Discriminant nonnegative tensor factorization algorithms

Multi-Branch Tensor Network Structure for Tensor-Train Discriminant Analysis

Higher-order data with high dimensionality arise in a diverse set of application areas such as computer vision, video analytics and medical imaging. Tensors provide a natural tool for representing these types of data. Two major challenges that confound current tensor based supervised learning algorithms are storage complexity and computational efficiency. In this paper, we address these problems by introducing a multi-branch tensor network structure. The multi-branch structure is a general tensor decomposition that includes Tucker and tensor-train (TT) as special cases and takes advantage of the flexibility of the tensor network to provide a better balance between storage and computational complexity. We then introduce a supervised discriminative tensor-train subspace learning approach referred to as tensor-train discriminant analysis (TTDA), and its implementations using the multi-branch tensor network structure. Multi-branch implementations of TTDA are shown to achieve lower storage and computational complexity while providing improved classification performance with respect to both Tucker and TT based supervised learning methods.

Pairwise Constraint Propagation-Induced Symmetric Nonnegative Matrix Factorization

As a variant of nonnegative matrix factorization (NMF), symmetric NMF (SNMF) has shown to be effective for capturing the cluster structure embedded in the graph representation. In contrast to the existing SNMF-based clustering methods that empirically construct the similarity matrix and rigidly introduce the supervisory information to the assignment matrix, in this paper, we propose a novel SNMF-based semisupervised clustering method, namely, pairwise constraint propagation-induced SNMF (PCPSNMF). By formulating a single-constrained optimization problem, PCPSNMF is capable of learning the similarity and assignment matrices adaptively and simultaneously, in which a small amount of supervisory information in the form of pairwise constraints is introduced in a flexible way to guide the construction of the similarity matrix, and the two matrices communicate with each other to achieve mutual refinement until convergence. In addition, we propose an efficient alternating iterative algorithm to solve the optimization problem, whose convergence is theoretically proven. Experimental results over several benchmark image data sets demonstrate that PCPSNMF is less sensitive to initialization and produces higher clustering performance, compared with the state-of-the-art methods.

A Deep Matrix Factorization Method for Learning Attribute Representations

Semi-Non-negative Matrix Factorization is a technique that learns a low-dimensional representation of a dataset that lends itself to a clustering interpretation. It is possible that the mapping between this new representation and our original data matrix contains rather complex hierarchical information with implicit lower-level hidden attributes, that classical one level clustering methodologies cannot interpret. In this work we propose a novel model, Deep Semi-NMF, that is able to learn such hidden representations that allow themselves to an interpretation of clustering according to different, unknown attributes of a given dataset. We also present a semi-supervised version of the algorithm, named Deep WSF, that allows the use of (partial) prior information for each of the known attributes of a dataset, that allows the model to be used on datasets with mixed attribute knowledge. Finally, we show that our models are able to learn low-dimensional representations that are better suited for clustering, but also classification, outperforming Semi-Non-negative Matrix Factorization, but also other state-of-the-art methodologies variants.

Sparse Canonical Temporal Alignment with Deep Tensor Decomposition for Action Recognition

In this paper we solve three problems in action recognition: sub-action, multi-subject, and multi-modality, by reducing the diversity of intra-class samples. The main stage contains canonical temporal alignment and key frames selection. As we know, temporal alignment aims to reduce the diversity of intra-class samples, however, dense frames may yield misalignment or overlapped alignment and decrease recognition performance. To overcome this problem, we propose a Sparse Canonical Temporal Alignment (SCTA) method which selects and aligns key frames from two sequences to reduce diversity. To extract better features from the key frames, we propose a Deep Non-negative Tensor Factorization (DNTF) method to find a tensor subspace integrated with SCTA scheme. First we model an action sequence as a third-order tensor with spatiotemporal structure. Then we design a DNTF scheme to find a tensor subspace in both spatial and temporal directions. Particularly, in the first layer the original tensor is decomposed into two lowrank tensors by Non-negative Tensor Factorization (NTF), and in the second layer each low-rank tensor is further decomposed by Tensor-Train (TT) for time efficiency. Finally, our framework composed of SCTA and DNTF could solve the three problems and extract effective features for action recognition. Experiments on synthetic data, MSRDailyActivity3D and MSRActionPairs datasets show that our method works better than competitive methods in terms of accuracy.

Decentralized Dimensionality Reduction for Distributed Tensor Data Across Sensor Networks

This paper develops a novel decentralized dimensionality reduction algorithm for the distributed tensor data across sensor networks. The main contributions of this paper are as follows. First, conventional centralized methods, which utilize entire data to simultaneously determine all the vectors of the projection matrix along each tensor mode, are not suitable for the network environment. Here, we relax the simultaneous processing manner into the one-vector-by-one-vector (OVBOV) manner, i.e., determining the projection vectors (PVs) related to each tensor mode one by one. Second, we prove that in the OVBOV manner each PV can be determined without modifying any tensor data, which simplifies corresponding computations. Third, we cast the decentralized PV determination problem as a set of subproblems with consensus constraints, so that it can be solved in the network environment only by local computations and information communications among neighboring nodes. Fourth, we introduce the null space and transform the PV determination problem with complex orthogonality constraints into an equivalent hidden convex one without any orthogonality constraint, which can be solved by the Lagrange multiplier method. Finally, experimental results are given to show that the proposed algorithm is an effective dimensionality reduction scheme for the distributed tensor data across the sensor networks.

Twin Bounded Support Tensor Machine for Classification

The traditional vector-based classifiers, such as support vector machine (SVM) and twin support vector machine (TSVM), cannot handle tensor data directly and may not utilize the data informations effectively. In this paper, we propose a novel classifier based on tensor data, called twin bounded support tensor machine (TBSTM) which is an extension of twin bounded support vector machine (TBSVM). Similar to TBSVM, TBSTM gets two hyperplanes and obtains the solution by solving two quadratic programming problems (QPPs). The computational complexity of each QPPs is smaller than that of support tensor machine (STM). TBSTM not only retains the advantage of TBSVM, but also has its unique superior characteristics: (1) it makes full use of the structure information of data; (2) it has acceptable or better classification accuracy compared to STM, TBSVM and SVM; (3) the computational cost is basically less than STM; (4) it can deal with large data that TBSVM is not easy to achieve, especially for small-sample-size (S3) problems; (5) it adopts alternating successive over relaxation iteration (ASOR) method to solve optimization problems which accelerates the pace of training. Finally, we demonstrate the effectiveness and superiority by the experiments based on vector and tensor data.

Projected gradients for subclass discriminant nonnegative subspace learning

Current discriminant nonnegative matrix factorization (NMF) methods either do not guarantee convergence to a stationary limit point or assume a compact data distribution inside classes, thus ignoring intra class variance in extracting discriminant data samples representations. To address both limitations, we regard that data inside each class has a multimodal distribution, forming various subclasses and perform optimization using a projected gradients framework to ensure limit point stationarity. The proposed method combines appropriate clustering-based discriminant criteria in the NMF decomposition cost function, in order to find discriminant projections that enhance class separability in the reduced dimensional projection space, thus improving classification performance. The developed algorithms have been applied to facial expression, face and object recognition, and experimental results verified that they successfully identified discriminant parts, thus enhancing recognition performance.

Generalized discriminant orthogonal nonnegative tensor factorization for facial expression recognition

In order to overcome the limitation of traditional nonnegative factorization algorithms, the paper presents a generalized discriminant orthogonal non-negative tensor factorization algorithm. At first, the algorithm takes the orthogonal constraint into account to ensure the nonnegativity of the low-dimensional features. Furthermore, the discriminant constraint is imposed on low-dimensional weights to strengthen the discriminant capability of the low-dimensional features. The experiments on facial expression recognition have demonstrated that the algorithm is superior to other non-negative factorization algorithms.

Hybrid dimensionality reduction method based on support vector machine and independent component analysis

This paper presents a new hybrid dimensionality reduction method to seek projection through optimization of both structural risk (supervised criterion) and data independence (unsupervised criterion). Classification accuracy is used as a metric to evaluate the performance of the method. By minimizing the structural risk, projection originated from the decision boundaries directly improves the classification performance from a supervised perspective. From an unsupervised perspective, projection can also be obtained based on maximum independence among features (or attributes) in data to indirectly achieve better classification accuracy over more intrinsic representation of the data. Orthogonality interrelates the two sets of projections such that minimum redundancy exists between the projections, leading to more effective dimensionality reduction. Experimental results show that the proposed hybrid dimensionality reduction method that satisfies both criteria simultaneously provides higher classification performance, especially for noisy data sets, in relatively lower dimensional space than various existing methods.

Tensor learning for regression

In this paper, we exploit the advantages of tensorial representations and propose several tensor learning models for regression. The model is based on the canonical/parallel-factor decomposition of tensors of multiple modes and allows the simultaneous projections of an input tensor to more than one direction along each mode. Two empirical risk functions are studied, namely, the square loss and ε -insensitive loss functions. The former leads to higher rank tensor ridge regression (TRR), and the latter leads to higher rank support tensor regression (STR), both formulated using the Frobenius norm for regularization. We also use the group-sparsity norm for regularization, favoring in that way the low rank decomposition of the tensorial weight. In that way, we achieve the automatic selection of the rank during the learning process and obtain the optimal-rank TRR and STR. Experiments conducted for the problems of head-pose, human-age, and 3-D body-pose estimations using real data from publicly available databases, verified not only the superiority of tensors over their vector counterparts but also the efficiency of the proposed algorithms.

Symmetric nonnegative matrix factorization: algorithms and applications to probabilistic clustering

Nonnegative matrix factorization (NMF) is an unsupervised learning method useful in various applications including image processing and semantic analysis of documents. This paper focuses on symmetric NMF (SNMF), which is a special case of NMF decomposition. Three parallel multiplicative update algorithms using level 3 basic linear algebra subprograms directly are developed for this problem. First, by minimizing the Euclidean distance, a multiplicative update algorithm is proposed, and its convergence under mild conditions is proved. Based on it, we further propose another two fast parallel methods: α-SNMF and β -SNMF algorithms. All of them are easy to implement. These algorithms are applied to probabilistic clustering. We demonstrate their effectiveness for facial image clustering, document categorization, and pattern clustering in gene expression.