Clustering multi-way data: a novel algebraic approach

arXiv: Computer Vision and Pattern Recognition, 2016

A new submodule clustering method via sparse and low-rank representation for multi-way data is proposed in this paper. Instead of reshaping multi-way data into vectors, this method maintains their natural orders to preserve data intrinsic structures, e.g., image data kept as matrices. To implement clustering, the multi-way data, viewed as tensors, are represented by the proposed tensor sparse and low-rank model to obtain its submodule representation, called a free module, which is finally used for spectral clustering. The proposed method extends the conventional subspace clustering method based on sparse and low-rank representation to multi-way data submodule clustering by combining t-product operator. The new method is tested on several public datasets, including synthetical data, video sequences and toy images. The experiments show that the new method outperforms the state-of-the-art methods, such as Sparse Subspace Clustering (SSC), Low-Rank Representation (LRR), Ordered Subspace...

Proceedings of the 13th annual ACM international conference on Multimedia, 2005

We consider the problem of image representation and clustering. Traditionally, an n1 × n2 image is represented by a vector in the Euclidean space R n 1 ×n 2. Some learning algorithms are then applied to these vectors in such a high dimensional space for dimensionality reduction, classification, and clustering. However, an image is intrinsically a matrix, or the second order tensor. The vector representation of the images ignores the spatial relationships between the pixels in an image. In this paper, we introduce a tensor framework for image analysis. We represent the images as points in the tensor space R n 1 ⊗ R n 2 which is a tensor product of two vector spaces. Based on the tensor representation, we propose a novel image representation and clustering algorithm which explicitly considers the manifold structure of the tensor space. By preserving the local structure of the data manifold, we can obtain a tensor subspace which is optimal for data representation in the sense of local isometry. We call it TensorImage approach. Traditional clustering algorithm such as k-means is then applied in the tensor subspace. Our algorithm shares many of the data representation and clustering properties of other techniques such as Locality Preserving Projections, Laplacian Eigenmaps, and spectral clustering, yet our algorithm is much more computationally efficient. Experimental results show the efficiency and effectiveness of our algorithm.

2014 International Joint Conference on Neural Networks (IJCNN), 2014

Subspace clustering groups a set of samples (vectors) into clusters by approximating this set with a mixture of several linear subspaces, so that the samples in the same cluster are drawn from the same linear subspace. In majority of existing works on subspace clustering, samples are simply regarded as being independent and identically distributed, that is, arbitrarily ordering samples when necessary. However, this setting ignores sample correlations in their original spatial structure. To address this issue, we propose a tensor low-rank representation (TLRR) for subspace clustering by keeping available spatial information of data. TLRR seeks a lowest-rank representation over all the candidates while maintaining the inherent spatial structures among samples, and the affinity matrix used for spectral clustering is built from the combination of similarities along all data spatial directions. TLRR better captures the global structures of data and provides a robust subspace segmentation from corrupted data. Experimental results on both synthetic and real-world datasets show that TLRR outperforms several established state-of-the-art methods.

ArXiv, 2016

A new submodule clustering method via sparse and low-rank representation for multi-way data is proposed in this paper. Instead of reshaping multi-way data into vectors, this method maintains their natural orders to preserve data intrinsic structures, e.g., image data kept as matrices. To implement clustering, the multi-way data, viewed as tensors, are represented by the proposed tensor sparse and low-rank model to obtain its submodule representation, called a free module, which is finally used for spectral clustering. The proposed method extends the conventional subspace clustering method based on sparse and low-rank representation to multi-way data submodule clustering by combining t-product operator. The new method is tested on several public datasets, including synthetical data, video sequences and toy images. The experiments show that the new method outperforms the state-of-the-art methods, such as Sparse Subspace Clustering (SSC), Low-Rank Representation (LRR), Ordered Subspace...

IEEE transactions on neural networks and learning systems, 2016

Subspace clustering groups a set of samples from a union of several linear subspaces into clusters, so that the samples in the same cluster are drawn from the same linear subspace. In the majority of the existing work on subspace clustering, clusters are built based on feature information, while sample correlations in their original spatial structure are simply ignored. Besides, original high-dimensional feature vector contains noisy/redundant information, and the time complexity grows exponentially with the number of dimensions. To address these issues, we propose a tensor low-rank representation (TLRR) and sparse coding-based (TLRRSC) subspace clustering method by simultaneously considering feature information and spatial structures. TLRR seeks the lowest rank representation over original spatial structures along all spatial directions. Sparse coding learns a dictionary along feature spaces, so that each sample can be represented by a few atoms of the learned dictionary. The affin...

ArXiv, 2016

The plenty information from multiple views data as well as the complementary information among different views are usually beneficial to various tasks, e.g., clustering, classification, de-noising. Multi-view subspace clustering is based on the fact that the multi-view data are generated from a latent subspace. To recover the underlying subspace structure, the success of the sparse and/or low-rank subspace clustering has been witnessed recently. Despite some state-of-the-art subspace clustering approaches can numerically handle multi-view data, by simultaneously exploring all possible pairwise correlation within views, the high order statistics is often disregarded which can only be captured by simultaneously utilizing all views. As a consequence, the clustering performance for multi-view data is compromised. To address this issue, in this paper, a novel multi-view clustering method is proposed by using \textit{t-product} in third-order tensor space. Based on the circular convolutio...

IEEE Transactions on Pattern Analysis and Machine Intelligence, 2015

Tensors or multiarray data are generalizations of matrices. Tensor clustering has become a very important research topic due to the intrinsically rich structures in real-world multiarray datasets. Subspace clustering based on vectorizing multiarray data has been extensively researched. However, vectorization of tensorial data does not exploit complete structure information. In this paper, we propose a subspace clustering algorithm without adopting any vectorization process. Our approach is based on a novel heterogeneous Tucker decomposition model. In contrast to existing techniques, we propose a new clustering algorithm that alternates between different modes of the proposed heterogeneous tensor model. All but the last mode have closed-form updates. Updating the last mode reduces to optimizing over the so-called multinomial manifold, for which we investigate second order Riemannian geometry and propose a trust-region algorithm. Numerical experiments show that our proposed algorithm compete effectively with state-of-the-art clustering algorithms that are based on tensor factorization.

IEEE Transactions on Pattern Analysis and Machine Intelligence, 2020

Clustering aims to separate observed data into different categories. The performance of popular clustering models relies on the sample-to-sample similarity. However, the pairwise similarity is prone to be corrupted by noise or outliers and thus deteriorates the subsequent clustering. A high-order relationship among samples-to-samples may elaborate the local manifold of the data and thus provide complementary information to guide the clustering. However, few studies have investigated the connection between high-order similarity and usual pairwise similarity. To fill this gap, we first define a high-order tensor similarity to exploit the samples-to-samples affinity relationship. We then establish the connection between tensor similarity and pairwise similarity, proving that the decomposable tensor similarity is the Kronecker product of the usual pairwise similarity and the non-decomposable tensor similarity is generalized to provide complementary information, which pairwise similarity fails to regard. Finally, the high-order tensor similarity and pairwise similarity (IPS2) were integrated collaboratively to enhance clustering performance by enjoying their merits. The proposed IPS2 is shown to perform superior or competitive to state-of-the-art methods on synthetic and real-world datasets. Extensive experiments demonstrated that tensor similarity is capable to boost the performance of the classical clustering method.

IEEE Transactions on Knowledge and Data Engineering, 2020

Subspace clustering assumes that the data is separable into separate subspaces. Such a simple assumption, does not always hold. We assume that, even if the raw data is not separable into subspaces, one can learn a representation (transform coefficients) such that the learnt representation is separable into subspaces. To achieve the intended goal, we embed subspace clustering techniques (locally linear manifold clustering, sparse subspace clustering and low rank representation) into transform learning. The entire formulation is jointly learnt; giving rise to a new class of methods called transformed subspace clustering (TSC). In order to account for non-linearity, kernelized extensions of TSC are also proposed. To test the performance of the proposed techniques, benchmarking is performed on image clustering and document clustering datasets. Comparison with state-of-the-art clustering techniques shows that our formulation improves upon them.

Neurocomputing, 2021

Tensor data analysis is the evolutionary step of data analysis to more than two dimensions. Dealing with tensor data is often based on tensor decomposition methods. The present paper focuses on unsupervised learning and provides a python package referred to as TensorClus including novel co-clustering algorithms of three-way data. All proposed algorithms are based on the latent block models and suitable to different types of data, sparse or not. They are successfully evaluated on challenges in text mining, recommender systems, and hyperspectral image clustering. TensorClus is an open-source Python package that allows easy interaction with other python packages such as NumPy and TensorFlow; it also offers an interface with some tensor decomposition packages namely Tensorly and TensorD on the one hand, and on the other, the co-clustering package Coclust. Finally, it provides CPU and GPU compatibility. The TensorClus library is available at https://pypi.org/project/TensorClus/ 1 .