3,088 research outputs found
The bottleneck degree of algebraic varieties
A bottleneck of a smooth algebraic variety is a pair
of distinct points such that the Euclidean normal spaces at
and contain the line spanned by and . The narrowness of bottlenecks
is a fundamental complexity measure in the algebraic geometry of data. In this
paper we study the number of bottlenecks of affine and projective varieties,
which we call the bottleneck degree. The bottleneck degree is a measure of the
complexity of computing all bottlenecks of an algebraic variety, using for
example numerical homotopy methods. We show that the bottleneck degree is a
function of classical invariants such as Chern classes and polar classes. We
give the formula explicitly in low dimension and provide an algorithm to
compute it in the general case.Comment: Major revision. New introduction. Added some new illustrative lemmas
and figures. Added pseudocode for the algorithm to compute bottleneck degree.
Fixed some typo
Numerical calculation of three-point branched covers of the projective line
We exhibit a numerical method to compute three-point branched covers of the
complex projective line. We develop algorithms for working explicitly with
Fuchsian triangle groups and their finite index subgroups, and we use these
algorithms to compute power series expansions of modular forms on these groups.Comment: 58 pages, 24 figures; referee's comments incorporate
Improved Asymmetric Locality Sensitive Hashing (ALSH) for Maximum Inner Product Search (MIPS)
Recently it was shown that the problem of Maximum Inner Product Search (MIPS)
is efficient and it admits provably sub-linear hashing algorithms. Asymmetric
transformations before hashing were the key in solving MIPS which was otherwise
hard. In the prior work, the authors use asymmetric transformations which
convert the problem of approximate MIPS into the problem of approximate near
neighbor search which can be efficiently solved using hashing. In this work, we
provide a different transformation which converts the problem of approximate
MIPS into the problem of approximate cosine similarity search which can be
efficiently solved using signed random projections. Theoretical analysis show
that the new scheme is significantly better than the original scheme for MIPS.
Experimental evaluations strongly support the theoretical findings.Comment: arXiv admin note: text overlap with arXiv:1405.586
Domain Adaptation for Statistical Classifiers
The most basic assumption used in statistical learning theory is that
training data and test data are drawn from the same underlying distribution.
Unfortunately, in many applications, the "in-domain" test data is drawn from a
distribution that is related, but not identical, to the "out-of-domain"
distribution of the training data. We consider the common case in which labeled
out-of-domain data is plentiful, but labeled in-domain data is scarce. We
introduce a statistical formulation of this problem in terms of a simple
mixture model and present an instantiation of this framework to maximum entropy
classifiers and their linear chain counterparts. We present efficient inference
algorithms for this special case based on the technique of conditional
expectation maximization. Our experimental results show that our approach leads
to improved performance on three real world tasks on four different data sets
from the natural language processing domain
Parallel Selective Algorithms for Big Data Optimization
We propose a decomposition framework for the parallel optimization of the sum
of a differentiable (possibly nonconvex) function and a (block) separable
nonsmooth, convex one. The latter term is usually employed to enforce structure
in the solution, typically sparsity. Our framework is very flexible and
includes both fully parallel Jacobi schemes and Gauss- Seidel (i.e.,
sequential) ones, as well as virtually all possibilities "in between" with only
a subset of variables updated at each iteration. Our theoretical convergence
results improve on existing ones, and numerical results on LASSO, logistic
regression, and some nonconvex quadratic problems show that the new method
consistently outperforms existing algorithms.Comment: This work is an extended version of the conference paper that has
been presented at IEEE ICASSP'14. The first and the second author contributed
equally to the paper. This revised version contains new numerical results on
non convex quadratic problem
Critical Phenomena with Linked Cluster Expansions in a Finite Volume
Linked cluster expansions are generalized from an infinite to a finite
volume. They are performed to 20th order in the expansion parameter to approach
the critical region from the symmetric phase. A new criterion is proposed to
distinguish 1st from 2nd order transitions within a finite size scaling
analysis. The criterion applies also to other methods for investigating the
phase structure such as Monte Carlo simulations. Our computational tools are
illustrated at the example of scalar O(N) models with four and six-point
couplings for and in three dimensions. It is shown how to localize
the tricritical line in these models. We indicate some further applications of
our methods to the electroweak transition as well as to models for
superconductivity.Comment: 36 pages, latex2e, 7 eps figures included, uuencoded, gzipped and
tarred tex file hdth9607.te
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