The widespread use of multisensor technology and the emergence of big
datasets have created the need to develop tools to reduce, approximate, and
classify large and multimodal data such as higher-order tensors. While early
approaches focused on matrix and vector based methods to represent these
higher-order data, more recently it has been shown that tensor decomposition
methods are better equipped to capture couplings across their different modes.
For these reasons, tensor decomposition methods have found applications in many
different signal processing problems including dimensionality reduction, signal
separation, linear regression, feature extraction, and classification. However,
most of the existing tensor decomposition methods are based on the principle of
finding a low-rank approximation in a linear subspace structure, where the
definition of the rank may change depending on the particular decomposition.
Since many datasets are not necessarily low-rank in a linear subspace, this
often results in high approximation errors or low compression rates. In this
paper, we introduce a new adaptive, multi-scale tensor decomposition method for
higher order data inspired by hybrid linear modeling and subspace clustering
techniques. In particular, we develop a multi-scale higher-order singular value
decomposition (MS-HoSVD) approach where a given tensor is first permuted and
then partitioned into several sub-tensors each of which can be represented as a
low-rank tensor with increased representational efficiency. The proposed
approach is evaluated for dimensionality reduction and classification for
several different real-life tensor signals with promising results.

介绍了一种基于混合线性建模和子空间聚类技术的自适应、多尺度张量分解方法，旨在降低大型和多模态数据的维度和表示复杂度。该方法在多个真实张量信号的维数约简和分类问题中表现良好。

高阶张量的多尺度分析

Multiscale Analysis for Higher-order Tensors

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.

本文从信号处理角度提供了关于高阶张量分解的全面介绍，包括基本的 CP 和 Tucker 模型，通过多种数据处理方法提供更加灵活和自然的潜在因素；研究表明张量分解可用作现代信号处理、数据分析和机器学习应用的最有效和最有前途的工具。

用于信号处理应用的张量分解：从双向到多向分量分析

Tensor Decompositions for Signal Processing Applications From Two-way to  Multiway Component Analysis

Gaussian processes (GPs) are a powerful tool for probabilistic inference over
functions. They have been applied to both regression and non-linear
dimensionality reduction, and offer desirable properties such as uncertainty
estimates, robustness to over-fitting, and principled ways for tuning
hyper-parameters. However the scalability of these models to big datasets
remains an active topic of research. We introduce a novel re-parametrisation of
variational inference for sparse GP regression and latent variable models that
allows for an efficient distributed algorithm. This is done by exploiting the
decoupling of the data given the inducing points to re-formulate the evidence
lower bound in a Map-Reduce setting. We show that the inference scales well
with data and computational resources, while preserving a balanced distribution
of the load among the nodes. We further demonstrate the utility in scaling
Gaussian processes to big data. We show that GP performance improves with
increasing amounts of data in regression (on flight data with 2 million
records) and latent variable modelling (on MNIST). The results show that GPs
perform better than many common models often used for big data.

本论文介绍了一种基于稀疏高斯过程回归和潜变量模型的重新参数化变分推断方法，可以有效解决大规模数据集下高斯过程模型的可扩展性问题，并在飞行数据和 MNIST 数据集上证明了其优越性。