Factorization of multiple tensors for supervised feature extraction

© Springer International Publishing AG 2016. Tensors are effective representations for complex and timevarying networks. The factorization of a tensor provides a high-quality low-rank compact basis for each dimension of the tensor, which facilitates the interpretation of important structures of the represented data. Many existing tensor factorization (TF) methods assume there is one tensor that needs to be decomposed to low-rank factors. However in practice, data are usually generated from different time periods or by different class labels, which are represented by a sequence of multiple tensors associated with different labels. When one needs to analyse and compare multiple tensors, existing TF methods are unsuitable for discovering all potentially useful patterns, as they usually fail to discover either common or unique factors among the tensors: (1) if each tensor is factorized separately, the factor matrices will fail to explicitly capture the common information shared by different tensors, and (2) if tensors are concatenated together to form a larger “overall” tensor and then factorize this concatenated tensor, the intrinsic unique subspaces that are specific to each tensor will be lost. The cause of such an issue is mainly from the fact that existing tensor factorization methods handle data observations in an unsupervised way, considering only features but not labels of the data. To tackle this problem, we design a novel probabilistic tensor factorization model that takes both features and class labels of tensors into account, and produces informative common and unique factors of all tensors simultaneously. Experiment results on feature extraction in classification problems demonstrate the effectiveness of the factors discovered by our method

Non-Negative Tensor Factorization Applied to Music Genre Classification

Music genre classification techniques are typically applied to the data matrix whose columns are the feature vectors extracted from music recordings. In this paper, a feature vector is extracted using a texture window of one sec, which enables the representation of any 30 sec long music recording as a time sequence of feature vectors, thus yielding a feature matrix. Consequently, by stacking the feature matrices associated to any dataset recordings, a tensor is created, a fact which necessitates studying music genre classification using tensors. First, a novel algorithm for non-negative tensor factorization (NTF) is derived that extends the non-negative matrix factorization. Several variants of the NTF algorithm emerge by employing different cost functions from the class of Bregman divergences. Second, a novel supervised NTF classifier is proposed, which trains a basis for each class separately and employs basis orthogonalization. A variety of spectral, temporal, perceptual, energy, and pitch descriptors is extracted from 1000 recordings of the GTZAN dataset, which are distributed across 10 genre classes. The NTF classifier performance is compared against that of the multilayer perceptron and the support vector machines by applying a stratified 10-fold cross validation. A genre classification accuracy of 78.9% is reported for the NTF classifier demonstrating the superiority of the aforementioned multilinear classifier over several data matrix-based state-of-the-art classifiers

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