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In recent years, sparse subspace tracking has attracted increasing attention in the signal processing community. In this paper, we propose a new provable effective method called OPIT (which stands for Online Power Iteration via Thresholding) for tracking the sparse principal subspace of data streams over time. Particularly, OPIT introduces a new adaptive variant of power iteration with space and computational complexity linear to the data dimension. In addition, a new column-based thresholding operator is developed to regularize the subspace sparsity. Utilizing both advantages of power iteration and thresholding operation, OPIT is capable of tracking the underlying subspace in both the classical regime and high dimensional regime. We also present a theoretical result on its convergence to verify its consistency in high dimensions. Several experiments are carried out on both synthetic and real data to demonstrate the performance of OPIT.

Tensor decomposition has been demonstrated to be successful in a wide range of applications, from neuroscience and wireless communications to social networks. In an online setting, factorizing tensors derived from multidimensional data streams is however nontrivial due to several inherent problems of real-time stream processing. In recent years, many research efforts have been dedicated to developing online techniques for decomposing such tensors, resulting in significant advances in streaming tensor decomposition or tensor tracking. This topic is emerging and enriches the literature on tensor decomposition, particularly from the data stream analystics perspective. Thus, it is imperative to carry out an overview of tensor tracking to help researchers and practitioners understand its developments and achievements, summarise the current trends and advances, and identify challenging problems. In this article, we provide a contemporary and comprehensive survey on different types of tensor tracking techniques. We particularly categorize the state-of-the-art methods into three main groups: streaming CP decompositions, streaming Tucker decompositions, and streaming decompositions under other tensor formats (i.e., tensor-train, t-SVD, and BTD). In each group, we further divide the existing algorithms into sub-categories based on their main optimization framework and model architectures. Finally, we present several applications, research challenges, open problems, and potential directions of tensor tracking in the future.

Exhibiting new features and likely related matching techniques to efficiently retrieve images from databases remains an open problem. This paper is first devoted to such a novel description of coloured textures by LRS (local relational string) so based on relative relations between neighbour pixels and on their distribution. It is illumination-invariant and able to capture some semantics local to objects through selected image stripes. Second, we propose a bi-directional query/candidate matching based on region--i.e. potential object--mutual preferences. The remaining third of the paper reports on the experimentation over 60,000 images, providing results in the TrecEval format about the performance compared with common available techniques and about the influence of key parameters in the proposed segmentation and pairing processes. Keywords: Image analysis; Symbolic representation; Relative relations; Bands based segmentation; Bipartite graph; Preference lists; Performances evaluation

In this paper, we propose two new provable algorithms for tracking online low-rank approximations of high-order streaming tensors with missing data. The first algorithm, dubbed adaptive Tucker decomposition (ATD), minimizes a weighted recursive least-squares cost function to obtain the tensor factors and the core tensor in an efficient way, thanks to an alternating minimization framework and a randomized sketching technique. Under the canonical polyadic (CP) model, the second algorithm, called ACP, is developed as a variant of ATD when the core tensor is imposed to be identity. Both algorithms are low-complexity tensor trackers that have fast convergence and low memory storage requirements. A unified convergence analysis is presented for ATD and ACP to justify their performance. Experiments indicate that the two proposed algorithms are capable of streaming tensor decomposition with competitive performance with respect to estimation accuracy and runtime on both synthetic and real data.