Principal Tensor Embedding for Unsupervised Tensor Learning

Claudio Turchetti,Lorenzo Manoni,Laura Falaschetti

doi:10.1109/access.2020.3044954

Claudio Turchetti, Lorenzo Manoni + Show 1 more

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https://doi.org/10.1109/access.2020.3044954

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Abstract

Tensors and multiway analysis aim to explore the relationships between the variables used to represent the data and find a summarization of the data with models of reduced dimensionality. However, although in this context a great attention was devoted to this problem, dimension reduction of high-order tensors remains a challenge. The aim of this article is to provide a nonlinear dimensionality reduction approach, named principal tensor embedding (PTE), for unsupervised tensor learning, that is able to derive an explicit nonlinear model of data. As in the standard manifold learning (ML) technique, it assumes multidimensional data lie close to a low-dimensional manifold embedded in a high-dimensional space. On the basis of this assumption a local parametrization of data that accurately captures its local geometry is derived. From this mathematical framework a nonlinear stochastic model of data that depends on a reduced set of latent variables is obtained. In this way the initial problem of unsupervised learning is reduced to the regression of a nonlinear input-output function, i.e. a supervised learning problem. Extensive experiments on several tensor datasets demonstrate that the proposed ML approach gives competitive performance when compared with other techniques used for data reconstruction and classification.

Highlights

T ENSORS, referred to as multiway arrays, are high-order generalizations of vectors and matrices and have been adopted in diverse branches of data analysis, to represent a wide range of real-world data
In this paper a nonlinear, explicit model of tensor data that depends on a reduced set of latent variables is derived
The main steps required for the estimation of the model from data are: i) compute a basis by a Gram-Schmidt procedure; ii) reorder the basis in such a way the variances of coefficients are in decreasing order; iii) estimate the intrinsic dimension d of data; iv) define a data parametrization of dimension d; v) approximate the nonlinearity in the parametrization by a regression model

Summary

Introduction

T ENSORS, referred to as multiway arrays, are high-order generalizations of vectors and matrices and have been adopted in diverse branches of data analysis, to represent a wide range of real-world data. Data modeling and classification of these data are important problems in several applications, such as human action and gesture recognition [8], tumor classifications [9], spatio-temporal analysis in climatology, geology and sociology [10], neuroimaging data analysis [11], big data representation [12], completion of big data [13], and so on To address these problems most previous works represent a tensor by a vector in high-dimensional space and apply ordinary learning methods for vectorial data. Representative techniques in this context include feature extraction and selection [14], [15], linear discriminant analysis (LDA) [16], and support vector machine (SVM) [17] This approach needs to arrange the tensor data into long vectors causing two main problems, i) loss of structural information of tensors, ii) vectors with very high dimensionality.

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