2,393 research outputs found
Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree
In this paper we study the problem of deterministic factorization of sparse
polynomials. We show that if is a
polynomial with monomials, with individual degrees of its variables bounded
by , then can be deterministically factored in time . Prior to our work, the only efficient factoring algorithms known for
this class of polynomials were randomized, and other than for the cases of
and , only exponential time deterministic factoring algorithms were
known.
A crucial ingredient in our proof is a quasi-polynomial sparsity bound for
factors of sparse polynomials of bounded individual degree. In particular we
show if is an -sparse polynomial in variables, with individual
degrees of its variables bounded by , then the sparsity of each factor of
is bounded by . This is the first nontrivial bound on
factor sparsity for . Our sparsity bound uses techniques from convex
geometry, such as the theory of Newton polytopes and an approximate version of
the classical Carath\'eodory's Theorem.
Our work addresses and partially answers a question of von zur Gathen and
Kaltofen (JCSS 1985) who asked whether a quasi-polynomial bound holds for the
sparsity of factors of sparse polynomials
Tropical Geometry of Statistical Models
This paper presents a unified mathematical framework for inference in
graphical models, building on the observation that graphical models are
algebraic varieties.
From this geometric viewpoint, observations generated from a model are
coordinates of a point in the variety, and the sum-product algorithm is an
efficient tool for evaluating specific coordinates. The question addressed here
is how the solutions to various inference problems depend on the model
parameters. The proposed answer is expressed in terms of tropical algebraic
geometry. A key role is played by the Newton polytope of a statistical model.
Our results are applied to the hidden Markov model and to the general Markov
model on a binary tree.Comment: 14 pages, 3 figures. Major revision. Applications now in companion
paper, "Parametric Inference for Biological Sequence Analysis
Phase retrieval from very few measurements
In many applications, signals are measured according to a linear process, but
the phases of these measurements are often unreliable or not available. To
reconstruct the signal, one must perform a process known as phase retrieval.
This paper focuses on completely determining signals with as few intensity
measurements as possible, and on efficient phase retrieval algorithms from such
measurements. For the case of complex M-dimensional signals, we construct a
measurement ensemble of size 4M-4 which yields injective intensity
measurements; this is conjectured to be the smallest such ensemble. For the
case of real signals, we devise a theory of "almost" injective intensity
measurements, and we characterize such ensembles. Later, we show that phase
retrieval from M+1 almost injective intensity measurements is NP-hard,
indicating that computationally efficient phase retrieval must come at the
price of measurement redundancy.Comment: 18 pages, 1 figur
Parametric Inference for Biological Sequence Analysis
One of the major successes in computational biology has been the unification,
using the graphical model formalism, of a multitude of algorithms for
annotating and comparing biological sequences. Graphical models that have been
applied towards these problems include hidden Markov models for annotation,
tree models for phylogenetics, and pair hidden Markov models for alignment. A
single algorithm, the sum-product algorithm, solves many of the inference
problems associated with different statistical models. This paper introduces
the \emph{polytope propagation algorithm} for computing the Newton polytope of
an observation from a graphical model. This algorithm is a geometric version of
the sum-product algorithm and is used to analyze the parametric behavior of
maximum a posteriori inference calculations for graphical models.Comment: 15 pages, 4 figures. See also companion paper "Tropical Geometry of
Statistical Models" (q-bio.QM/0311009
Explicit excluded volume of cylindrically symmetric convex bodies
We represent explicitly the excluded volume Ve{B1,B2} of two generic
cylindrically symmetric, convex rigid bodies, B1 and B2, in terms of a family
of shape functionals evaluated separately on B1 and B2. We show that Ve{B1,B2}
fails systematically to feature a dipolar component, thus making illusory the
assignment of any shape dipole to a tapered body in this class. The method
proposed here is applied to cones and validated by a shape-reconstruction
algorithm. It is further applied to spheroids (ellipsoids of revolution), for
which it shows how some analytic estimates already regarded as classics should
indeed be emended
Phase Transitions in Phase Retrieval
Consider a scenario in which an unknown signal is transformed by a known
linear operator, and then the pointwise absolute value of the unknown output
function is reported. This scenario appears in several applications, and the
goal is to recover the unknown signal -- this is called phase retrieval. Phase
retrieval has been a popular subject of research in the last few years, both in
determining whether complete information is available with a given linear
operator, and in finding efficient and stable phase retrieval algorithms in the
cases where complete information is available. Interestingly, there are a few
ways to measure information completeness, and each way appears to be governed
by a phase transition of sorts. This chapter will survey the state of the art
with some of these phase transitions, and identify a few open problems for
further research.Comment: Book chapter, survey of recent literature, submitted to Excursions in
Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Cente
Efficient Algorithms for Privately Releasing Marginals via Convex Relaxations
Consider a database of people, each represented by a bit-string of length
corresponding to the setting of binary attributes. A -way marginal
query is specified by a subset of attributes, and a -dimensional
binary vector specifying their values. The result for this query is a
count of the number of people in the database whose attribute vector restricted
to agrees with .
Privately releasing approximate answers to a set of -way marginal queries
is one of the most important and well-motivated problems in differential
privacy. Information theoretically, the error complexity of marginal queries is
well-understood: the per-query additive error is known to be at least
and at most
. However, no polynomial
time algorithm with error complexity as low as the information theoretic upper
bound is known for small . In this work we present a polynomial time
algorithm that, for any distribution on marginal queries, achieves average
error at most . This error
bound is as good as the best known information theoretic upper bounds for
. This bound is an improvement over previous work on efficiently releasing
marginals when is small and when error is desirable. Using private
boosting we are also able to give nearly matching worst-case error bounds.
Our algorithms are based on the geometric techniques of Nikolov, Talwar, and
Zhang. The main new ingredients are convex relaxations and careful use of the
Frank-Wolfe algorithm for constrained convex minimization. To design our
relaxations, we rely on the Grothendieck inequality from functional analysis
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