Dimensionality reduction and topographic mapping of binary tensors

Mažgut, Jakub; Tiňo, Peter; Bodén, Mikael; Yan, Hong

doi:10.1007/s10044-013-0317-y

Cited by 9 publications

(4 citation statements)

References 30 publications

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“…The methods were then later generalized to handle tensors of any order [22,83]. Subsequent work then tailored these general methods to several different applications including variants based on non-negative factorizations for audio engineering [88], weighted versions for EEG signal classification [89], online versions for tracking [90], variants for binary tensors [91], and incremental versions for streamed tensor data [92].…”

Section: Multilinear Principal Component Analysis (Mpca)mentioning

confidence: 99%

Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA

et al. 2018

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The widespread use of multisensor technology and the emergence of big data sets have brought the necessity to develop more versatile tools to represent higher-order data with multiple aspects and high dimensionality. Data in the form of multidimensional arrays, also referred to as tensors, arises in a variety of applications including chemometrics, hyperspectral imaging, high resolution videos, neuroimaging, biometrics, and social network analysis. Early multiway data analysis approaches reformatted such tensor data as large vectors or matrices and then resorted to dimensionality reduction methods developed for classical two-way analysis such as PCA. However, one cannot discover hidden components within multiway data using conventional PCA. To this end, tensor decomposition methods which are flexible in the choice of the constraints and that extract more general latent components have been proposed. In this paper, we review the major tensor decomposition methods with a focus on problems targeted by classical PCA. In particular, we present tensor methods that aim to solve three important challenges typically addressed by PCA: dimensionality reduction, i.e. low-rank tensor approximation, supervised learning, i.e. learning linear subspaces for feature extraction, and robust low-rank tensor recovery. We also provide experimental results to compare different tensor models for both dimensionality reduction and supervised learning applications.

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Section: Multilinear Principal Component Analysis (Mpca)mentioning

confidence: 99%

Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA

et al. 2018

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“…Binary tensor decomposition. More recently, Mažgut et al (2014); Rai et al (2015); Hong et al (2018) studied higher-order binary tensor decomposition, and we target the same problem.…”

Section: Related Workmentioning

confidence: 99%

Learning from Binary Multiway Data: Probabilistic Tensor Decomposition and its Statistical Optimality

Wang,

2018

Preprint

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We consider the problem of decomposition of multiway tensor with binary entries. Such data problems arise frequently in numerous applications such as neuroimaging, recommendation system, topic modeling, and sensor network localization. We propose that the observed binary entries follow a Bernoulli model, develop a rank-constrained likelihood-based estimation procedure, and obtain the theoretical accuracy guarantees. Specifically, we establish the error bound of the tensor estimation, and show that the obtained rate is minimax optimal under the considered model. We demonstrate the efficacy of our approach through both simulations and analyses of multiple real-world datasets on the tasks of tensor completion and clustering.

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“…As a closely related problem, various types of principal component analysis (PCA) and matrix factorization methods have been developed for discrete non-Gaussian matrix data (Collins et al, 2002;de Leeuw, 2006;Udell et al, 2016;Landgraf and Lee, 2020b,a). Extending the matrix factorization approach in Collins et al (2002) to binary tensor data, Mažgut et al (2014) considered a Tucker decomposition of the logit parameter tensor, and Wang and Li (2020) considered a CP decomposition of the logit parameter tensor with max-norm constraint and investigated its statistical optimality. More generally, Hong et al (2020) proposed a CP decomposition of the natural parameter tensor for exponential family data.…”

Section: Introductionmentioning

confidence: 99%

Sparse Logistic Tensor Decomposition for Binary Data

Zhang¹,

Lee²

2021

Preprint

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Tensor data are increasingly available in many application domains. We develop several tensor decomposition methods for binary tensor data. Different from classical tensor decompositions for continuous-valued data with squared error loss, we formulate logistic tensor decompositions for binary data with a Bernoulli likelihood. To enhance the interpretability of estimated factors and improve their stability further, we propose sparse formulations of logistic tensor decomposition by considering 1 -norm and 0 -norm regularized likelihood. To handle the resulting optimization problems, we develop computational algorithms which combine the strengths of tensor power method and majorization-minimization (MM) algorithm. Through simulation studies, we demonstrate the utility of our methods in analysis of binary tensor data. To illustrate the effectiveness of the proposed methods, we analyze a dataset concerning nations and their political relations and perform co-clustering of estimated factors to find associations between the nations and political relations.

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Dimensionality reduction and topographic mapping of binary tensors

Cited by 9 publications

References 30 publications

Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA

Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA

Learning from Binary Multiway Data: Probabilistic Tensor Decomposition and its Statistical Optimality

Sparse Logistic Tensor Decomposition for Binary Data

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