1,785 research outputs found
Tensor Analysis and Fusion of Multimodal Brain Images
Current high-throughput data acquisition technologies probe dynamical systems
with different imaging modalities, generating massive data sets at different
spatial and temporal resolutions posing challenging problems in multimodal data
fusion. A case in point is the attempt to parse out the brain structures and
networks that underpin human cognitive processes by analysis of different
neuroimaging modalities (functional MRI, EEG, NIRS etc.). We emphasize that the
multimodal, multi-scale nature of neuroimaging data is well reflected by a
multi-way (tensor) structure where the underlying processes can be summarized
by a relatively small number of components or "atoms". We introduce
Markov-Penrose diagrams - an integration of Bayesian DAG and tensor network
notation in order to analyze these models. These diagrams not only clarify
matrix and tensor EEG and fMRI time/frequency analysis and inverse problems,
but also help understand multimodal fusion via Multiway Partial Least Squares
and Coupled Matrix-Tensor Factorization. We show here, for the first time, that
Granger causal analysis of brain networks is a tensor regression problem, thus
allowing the atomic decomposition of brain networks. Analysis of EEG and fMRI
recordings shows the potential of the methods and suggests their use in other
scientific domains.Comment: 23 pages, 15 figures, submitted to Proceedings of the IEE
Multidimensional approximation of nonlinear dynamical systems
A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method multidimensional approximation of nonlinear dynamical systems (MANDy) which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
We present a compendium of numerical simulation techniques, based on tensor
network methods, aiming to address problems of many-body quantum mechanics on a
classical computer. The core setting of this anthology are lattice problems in
low spatial dimension at finite size, a physical scenario where tensor network
methods, both Density Matrix Renormalization Group and beyond, have long proven
to be winning strategies. Here we explore in detail the numerical frameworks
and methods employed to deal with low-dimension physical setups, from a
computational physics perspective. We focus on symmetries and closed-system
simulations in arbitrary boundary conditions, while discussing the numerical
data structures and linear algebra manipulation routines involved, which form
the core libraries of any tensor network code. At a higher level, we put the
spotlight on loop-free network geometries, discussing their advantages, and
presenting in detail algorithms to simulate low-energy equilibrium states.
Accompanied by discussions of data structures, numerical techniques and
performance, this anthology serves as a programmer's companion, as well as a
self-contained introduction and review of the basic and selected advanced
concepts in tensor networks, including examples of their applications.Comment: 115 pages, 56 figure
Nonparametric Estimation of Multi-View Latent Variable Models
Spectral methods have greatly advanced the estimation of latent variable
models, generating a sequence of novel and efficient algorithms with strong
theoretical guarantees. However, current spectral algorithms are largely
restricted to mixtures of discrete or Gaussian distributions. In this paper, we
propose a kernel method for learning multi-view latent variable models,
allowing each mixture component to be nonparametric. The key idea of the method
is to embed the joint distribution of a multi-view latent variable into a
reproducing kernel Hilbert space, and then the latent parameters are recovered
using a robust tensor power method. We establish that the sample complexity for
the proposed method is quadratic in the number of latent components and is a
low order polynomial in the other relevant parameters. Thus, our non-parametric
tensor approach to learning latent variable models enjoys good sample and
computational efficiencies. Moreover, the non-parametric tensor power method
compares favorably to EM algorithm and other existing spectral algorithms in
our experiments
- …