A new projection method for finding the closest point in the intersection of convex sets

In this paper we present a new iterative projection method for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method, termed AAMR for averaged alternating modified reflections, can be viewed as an adequate modification of the Douglas--Rachford method that yields a solution to the best approximation problem. Under a constraint qualification at the point of interest, we show strong convergence of the method. In fact, the so-called strong CHIP fully characterizes the convergence of the AAMR method for every point in the space. We report some promising numerical experiments where we compare the performance of AAMR against other projection methods for finding the closest point in the intersection of pairs of finite dimensional subspaces

Computing the Resolvent of the Sum of Maximally Monotone Operators with the Averaged Alternating Modified Reflections Algorithm

The averaged alternating modified reflections algorithm is a projection method for finding the closest point in the intersection of closed and convex sets to a given point in a Hilbert space. In this work, we generalize the scheme so that it can be used to compute the resolvent of the sum of two maximally monotone operators. This gives rise to a new splitting method, which is proved to be strongly convergent. A standard product space reformulation permits to apply the method for computing the resolvent of a finite sum of maximally monotone operators. Based on this, we propose two variants of such parallel splitting method.This work was partially supported by Ministerio de Economía, Industria y Competitividad (MINECO) of Spain and European Regional Development Fund (ERDF), grant MTM2014-59179-C2-1-P. FJAA was supported by the Ramón y Cajal program by MINECO and ERDF (RYC-2013-13327) and RC was supported by MINECO and European Social Fund (BES-2015-073360) under the program “Ayudas para contratos predoctorales para la formación de doctores 2015”

Computing the Resolvent of the Sum of Maximally Monotone Operators with the Averaged Alternating Modified Reflections Algorithm

The averaged alternating modified reflections algorithm is a projection method for finding the closest point in the intersection of closed and convex sets to a given point in a Hilbert space. In this work, we generalize the scheme so that it can be used to compute the resolvent of the sum of two maximally monotone operators. This gives rise to a new splitting method, which is proved to be strongly convergent. A standard product space reformulation permits to apply the method for computing the resolvent of a finite sum of maximally monotone operators. Based on this, we propose two variants of such parallel splitting method.This work was partially supported by Ministerio de Economía, Industria y Competitividad (MINECO) of Spain and European Regional Development Fund (ERDF), grant MTM2014-59179-C2-1-P. FJAA was supported by the Ramón y Cajal program by MINECO and ERDF (RYC-2013-13327) and RC was supported by MINECO and European Social Fund (BES-2015-073360) under the program “Ayudas para contratos predoctorales para la formación de doctores 2015”

Analysis of Incomplete Data and an Intrinsic-Dimension Helly Theorem

The analysis of incomplete data is a long-standing challenge in practical statistics. When, as is typical, data objects are represented by points in R^d , incomplete data objects correspond to affine subspaces (lines or Δ-flats).With this motivation we study the problem of finding the minimum intersection radius r(L) of a set of lines or Δ-flats L: the least r such that there is a ball of radius r intersecting every flat in L. Known algorithms for finding the minimum enclosing ball for a point set (or clustering by several balls) do not easily extend to higher dimensional flats, primarily because “distances” between flats do not satisfy the triangle inequality. In this paper we show how to restore geometry (i.e., a substitute for the triangle inequality) to the problem, through a new analog of Helly’s theorem. This “intrinsic-dimension” Helly theorem states: for any family L of Δ-dimensional convex sets in a Hilbert space, there exist Δ + 2 sets L' ⊆ L such that r(L) ≤ 2r(L'). Based upon this we present an algorithm that computes a (1+ε)-core set L' ⊆ L, |L'| = O(Δ^4/ε), such that the ball centered at a point c with radius (1 +ε)r(L') intersects every element of L. The running time of the algorithm is O(n^(Δ+1)dpoly(Δ/ε)). For the case of lines or line segments (Δ = 1), the (expected) running time of the algorithm can be improved to O(ndpoly(1/ε)).We note that the size of the core set depends only on the dimension of the input objects and is independent of the input size n and the dimension d of the ambient space

Designing structured tight frames via an alternating projection method

Tight frames, also known as general Welch-bound- equality sequences, generalize orthonormal systems. Numerous applications - including communications, coding, and sparse approximation- require finite-dimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems (IEPs), which includes the frame design problem. To apply this method, one needs only to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is the fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate that alternating projection is an effective tool for frame design, the paper studies some important structural properties in detail. First, it addresses the most basic design problem: constructing tight frames with prescribed vector norms. Then, it discusses equiangular tight frames, which are natural dictionaries for sparse approximation. Finally, it examines tight frames whose individual vectors have low peak-to-average-power ratio (PAR), which is a valuable property for code-division multiple-access (CDMA) applications. Numerical experiments show that the proposed algorithm succeeds in each of these three cases. The appendices investigate the convergence properties of the algorithm

Projections Onto Convex Sets (POCS) Based Optimization by Lifting

Two new optimization techniques based on projections onto convex space (POCS) framework for solving convex and some non-convex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. If the cost function is a convex function in R^N the corresponding set is a convex set in R^(N+1). The iterative optimization approach starts with an arbitrary initial estimate in R^(N+1) and an orthogonal projection is performed onto one of the sets in a sequential manner at each step of the optimization problem. The method provides globally optimal solutions in total-variation, filtered variation, l1, and entropic cost functions. It is also experimentally observed that cost functions based on lp, p<1 can be handled by using the supporting hyperplane concept