We consider adaptations of the Mumford-Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford-Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the graph constructions under which the minimizers of graph Mumford-Shah functionals converge to a minimizer of a continuum Mumford-Shah functional. Furthermore we explicitly identify the limiting functional. Moreover we describe an efficient algorithm for computing the approximate minimizers of the graph Mumford-Shah functional.
TV based regularizations have also been developed in statistics community. Mammen and van de Geer @cite_4 have considered it is the setting of nonparametric regression and have shown that the TV regularization provides an estimator that achieves the optimal min-max recovery rate in one dimension over noisy samples of functions in unit ball with respect to the BV norm. TV based regularizations in hifher dimensions have been considered by by Tibshirani, Saunders, Rosset, Zhu, and Knight @cite_14 who call the functional fused LASSO. H "utter and Rigollet @cite_23 , show that, up to logarithms, in dimension @math , the TV regularization on grids achieves the optimal min-max rate over the unit ball with respect to the BV norm. Recently, Padilla, Sharpnack, Chen and Witten @cite_36 show for random. geometric graphs and for KNN graphs that up to logarithms, in dimension @math , TV regularization again achieves the optimal min-max rate.
@cite_36 @cite_14 @cite_4 @cite_23
1997409415 2963583445 1874232560 2011710850
TV regularization is an L1 penalization of the flow gradient magnitudes, and due to the tendency of the L1 norm to favor sparse solutions (i.e. lots of ‘zeros’), the fill-in effect caused by the regularizer leads to piecewise constant solutions in weakly textured areas. This effect, known as ‘staircasing’ in a 1D setting, can be reduced significantly by using a quadratic penalization for small gradient magnitudes while sticking to linear penalization for larger magnitudes to maintain the discontinuity preserving properties known from TV. A comparison of isotropic TV and isotropic Huber regularity is shown in Fig. 1 by means of rendering the disparities u1 of the Dimetrodon dataset. The color coded flow (cf. Fig. 1(a)) is superimposed as texture. Based on the two observations that motion discontinuities often occur along object boundaries and that in turn object boundaries often coincide Abstract We study the recovery of Hermitian low rank matrices X ∈ C n × n from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form a j a j ⁎ for some measurement vectors a 1 , … , a m , i.e., the measurements are given by b j = tr ( X a j a j ⁎ ) . The case where the matrix X = x x ⁎ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements b j = | 〈 x , a j 〉 | 2 ) via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors a j , j = 1 , … , m , being chosen independently at random according to a standard Gaussian distribution, or a j being sampled independently from an (approximate) complex projective t-design with t = 4 . In the Gaussian case, we require m ≥ C r n measurements, while in the case of 4-designs we need m ≥ Cr n log ⁡ ( n ) . Our results are uniform in the sense that one random choice of the measurement vectors a j guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Krahmer and Kueng. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii. Many statistical M-estimators are based on convex optimization problems formed by the weighted sum of a loss function with a norm-based regularizes We analyze the convergence rates of first-order gradient methods for solving such problems within a high-dimensional framework that allows the data dimension d to grow with (and possibly exceed) the sample size n. This high-dimensional structure precludes the usual global assumptions— namely, strong convexity and smoothness conditions—that underlie classical optimization analysis. We define appropriately restricted versions of these conditions, and show that they are satisfied with high probability for various statistical models. Under these conditions, our theory guarantees that Nesterov's first-order method [12] has a globally geometric rate of convergence up to the statistical precision of the model, meaning the typical Euclidean distance between the true unknown parameter θ* and the optimal solution ^θ. This globally linear rate is substantially faster than previous analyses of global convergence for specific methods that yielded only sublinear rates. Our analysis applies to a wide range of M-estimators and statistical models, including sparse linear regression using Lasso (l1-regularized regression), group Lasso, block sparsity, and low-rank matrix recovery using nuclear norm regularization. Overall, this result reveals an interesting connection between statistical precision and computational efficiency in high-dimensional estimation. Abstract In this paper, we propose an efficient algorithm for MR image reconstruction. The algorithm minimizes a linear combination of three terms corresponding to a least square data fitting, total variation (TV) and L 1 norm regularization. This has been shown to be very powerful for the MR image reconstruction. First, we decompose the original problem into L 1 and TV norm regularization subproblems respectively. Then, these two subproblems are efficiently solved by existing techniques. Finally, the reconstructed image is obtained from the weighted average of solutions from two subproblems in an iterative framework. We compare the proposed algorithm with previous methods in term of the reconstruction accuracy and computation complexity. Numerous experiments demonstrate the superior performance of the proposed algorithm for compressed MR image reconstruction.