789 research outputs found
A note on a problem in communication complexity
In this note, we prove a version of Tarui's Theorem in communication
complexity, namely . Consequently, every
measure for leads to a measure for , subsuming a result of
Linial and Shraibman that problems with high mc-rigidity lie outside the
polynomial hierarchy. By slightly changing the definition of mc-rigidity
(arbitrary instead of uniform distribution), it is then evident that the class
of problems with low mc-rigidity equals . As , this rules out the possibility, that had been
left open, that even polynomial space is contained in
Lower Bounds on the Bounded Coefficient Complexity of Bilinear Maps
We prove lower bounds of order for both the problem to multiply
polynomials of degree , and to divide polynomials with remainder, in the
model of bounded coefficient arithmetic circuits over the complex numbers.
These lower bounds are optimal up to order of magnitude. The proof uses a
recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix
multiplication. It reduces the linear problem to multiply a random circulant
matrix with a vector to the bilinear problem of cyclic convolution. We treat
the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp.
305-306, 1973] in a unitarily invariant way. This establishes a new lower bound
on the bounded coefficient complexity of linear forms in terms of the singular
values of the corresponding matrix. In addition, we extend these lower bounds
for linear and bilinear maps to a model of circuits that allows a restricted
number of unbounded scalar multiplications.Comment: 19 page
Rank-Sparsity Incoherence for Matrix Decomposition
Suppose we are given a matrix that is formed by adding an unknown sparse
matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix
into its sparse and low-rank components. Such a problem arises in a number of
applications in model and system identification, and is NP-hard in general. In
this paper we consider a convex optimization formulation to splitting the
specified matrix into its components, by minimizing a linear combination of the
norm and the nuclear norm of the components. We develop a notion of
\emph{rank-sparsity incoherence}, expressed as an uncertainty principle between
the sparsity pattern of a matrix and its row and column spaces, and use it to
characterize both fundamental identifiability as well as (deterministic)
sufficient conditions for exact recovery. Our analysis is geometric in nature,
with the tangent spaces to the algebraic varieties of sparse and low-rank
matrices playing a prominent role. When the sparse and low-rank matrices are
drawn from certain natural random ensembles, we show that the sufficient
conditions for exact recovery are satisfied with high probability. We conclude
with simulation results on synthetic matrix decomposition problems
Using Elimination Theory to construct Rigid Matrices
The rigidity of a matrix A for target rank r is the minimum number of entries
of A that must be changed to ensure that the rank of the altered matrix is at
most r. Since its introduction by Valiant (1977), rigidity and similar
rank-robustness functions of matrices have found numerous applications in
circuit complexity, communication complexity, and learning complexity. Almost
all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a
long-standing open question to construct infinite families of explicit matrices
even with superlinear rigidity when r = Omega(n).
In this paper, we construct an infinite family of complex matrices with the
largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in
this family are distinct primitive roots of unity of orders roughly exp(n^2 log
n). To the best of our knowledge, this is the first family of concrete (but not
entirely explicit) matrices having maximal rigidity and a succinct algebraic
description.
Our construction is based on elimination theory of polynomial ideals. In
particular, we use results on the existence of polynomials in elimination
ideals with effective degree upper bounds (effective Nullstellensatz). Using
elementary algebraic geometry, we prove that the dimension of the affine
variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we
use elimination theory to examine whether the rigidity function is
semi-continuous.Comment: 25 Pages, minor typos correcte
Block Rigidity: Strong Multiplayer Parallel Repetition Implies Super-Linear Lower Bounds for Turing Machines
We prove that a sufficiently strong parallel repetition theorem for a special
case of multiplayer (multiprover) games implies super-linear lower bounds for
multi-tape Turing machines with advice. To the best of our knowledge, this is
the first connection between parallel repetition and lower bounds for time
complexity and the first major potential implication of a parallel repetition
theorem with more than two players.
Along the way to proving this result, we define and initiate a study of block
rigidity, a weakening of Valiant's notion of rigidity. While rigidity was
originally defined for matrices, or, equivalently, for (multi-output) linear
functions, we extend and study both rigidity and block rigidity for general
(multi-output) functions. Using techniques of Paul, Pippenger, Szemer\'edi and
Trotter, we show that a block-rigid function cannot be computed by multi-tape
Turing machines that run in linear (or slightly super-linear) time, even in the
non-uniform setting, where the machine gets an arbitrary advice tape.
We then describe a class of multiplayer games, such that, a sufficiently
strong parallel repetition theorem for that class of games implies an explicit
block-rigid function. The games in that class have the following property that
may be of independent interest: for every random string for the verifier
(which, in particular, determines the vector of queries to the players), there
is a unique correct answer for each of the players, and the verifier accepts if
and only if all answers are correct. We refer to such games as independent
games. The theorem that we need is that parallel repetition reduces the value
of games in this class from to , where is the number of
repetitions.
As another application of block rigidity, we show conditional size-depth
tradeoffs for boolean circuits, where the gates compute arbitrary functions
over large sets.Comment: 17 pages, ITCS 202
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The Preliminary Design of a Standardized Spacecraft Bus for Small Tactical Satellites
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Feasibility study of a 200 watt per kilogram lightweight solar array system
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