Abstract Most existing approaches address multi-view subspace clustering problem by constructing the affinity matrix on each view separately and afterwards propose how to extend spectral clustering algorithm to handle multi-view data. This paper presents an approach to multi-view subspace clustering that learns a joint subspace representation by constructing affinity matrix shared among all views. Relying on the importance of both low-rank and sparsity constraints in the construction of the affinity matrix, we introduce the objective that balances between the agreement across different views, while at the same time encourages sparsity and low-rankness of the solution. Related low-rank and sparsity constrained optimization problem is for each view solved using the alternating direction method of multipliers. Furthermore, we extend our approach to cluster data drawn from nonlinear subspaces by solving the corresponding problem in a reproducing kernel Hilbert space. The proposed algorithm outperforms state-of-the-art multi-view subspace clustering algorithms on one synthetic and four real-world datasets.
If data is contaminated by noise, the following minimization problem needs to be solved: This problem can be efficiently solved using ADMM optimization procedure @cite_41 .
@cite_41
2952716509 2047071281 2011194005 2114147096
On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries, and comes up in many areas of science and engineering including collaborative filtering, machine learning, control, remote sensing, and computer vision to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown low-rank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclear-norm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown n x n matrix of low rank r from just about nr log^2 n noisy samples with an error which is proportional to the noise level. We present numerical results which complement our quantitative analysis and show that, in practice, nuclear norm minimization accurately fills in the many missing entries of large low-rank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout. On the heels of compressed sensing, a new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries. It comes up in many areas of science and engineering, including collaborative filtering, machine learning, control, remote sensing, and computer vision, to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown low-rank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclear-norm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown matrix of low rank from just about log noisy samples with an error that is proportional to the noise level. We present numerical results that complement our quantitative analysis and show that, in practice, nuclear-norm minimization accurately fills in the many missing entries of large low-rank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout. In this paper, we propose a Bayesian MAP estimator for solving the deconvolution problems when the observations are corrupted by Poisson noise. Towards this goal, a proper data fidelity term (log-likelihood) is introduced to reflect the Poisson statistics of the noise. On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms such as wavelets or curvelets. Both analysis and synthesis-type sparsity priors are considered. Piecing together the data fidelity and the prior terms, the deconvolution problem boils down to the minimization of non-smooth convex functionals (for each prior). We establish the well-posedness of each optimization problem, characterize the corresponding minimizers, and solve them by means of proximal splitting algorithms originating from the realm of non-smooth convex optimization theory. Experimental results are conducted to demonstrate the potential applicability of the proposed algorithms to astronomical imaging datasets. We consider inexact linear equations y ≈ Φx where y is a given vector in R n , Φ is a given n x m matrix, and we wish to find x 0,∈ as sparse as possible while obeying ∥y - Φx 0,∈ ∥ 2 ≤ ∈. In general, this requires combinatorial optimization and so is considered intractable. On the other hand, the l 1 -minimization problem min ∥x∥ 1 subject to ∥y - Φx∥ 2 ≤ e is convex and is considered tractable. We show that for most Φ, if the optimally sparse approximation x 0,∈ is sufficiently sparse, then the solution x 1,∈ of the l 1 -minimization problem is a good approximation to x 0,∈ . We suppose that the columns of Φ are normalized to the unit l 2 -norm, and we place uniform measure on such Φ. We study the underdetermined case where m ∼ τn and τ > 1, and prove the existence of p = p(r) > 0 and C = C(p, τ) so that for large n and for all Φ's except a negligible fraction, the following approximate sparse solution property of Φ holds: for every y having an approximation ∥y - Φx 0 ∥ 2 ≤ ∈ by a coefficient vector x 0 e R m with fewer than ρ · n nonzeros, ∥x 1,∈ - x 0 ∥ 2 ≤ C ≤ ∈. This has two implications. First, for most Φ, whenever the combinatorial optimization result x 0,∈ would be very sparse, x 1,∈ is a good approximation to x 0,∈ . Second, suppose we are given noisy data obeying y = Φx 0 + z where the unknown x 0 is known to be sparse and the noise ∥z∥ 2 ≤ ∈. For most Φ, noise-tolerant l 1 -minimization will stably recover x 0 from y in the presence of noise z. We also study the barely determined case m = n and reach parallel conclusions by slightly different arguments. Proof techniques include the use of almost-spherical sections in Banach space theory and concentration of measure for eigenvalues of random matrices.