143,747 research outputs found
Fast truncation of mode ranks for bilinear tensor operations
We propose a fast algorithm for mode rank truncation of the result of a
bilinear operation on 3-tensors given in the Tucker or canonical form. If the
arguments and the result have mode sizes n and mode ranks r, the computation
costs . The algorithm is based on the cross approximation of
Gram matrices, and the accuracy of the resulted Tucker approximation is limited
by square root of machine precision.Comment: 9 pages, 2 tables. Submitted to Numerical Linear Algebra and
Applications, special edition for ICSMT conference, Hong Kong, January 201
Fast algorithms for solving H∞-norm minimization problems
We propose an efficient computational approach to minimize the H ∞-norm of a transfer-function matrix depending affinely on a set of free parameters. The minimization problem, formulated as a semi-infinite convex programming problem, is solved via a relaxation approach over a finite set of frequency values. In this way, a significant speed up is achieved by avoiding the solution of high order LMIs resulting by equivalently formulating the minimization problem as a high dimensional semidefinite programming problem. Numerical results illustrate the superiority of proposed approach over LMIs based techniques in solving zero order H∞-norm approximation problems
An efficient high-order Nystr\"om scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface
This text proposes a fast, rapidly convergent Nystr\"{o}m method for the
solution of the Lippmann-Schwinger integral equation that mathematically models
the scattering of time-harmonic acoustic waves by inhomogeneous obstacles,
while allowing the material properties to jump across the interface. The method
works with overlapping coordinate charts as a description of the given
scatterer. In particular, it employs "partitions of unity" to simplify the
implementation of high-order quadratures along with suitable changes of
parametric variables to analytically resolve the singularities present in the
integral operator to achieve desired accuracies in approximations. To deal with
the discontinuous material interface in a high-order manner, a specialized
quadrature is used in the boundary region. The approach further utilizes an FFT
based strategy that uses equivalent source approximations to accelerate the
evaluation of large number of interactions that arise in the approximation of
the volumetric integral operator and thus achieves a reduced computational
complexity of for an -point discretization. A detailed
discussion on the solution methodology along with a variety of numerical
experiments to exemplify its performance in terms of both speed and accuracy
are presented in this paper
Bolt: Accelerated Data Mining with Fast Vector Compression
Vectors of data are at the heart of machine learning and data mining.
Recently, vector quantization methods have shown great promise in reducing both
the time and space costs of operating on vectors. We introduce a vector
quantization algorithm that can compress vectors over 12x faster than existing
techniques while also accelerating approximate vector operations such as
distance and dot product computations by up to 10x. Because it can encode over
2GB of vectors per second, it makes vector quantization cheap enough to employ
in many more circumstances. For example, using our technique to compute
approximate dot products in a nested loop can multiply matrices faster than a
state-of-the-art BLAS implementation, even when our algorithm must first
compress the matrices.
In addition to showing the above speedups, we demonstrate that our approach
can accelerate nearest neighbor search and maximum inner product search by over
100x compared to floating point operations and up to 10x compared to other
vector quantization methods. Our approximate Euclidean distance and dot product
computations are not only faster than those of related algorithms with slower
encodings, but also faster than Hamming distance computations, which have
direct hardware support on the tested platforms. We also assess the errors of
our algorithm's approximate distances and dot products, and find that it is
competitive with existing, slower vector quantization algorithms.Comment: Research track paper at KDD 201
Fast Routing Table Construction Using Small Messages
We describe a distributed randomized algorithm computing approximate
distances and routes that approximate shortest paths. Let n denote the number
of nodes in the graph, and let HD denote the hop diameter of the graph, i.e.,
the diameter of the graph when all edges are considered to have unit weight.
Given 0 < eps <= 1/2, our algorithm runs in weak-O(n^(1/2 + eps) + HD)
communication rounds using messages of O(log n) bits and guarantees a stretch
of O(eps^(-1) log eps^(-1)) with high probability. This is the first
distributed algorithm approximating weighted shortest paths that uses small
messages and runs in weak-o(n) time (in graphs where HD in weak-o(n)). The time
complexity nearly matches the lower bounds of weak-Omega(sqrt(n) + HD) in the
small-messages model that hold for stateless routing (where routing decisions
do not depend on the traversed path) as well as approximation of the weigthed
diameter. Our scheme replaces the original identifiers of the nodes by labels
of size O(log eps^(-1) log n). We show that no algorithm that keeps the
original identifiers and runs for weak-o(n) rounds can achieve a
polylogarithmic approximation ratio.
Variations of our techniques yield a number of fast distributed approximation
algorithms solving related problems using small messages. Specifically, we
present algorithms that run in weak-O(n^(1/2 + eps) + HD) rounds for a given 0
< eps <= 1/2, and solve, with high probability, the following problems:
- O(eps^(-1))-approximation for the Generalized Steiner Forest (the running
time in this case has an additive weak-O(t^(1 + 2eps)) term, where t is the
number of terminals);
- O(eps^(-2))-approximation of weighted distances, using node labels of size
O(eps^(-1) log n) and weak-O(n^(eps)) bits of memory per node;
- O(eps^(-1))-approximation of the weighted diameter;
- O(eps^(-3))-approximate shortest paths using the labels 1,...,n.Comment: 40 pages, 2 figures, extended abstract submitted to STOC'1
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