4,582 research outputs found
Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis
The widespread use of multi-sensor technology and the emergence of big
datasets has highlighted the limitations of standard flat-view matrix models
and the necessity to move towards more versatile data analysis tools. We show
that higher-order tensors (i.e., multiway arrays) enable such a fundamental
paradigm shift towards models that are essentially polynomial and whose
uniqueness, unlike the matrix methods, is guaranteed under verymild and natural
conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical
backbone, data analysis techniques using tensor decompositions are shown to
have great flexibility in the choice of constraints that match data properties,
and to find more general latent components in the data than matrix-based
methods. A comprehensive introduction to tensor decompositions is provided from
a signal processing perspective, starting from the algebraic foundations, via
basic Canonical Polyadic and Tucker models, through to advanced cause-effect
and multi-view data analysis schemes. We show that tensor decompositions enable
natural generalizations of some commonly used signal processing paradigms, such
as canonical correlation and subspace techniques, signal separation, linear
regression, feature extraction and classification. We also cover computational
aspects, and point out how ideas from compressed sensing and scientific
computing may be used for addressing the otherwise unmanageable storage and
manipulation problems associated with big datasets. The concepts are supported
by illustrative real world case studies illuminating the benefits of the tensor
framework, as efficient and promising tools for modern signal processing, data
analysis and machine learning applications; these benefits also extend to
vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker
decomposition, HOSVD, tensor networks, Tensor Train
Tensor Computation: A New Framework for High-Dimensional Problems in EDA
Many critical EDA problems suffer from the curse of dimensionality, i.e. the
very fast-scaling computational burden produced by large number of parameters
and/or unknown variables. This phenomenon may be caused by multiple spatial or
temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit
simulation), nonlinearity of devices and circuits, large number of design or
optimization parameters (e.g. full-chip routing/placement and circuit sizing),
or extensive process variations (e.g. variability/reliability analysis and
design for manufacturability). The computational challenges generated by such
high dimensional problems are generally hard to handle efficiently with
traditional EDA core algorithms that are based on matrix and vector
computation. This paper presents "tensor computation" as an alternative general
framework for the development of efficient EDA algorithms and tools. A tensor
is a high-dimensional generalization of a matrix and a vector, and is a natural
choice for both storing and solving efficiently high-dimensional EDA problems.
This paper gives a basic tutorial on tensors, demonstrates some recent examples
of EDA applications (e.g., nonlinear circuit modeling and high-dimensional
uncertainty quantification), and suggests further open EDA problems where the
use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and
System
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Generating realistic scaled complex networks
Research on generative models is a central project in the emerging field of
network science, and it studies how statistical patterns found in real networks
could be generated by formal rules. Output from these generative models is then
the basis for designing and evaluating computational methods on networks, and
for verification and simulation studies. During the last two decades, a variety
of models has been proposed with an ultimate goal of achieving comprehensive
realism for the generated networks. In this study, we (a) introduce a new
generator, termed ReCoN; (b) explore how ReCoN and some existing models can be
fitted to an original network to produce a structurally similar replica, (c)
use ReCoN to produce networks much larger than the original exemplar, and
finally (d) discuss open problems and promising research directions. In a
comparative experimental study, we find that ReCoN is often superior to many
other state-of-the-art network generation methods. We argue that ReCoN is a
scalable and effective tool for modeling a given network while preserving
important properties at both micro- and macroscopic scales, and for scaling the
exemplar data by orders of magnitude in size.Comment: 26 pages, 13 figures, extended version, a preliminary version of the
paper was presented at the 5th International Workshop on Complex Networks and
their Application
Exact and efficient top-K inference for multi-target prediction by querying separable linear relational models
Many complex multi-target prediction problems that concern large target
spaces are characterised by a need for efficient prediction strategies that
avoid the computation of predictions for all targets explicitly. Examples of
such problems emerge in several subfields of machine learning, such as
collaborative filtering, multi-label classification, dyadic prediction and
biological network inference. In this article we analyse efficient and exact
algorithms for computing the top- predictions in the above problem settings,
using a general class of models that we refer to as separable linear relational
models. We show how to use those inference algorithms, which are modifications
of well-known information retrieval methods, in a variety of machine learning
settings. Furthermore, we study the possibility of scoring items incompletely,
while still retaining an exact top-K retrieval. Experimental results in several
application domains reveal that the so-called threshold algorithm is very
scalable, performing often many orders of magnitude more efficiently than the
naive approach
PReaCH: A Fast Lightweight Reachability Index using Pruning and Contraction Hierarchies
We develop the data structure PReaCH (for Pruned Reachability Contraction
Hierarchies) which supports reachability queries in a directed graph, i.e., it
supports queries that ask whether two nodes in the graph are connected by a
directed path. PReaCH adapts the contraction hierarchy speedup techniques for
shortest path queries to the reachability setting. The resulting approach is
surprisingly simple and guarantees linear space and near linear preprocessing
time. Orthogonally to that, we improve existing pruning techniques for the
search by gathering more information from a single DFS-traversal of the graph.
PReaCH-indices significantly outperform previous data structures with
comparable preprocessing cost. Methods with faster queries need significantly
more preprocessing time in particular for the most difficult instances
Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio
Many problems in fluid modelling require the efficient solution of highly
anisotropic elliptic partial differential equations (PDEs) in "flat" domains.
For example, in numerical weather- and climate-prediction an elliptic PDE for
the pressure correction has to be solved at every time step in a thin spherical
shell representing the global atmosphere. This elliptic solve can be one of the
computationally most demanding components in semi-implicit semi-Lagrangian time
stepping methods which are very popular as they allow for larger model time
steps and better overall performance. With increasing model resolution,
algorithmically efficient and scalable algorithms are essential to run the code
under tight operational time constraints. We discuss the theory and practical
application of bespoke geometric multigrid preconditioners for equations of
this type. The algorithms deal with the strong anisotropy in the vertical
direction by using the tensor-product approach originally analysed by B\"{o}rm
and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the
analysis to three dimensions under slightly weakened assumptions, and
numerically demonstrate its efficiency for the solution of the elliptic PDE for
the global pressure correction in atmospheric forecast models. For this we
compare the performance of different multigrid preconditioners on a
tensor-product grid with a semi-structured and quasi-uniform horizontal mesh
and a one dimensional vertical grid. The code is implemented in the Distributed
and Unified Numerics Environment (DUNE), which provides an easy-to-use and
scalable environment for algorithms operating on tensor-product grids. Parallel
scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR
supercomputer.Comment: 22 pages, 6 Figures, 2 Table
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