Subspace clustering is a problem of exploring the low-dimensional subspaces of high-dimensional data. State-of-the-art approaches are designed by following the model of spectral clustering-based method. These methods pay much attention to learn the representation matrix to construct a suitable similarity matrix and overlook the influence of the noise term on subspace clustering. However, the real data are always contaminated by the noise and the noise usually has a complicated statistical distribution. To alleviate this problem, in this paper, we propose a subspace clustering method based on Cauchy loss function (CLF). Particularly, it uses CLF to penalize the noise term for suppressing the large noise mixed in the real data. This is due to that the CLF’s influence function has an upper bound that can alleviate the influence of a single sample, especially the sample with a large noise, on estimating the residuals. Furthermore, we theoretically prove the grouping effect of our proposed method, which means that highly correlated data can be grouped together. Finally, experimental results on five real data sets reveal that our proposed method outperforms several representative clustering methods.
Least Squares Regression (LSR) @cite_15 employs the Frobenius norm to handle the representation matrix and the noise matrix simultaneously. The corresponding optimization problem is defined as Note that the above problem can be efficiently solved. The main contribution of LSR is that it encourages grouping effect which can group highly correlated data together.
@cite_15
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The problem of minimizing the rank of a matrix subject to affine constraints has applications in several areas including machine learning, and is known to be NP-hard. A tractable relaxation for this problem is nuclear norm (or trace norm) minimization, which is guaranteed to find the minimum rank matrix under suitable assumptions. In this paper, we propose a family of Iterative Reweighted Least Squares algorithms IRLS-p (with 0 ≤ p ≤ 1), as a computationally efficient way to improve over the performance of nuclear norm minimization. The algorithms can be viewed as (locally) minimizing certain smooth approximations to the rank function. When p = 1, we give theoretical guarantees similar to those for nuclear norm minimization, that is, recovery of low-rank matrices under certain assumptions on the operator defining the constraints. For p < 1, IRLS-p shows better empirical performance in terms of recovering low-rank matrices than nuclear norm minimization. We provide an efficient implementation for IRLS-p, and also present a related family of algorithms, sIRLS-p. These algorithms exhibit competitive run times and improved recovery when compared to existing algorithms for random instances of the matrix completion problem, as well as on the MovieLens movie recommendation data set. We consider in this paper the problem of noisy 1-bit matrix completion under a general non-uniform sampling distribution using the max-norm as a convex relaxation for the rank. A max-norm constrained maximum likelihood estimate is introduced and studied. The rate of convergence for the estimate is obtained. Information-theoretical methods are used to establish a minimax lower bound under the general sampling model. The minimax upper and lower bounds together yield the optimal rate of convergence for the Frobenius norm loss. Computational algorithms and numerical performance are also discussed. Low-rank representation (LRR) intends to find the representation with lowest rank of a given data set, which can be formulated as a rank-minimisation problem. Since the rank operator is non-convex and discontinuous, most of the recent works use the nuclear norm as a convex relaxation. It is theoretically shown that, under some conditions, the Frobenius-norm-based optimisation problem has a unique solution that is also a solution of the original LRR optimisation problem. In other words, it is feasible to apply the Frobenius norm as a surrogate of the non-convex matrix rank function. This replacement will largely reduce the time costs for obtaining the lowest-rank solution. Experimental results show that the method (i.e. fast LRR (fLRR)) performs well in terms of accuracy and computation speed in image clustering and motion segmentation compared with nuclear-norm-based LRR algorithm. Sparse regression often uses l p norm priors (with p < 2). This paper demonstrates that the introduction of mixed-norms in such contexts allows one to go one step beyond in signal models, and promote some different, structured, forms of sparsity. It is shown that the particular case of the l1,2 and l2,1 norms leads to new group shrinkage operators. Mixed norm priors are shown to be particularly efficient in a generalized basis pursuit denoising approach, and are also used in a context of morphological component analysis. A suitable version of the Block Coordinate Relaxation algorithm is derived for the latter. The group-shrinkage operators are then modified to overcome some limitations of the mixed-norms. The proposed group shrinkage operators are tested on simulated signals in specific situations, to illustrate and compare their different behaviors. Results on real data are also used to illustrate the relevance of the approach.