Subspace clustering is a useful technique for many computer vision applications in which the intrinsic dimension of high-dimensional data is often smaller than the ambient dimension. Spectral clustering, as one of the main approaches to subspace clustering, often takes on a sparse representation or a low-rank representation to learn a block diagonal self-representation matrix for subspace generation. However, existing methods require solving a large scale convex optimization problem with a large set of data, with computational complexity reaches O(N^3) for N data points. Therefore, the efficiency and scalability of traditional spectral clustering methods can not be guaranteed for large scale datasets. In this paper, we propose a subspace clustering model based on the Kronecker product. Due to the property that the Kronecker product of a block diagonal matrix with any other matrix is still a block diagonal matrix, we can efficiently learn the representation matrix which is formed by the Kronecker product of k smaller matrices. By doing so, our model significantly reduces the computational complexity to O(kN^ 3 k ). Furthermore, our model is general in nature, and can be adapted to different regularization based subspace clustering methods. Experimental results on two public datasets show that our model significantly improves the efficiency compared with several state-of-the-art methods. Moreover, we have conducted experiments on synthetic data to verify the scalability of our model for large scale datasets.
Based on TRR, a 2D nonlinear variance regularized ridge regression (NVR3) @cite_28 was proposed to directly use 2D data, and thus the spatial information is maximally retained.
@cite_28
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In this paper, we analyze the spatial information of deep features, and propose two complementary regressions for robust visual tracking. First, we propose a kernelized ridge regression model wherein the kernel value is defined as the weighted sum of similarity scores of all pairs of patches between two samples. We show that this model can be formulated as a neural network and thus can be efficiently solved. Second, we propose a fully convolutional neural network with spatially regularized kernels, through which the filter kernel corresponding to each output channel is forced to focus on a specific region of the target. Distance transform pooling is further exploited to determine the effectiveness of each output channel of the convolution layer. The outputs from the kernelized ridge regression model and the fully convolutional neural network are combined to obtain the ultimate response. Experimental results on two benchmark datasets validate the effectiveness of the proposed method. Given a data set from a union of multiple linear sub-spaces, a robust subspace clustering algorithm fits each group of data points with a low-dimensional subspace and then clusters these data even though they are grossly corrupted or sampled from the union of dependent sub-spaces. Under the framework of spectral clustering, recent works using sparse representation, low rank representation and their extensions achieve robust clustering results by formulating the errors (e.g., corruptions) into their objective functions so that the errors can be removed from the inputs. However, these approaches have suffered from the limitation that the structure of the errors should be known as the prior knowledge. In this paper, we present a new method of robust sub-space clustering by eliminating the effect of the errors from the projection space (representation) rather than from the input space. We firstly prove that l1-, l2-, and l∞ -norm-based linear projection spaces share the property of intra-subspace projection dominance, i.e., the coefficients over intra-subspace data points are larger than those over inter-subspace data points. Based on this property, we propose a robust and efficient sub-space clustering algorithm, called Thresholding Ridge Regression (TRR). TRR calculates the ∞2 -norm-based coefficients of a given data set and performs a hard thresholding operator; and then the coefficients are used to build a similarity graph for clustering. Experimental studies show that TRR outperforms the state-of-the-art methods with respect to clustering quality, robustness, and time-saving. We propose a fast algorithm for ridge regression when the number of features is much larger than the number of observations (p ≫ n). The standard way to solve ridge regression in this setting works in the dual space and gives a running time of O(n2p). Our algorithm Subsampled Randomized Hadamard Transform- Dual Ridge Regression (SRHT-DRR) runs in time O(np log(n)) and works by preconditioning the design matrix by a Randomized Walsh-Hadamard Transform with a subsequent subsampling of features. We provide risk bounds for our SRHT-DRR algorithm in the fixed design setting and show experimental results on synthetic and real datasets. 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 e-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.