Fast learning rate of multiple kernel learning: Trade-off between sparsity and smoothness

We investigate the learning rate of multiple kernel learning (MKL) with $\ell_1$ and elastic-net regularizations. The elastic-net regularization is a composition of an $\ell_1$-regularizer for inducing the sparsity and an $\ell_2$-regularizer for controlling the smoothness. We focus on a sparse setting where the total number of kernels is large, but the number of nonzero components of the ground truth is relatively small, and show sharper convergence rates than the learning rates have ever shown for both $\ell_1$ and elastic-net regularizations. Our analysis reveals some relations between the choice of a regularization function and the performance. If the ground truth is smooth, we show a faster convergence rate for the elastic-net regularization with less conditions than $\ell_1$-regularization; otherwise, a faster convergence rate for the $\ell_1$-regularization is shown.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1095 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1103.043

A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure

We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge

Efficient Semidefinite Spectral Clustering via Lagrange Duality

We propose an efficient approach to semidefinite spectral clustering (SSC), which addresses the Frobenius normalization with the positive semidefinite (p.s.d.) constraint for spectral clustering. Compared with the original Frobenius norm approximation based algorithm, the proposed algorithm can more accurately find the closest doubly stochastic approximation to the affinity matrix by considering the p.s.d. constraint. In this paper, SSC is formulated as a semidefinite programming (SDP) problem. In order to solve the high computational complexity of SDP, we present a dual algorithm based on the Lagrange dual formalization. Two versions of the proposed algorithm are proffered: one with less memory usage and the other with faster convergence rate. The proposed algorithm has much lower time complexity than that of the standard interior-point based SDP solvers. Experimental results on both UCI data sets and real-world image data sets demonstrate that 1) compared with the state-of-the-art spectral clustering methods, the proposed algorithm achieves better clustering performance; and 2) our algorithm is much more efficient and can solve larger-scale SSC problems than those standard interior-point SDP solvers.Comment: 13 page

Weak Lensing Mass Reconstruction using Wavelets

This paper presents a new method for the reconstruction of weak lensing mass maps. It uses the multiscale entropy concept, which is based on wavelets, and the False Discovery Rate which allows us to derive robust detection levels in wavelet space. We show that this new restoration approach outperforms several standard techniques currently used for weak shear mass reconstruction. This method can also be used to separate E and B modes in the shear field, and thus test for the presence of residual systematic effects. We concentrate on large blind cosmic shear surveys, and illustrate our results using simulated shear maps derived from N-Body Lambda-CDM simulations with added noise corresponding to both ground-based and space-based observations.Comment: Accepted manuscript with all figures can be downloaded at: http://jstarck.free.fr/aa_wlens05.pdf and software can be downloaded at http://jstarck.free.fr/mrlens.htm