The Geometry of Uniqueness, Sparsity and Clustering in Penalized Estimation

We provide a necessary and sufficient condition for the uniqueness of penalized least-squares estimators whose penalty term is given by a norm with a polytope unit ball, covering a wide range of methods including SLOPE and LASSO, as well as the related method of basis pursuit. We consider a strong type of uniqueness that is relevant for statistical problems. The uniqueness condition is geometric and involves how the row span of the design matrix intersects the faces of the dual norm unit ball, which for SLOPE is given by the sign permutahedron. Further considerations based this condition also allow to derive results on sparsity and clustering features. In particular, we define the notion of a SLOPE model to describe both sparsity and clustering properties of this method and also provide a geometric characterization of accessible SLOPE models.Comment: new title, minor change

Effective Condition Number Bounds for Convex Regularization

We derive bounds relating Renegar's condition number to quantities that govern the statistical performance of convex regularization in settings that include the $\ell_1$-analysis setting. Using results from conic integral geometry, we show that the bounds can be made to depend only on a random projection, or restriction, of the analysis operator to a lower dimensional space, and can still be effective if these operators are ill-conditioned. As an application, we get new bounds for the undersampling phase transition of composite convex regularizers. Key tools in the analysis are Slepian's inequality and the kinematic formula from integral geometry.Comment: 17 pages, 4 figures . arXiv admin note: text overlap with arXiv:1408.301

Projection Methods in Sparse and Low Rank Feasibility

In this thesis, we give an analysis of fixed point algorithms involving projections onto closed, not necessarily convex, subsets of finite dimensional vector spaces. These methods are used in applications such as imaging science, signal processing, and inverse problems. The tools used in the analysis place this work at the intersection of optimization and variational analysis. Based on the underlying optimization problems, this work is devided into two main parts. The first one is the compressed sensing problem. Because the problem is NP-hard, we relax it to a feasibility problem with two sets, namely, the set of vectors with at most s nonzero entries and, for a linear mapping M the affine subspace B of vectors satisfying Mx=p for p given. This problem will be referred to as the sparse-affine-feasibility problem. For the Douglas-Rachford algorithm, we give the proof of linear convergence to a fixed point in the case of a feasibility problem of two affine subspaces. It allows us to conclude a result of local linear convergence of the Douglas-Rachford algorithm in the sparse affine feasibility problem. Proceeding, we name sufficient conditions for the alternating projections algorithm to converge to the intersection of an affine subspace with lower level sets of point symmetric, lower semicontinuous, subadditive functions. This implies convergence of alternating projections to a solution of the sparse affine feasibility problem. Together with a result of local linear convergence of the alternating projections algorithm, this allows us to deduce linear convergence after finitely many steps for any initial point of a sequence of points generated by the alternating projections algorithm. The second part of this dissertation deals with the minimization of the rank of matrices satisfying a set of linear equations. This problem will be called rank-constrained-affine-feasibility problem. The motivation for the analysis of the rank minimization problem comes from the physical application of phase retrieval and a reformulation of the same as a rank minimization problem. We show that, locally, the method of alternating projections must converge at linear rate to a solution of the rank constrained affine feasibility problem

Extremal Area of Polygons sliding along Curves

In this paper we study the area function of polygons, where the vertices are sliding along curves. We give geometric criteria for the critical points and determine also the Hesse matrix at those points. This is the starting point for a Morse-theoretic approach, which includes the relation with the topology of the configuration spaces. Moreover the condition for extremal area gives rise to a new type of billiard: the inner area billiard.Comment: 20 pages, 12 figure

L1L_1-Penalization in Functional Linear Regression with Subgaussian Design

We study functional regression with random subgaussian design and real-valued response. The focus is on the problems in which the regression function can be well approximated by a functional linear model with the slope function being "sparse" in the sense that it can be represented as a sum of a small number of well separated "spikes". This can be viewed as an extension of now classical sparse estimation problems to the case of infinite dictionaries. We study an estimator of the regression function based on penalized empirical risk minimization with quadratic loss and the complexity penalty defined in terms of $L_1$-norm (a continuous version of LASSO). The main goal is to introduce several important parameters characterizing sparsity in this class of problems and to prove sharp oracle inequalities showing how the $L_2$-error of the continuous LASSO estimator depends on the underlying sparsity of the problem