Friday, August 19, 2016

Reviews: Low-Rank Semidefinite Programming:Theory and Applications / A Primer on Reproducing Kernel Hilbert Spaces

In the same spirit as the Highly Technical Reference pages, I have created an "Overviews" tag that links back to all the entries that provide a review or an overview of a specific field. Today we have two:
Finding low-rank solutions of semidefinite programs is important in many applications. For example, semidefinite programs that arise as relaxations of polynomial optimization problems are exact relaxations when the semidefinite program has a rank-1 solution. Unfortunately, computing a minimum-rank solution of a semidefinite program is an NP-hard problem. In this paper we review the theory of low-rank semidefinite programming, presenting theorems that guarantee the existence of a low-rank solution, heuristics for computing low-rank solutions, and algorithms for finding low-rank approximate solutions. Then we present applications of the theory to trust-region problems and signal processing.

A Primer on Reproducing Kernel Hilbert Spaces by Jonathan H. Manton, Pierre-Olivier Amblard
Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and explaining when and why these spaces are efficacious. The novel viewpoint is that reproducing kernel Hilbert space theory studies extrinsic geometry, associating with each geometric configuration a canonical overdetermined coordinate system. This coordinate system varies continuously with changing geometric configurations, making it well-suited for studying problems whose solutions also vary continuously with changing geometry. This primer can also serve as an introduction to infinite-dimensional linear algebra because reproducing kernel Hilbert spaces have more properties in common with Euclidean spaces than do more general Hilbert spaces.

Image Credit: NASA/JPL-Caltech/Space Science Institute
W00100570.jpg was taken on 2016-08-16 22:59 (UTC) and received on Earth 2016-08-17 07:42 (UTC). The camera was pointing toward SATURN, and the image was taken using the MT3 and CL2 filters.

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