Multi-index fusion has demonstrated impressive performances in retrieval task by integrating different visual representations in a unified framework. However, previous works mainly consider propagating similarities via neighbor structure, ignoring the high order information among different visual representations. In this paper, we propose a new multi-index fusion scheme for image retrieval. By formulating this procedure as a multilinear based optimization problem, the complementary information hidden in different indexes can be explored more thoroughly. Specially, we first build our multiple indexes from various visual representations. Then a so-called index-specific functional matrix, which aims to propagate similarities, is introduced for updating the original index. The functional matrices are then optimized in a unified tensor space to achieve a refinement, such that the relevant images can be pushed more closer. The optimization problem can be efficiently solved by the augmented Lagrangian method with theoretical convergence guarantee. Unlike the traditional multi-index fusion scheme, our approach embeds the multi-index subspace structure into the new indexes with sparse constraint, thus it has little additional memory consumption in online query stage. Experimental evaluation on three benchmark datasets reveals that the proposed approach achieves the state-of-the-art performance, i.e., N-score 3.94 on UKBench, mAP 94.1 on Holiday and 62.39 on Market-1501.
Our work is also related with the multi-view subspace learning methods, especially the subspace clustering methods. Sparse subspace clustering @cite_26 and low-rank representation @cite_47 are most popular subspace clustering methods, which explore the relationships between samples via self-representation matrix. Zhang @cite_28 extend the low-rank representation to the multi-view setting via imposing a unfolding high-order norm to the subspace coefficient tensor. However, this tensor constraint can not explore the complementary information thoroughly, due to the fact that the low-rank norm penalize each view equally. By using a new tensor construction method, Xie @cite_10 replace the unfolding tensor norm with a recently proposed t-SVD based norm @cite_35 @cite_52 , which is based on a new tensor computational framework @cite_27 . This framework provides a closed multiplication operation between tensors @cite_19 , where the familiar tools used in matrix case can be directly extended to tensor case. Hence, it has good theoretical properties for handling complicated relationship among different views. For more detail information, we refer readers to read the section .
@cite_35 @cite_26 @cite_28 @cite_52 @cite_19 @cite_27 @cite_47 @cite_10
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In this paper we propose novel methods for completion (from limited samples) and de-noising of multilinear (tensor) data and as an application consider 3-D and 4- D (color) video data completion and de-noising. We exploit the recently proposed tensor-Singular Value Decomposition (t-SVD)[11]. Based on t-SVD, the notion of multilinear rank and a related tensor nuclear norm was proposed in [11] to characterize informational and structural complexity of multilinear data. We first show that videos with linear camera motion can be represented more efficiently using t-SVD compared to the approaches based on vectorizing or flattening of the tensors. Since efficiency in representation implies efficiency in recovery, we outline a tensor nuclear norm penalized algorithm for video completion from missing entries. Application of the proposed algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. We also consider the problem of tensor robust Principal Component Analysis (PCA) for de-noising 3-D video data from sparse random corruptions. We show superior performance of our method compared to the matrix robust PCA adapted to this setting as proposed in [4]. Many real-world problems deal with collections of high-dimensional data, such as images, videos, text, and web documents, DNA microarray data, and more. Often, such high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories to which the data belong. In this paper, we propose and study an algorithm, called sparse subspace clustering, to cluster data points that lie in a union of low-dimensional subspaces. The key idea is that, among the infinitely many possible representations of a data point in terms of other points, a sparse representation corresponds to selecting a few points from the same subspace. This motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of the data into subspaces. Since solving the sparse optimization program is in general NP-hard, we consider a convex relaxation and show that, under appropriate conditions on the arrangement of the subspaces and the distribution of the data, the proposed minimization program succeeds in recovering the desired sparse representations. The proposed algorithm is efficient and can handle data points near the intersections of subspaces. Another key advantage of the proposed algorithm with respect to the state of the art is that it can deal directly with data nuisances, such as noise, sparse outlying entries, and missing entries, by incorporating the model of the data into the sparse optimization program. We demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real-world problems of motion segmentation and face clustering. In this paper, we explore the problem of multiview subspace clustering. We introduce a low-rank tensor constraint to explore the complementary information from multiple views and, accordingly, establish a novel method called Low-rank Tensor constrained Multiview Subspace Clustering (LT-MSC). Our method regards the subspace representation matrices of different views as a tensor, which captures dexterously the high order correlations underlying multiview data. Then the tensor is equipped with a low-rank constraint, which models elegantly the cross information among different views, reduces effectually the redundancy of the learned subspace representations, and improves the accuracy of clustering as well. The inference process of the affinity matrix for clustering is formulated as a tensor nuclear norm minimization problem, constrained with an additional L2,1-norm regularizer and some linear equalities. The minimization problem is convex and thus can be solved efficiently by an Augmented Lagrangian Alternating Direction Minimization (AL-ADM) method. Extensive experimental results on four benchmark datasets show the effectiveness of our proposed LT-MSC method. The development of energy selective, photon counting X-ray detectors allows for a wide range of new possibilities in the area of computed tomographic image formation. Under the assumption of perfect energy resolution, here we propose a tensor-based iterative algorithm that simultaneously reconstructs the X-ray attenuation distribution for each energy. We use a multilinear image model rather than a more standard stacked vector representation in order to develop novel tensor-based regularizers. In particular, we model the multispectral unknown as a three-way tensor where the first two dimensions are space and the third dimension is energy. This approach allows for the design of tensor nuclear norm regularizers, which like its 2D counterpart, is a convex function of the multispectral unknown. The solution to the resulting convex optimization problem is obtained using an alternating direction method of multipliers approach. Simulation results show that the generalized tensor nuclear norm can be used as a standalone regularization technique for the energy selective (spectral) computed tomography problem and when combined with total variation regularization it enhances the regularization capabilities especially at low energy images where the effects of noise are most prominent. Recent work by Kilmer and Martin [Linear Algebra Appl., 435 (2011), pp. 641--658] and Braman [Linear Algebra Appl., 433 (2010), pp. 1241--1253] provides a setting in which the familiar tools of linear algebra can be extended to better understand third-order tensors. Continuing along this vein, this paper investigates further implications including (1) a bilinear operator on the matrices which is nearly an inner product and which leads to definitions for length of matrices, angle between two matrices, and orthogonality of matrices, and (2) the use of t-linear combinations to characterize the range and kernel of a mapping defined by a third-order tensor and the t-product and the quantification of the dimensions of those sets. These theoretical results lead to the study of orthogonal projections as well as an effective Gram--Schmidt process for producing an orthogonal basis of matrices. The theoretical framework also leads us to consider the notion of tensor polynomials and their relation to tensor eigentupl... In this paper, we address the subspace clustering problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to cluster the samples into their respective subspaces and remove possible outliers as well. To this end, we propose a novel objective function named Low-Rank Representation (LRR), which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that the convex program associated with LRR solves the subspace clustering problem in the following sense: When the data is clean, we prove that LRR exactly recovers the true subspace structures; when the data are contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for data corrupted by arbitrary sparse errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace clustering and error correction in an efficient and effective way. In this paper, we address the multi-view subspace clustering problem. Our method utilizes the circulant algebra for tensor, which is constructed by stacking the subspace representation matrices of different views and then rotating, to capture the low rank tensor subspace so that the refinement of the view-specific subspaces can be achieved, as well as the high order correlations underlying multi-view data can be explored. By introducing a recently proposed tensor factorization, namely tensor-Singular Value Decomposition (t-SVD) kilmer13 , we can impose a new type of low-rank tensor constraint on the rotated tensor to capture the complementary information from multiple views. Different from traditional unfolding based tensor norm, this low-rank tensor constraint has optimality properties similar to that of matrix rank derived from SVD, so the complementary information among views can be explored more efficiently and thoroughly. The established model, called t-SVD based Multi-view Subspace Clustering (t-SVD-MSC), falls into the applicable scope of augmented Lagrangian method, and its minimization problem can be efficiently solved with theoretical convergence guarantee and relatively low computational complexity. Extensive experimental testing on eight challenging image dataset shows that the proposed method has achieved highly competent objective performance compared to several state-of-the-art multi-view clustering methods.