and lower bounds. Our second motivation is to explore the connection
between the expressive power of the circuits and the complexity of the
corresponding circuit satisfiability problem
String Matching: Communication, Circuits, and Learning
String matching is the problem of deciding whether a given n-bit string contains a given k-bit pattern. We study the complexity of this problem in three settings.
- Communication complexity. For small k, we provide near-optimal upper and lower bounds on the communication complexity of string matching. For large k, our bounds leave open an exponential gap; we exhibit some evidence for the existence of a better protocol.
- Circuit complexity. We present several upper and lower bounds on the size of circuits with threshold and DeMorgan gates solving the string matching problem. Similarly to the above, our bounds are near-optimal for small k.
- Learning. We consider the problem of learning a hidden pattern of length at most k relative to the classifier that assigns 1 to every string that contains the pattern. We prove optimal bounds on the VC dimension and sample complexity of this problem
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Quantified Derandomization of Linear Threshold Circuits
One of the prominent current challenges in complexity theory is the attempt
to prove lower bounds for , the class of constant-depth, polynomial-size
circuits with majority gates. Relying on the results of Williams (2013), an
appealing approach to prove such lower bounds is to construct a non-trivial
derandomization algorithm for . In this work we take a first step towards
the latter goal, by proving the first positive results regarding the
derandomization of circuits of depth .
Our first main result is a quantified derandomization algorithm for
circuits with a super-linear number of wires. Specifically, we construct an
algorithm that gets as input a circuit over input bits with
depth and wires, runs in almost-polynomial-time, and
distinguishes between the case that rejects at most inputs
and the case that accepts at most inputs. In fact, our
algorithm works even when the circuit is a linear threshold circuit, rather
than just a circuit (i.e., is a circuit with linear threshold gates,
which are stronger than majority gates).
Our second main result is that even a modest improvement of our quantified
derandomization algorithm would yield a non-trivial algorithm for standard
derandomization of all of , and would consequently imply that
. Specifically, if there exists a quantified
derandomization algorithm that gets as input a circuit with depth
and wires (rather than wires), runs in time at
most , and distinguishes between the case that rejects at
most inputs and the case that accepts at most
inputs, then there exists an algorithm with running time
for standard derandomization of .Comment: Changes in this revision: An additional result (a PRG for quantified
derandomization of depth-2 LTF circuits); rewrite of some of the exposition;
minor correction
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