Sparse convex optimization methods for machine learning

Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 20013, 201

Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

Doctor of Philosophy

dissertationKernel smoothing provides a simple way of finding structures in data sets without the imposition of a parametric model, for example, nonparametric regression and density estimates. However, in many data-intensive applications, the data set could be large. Thus, evaluating a kernel density estimate or kernel regression over the data set directly can be prohibitively expensive in big data. This dissertation is working on how to efficiently find a smaller data set that can approximate the original data set with a theoretical guarantee in the kernel smoothing setting and how to extend it to more general smooth range spaces. For kernel density estimates, we propose randomized and deterministic algorithms with quality guarantees that are orders of magnitude more efficient than previous algorithms, which do not require knowledge of the kernel or its bandwidth parameter and are easily parallelizable. Our algorithms are applicable to any large-scale data processing framework. We then further investigate how to measure the error between two kernel density estimates, which is usually measured either in L1 or L2 error. In this dissertation, we investigate the challenges in using a stronger error, L ∞ (or worst case) error. We present efficient solutions for how to estimate the L∞ error and how to choose the bandwidth parameter for a kernel density estimate built on a subsample of a large data set. We next extend smoothed versions of geometric range spaces from kernel range spaces to more general types of ranges, so that an element of the ground set can be contained in a range with a non-binary value in [0,1]. We investigate the approximation of these range spaces through ϵ-nets and ϵ-samples. Finally, we study coresets algorithms for kernel regression. The size of the coresets are independent of the size of the data set, rather they only depend on the error guarantee, and in some cases the size of domain and amount of smoothing. We evaluate our methods on very large time series and spatial data, demonstrate that they can be constructed extremely efficiently, and allow for great computational gains