94 KAM Mathematical Colloquium
Emmanuel Candes
Stanford University
ROBUST PRINCIPAL COMPONENT ANALYSIS?
Wednesday February 4 2015 at 14:00, lecture room S5, second floor
KAM MFF UK
Malostranske nam. 25
118 00 Praha 1
Abstract
This talk is about a curious phenomenon, which concerns the reliable estimation of principal components in the face of severe corruptions. Here, the scientist is given a data matrix which is the sum of an approximately low-rank matrix and a sparse matrix modeling corrupted entries. In addition, many entries may be missing. Hence, we have a blind de-mixing problem in which the goal is to recover the low-rank structure and find out which entries have been corrupted. We present a novel approach to this problem with very surprising performance guarantees as well as a few applications in computer vision and biomedical imaging, where this technique opens new perspectives.
MODERN OPTIMIZATION MEETS PHYSICS: RECENT PROGRESS ON THE PHASE RETRIEVAL PROBLEM
Thursday February 5 2015 at 14:00, lecture room S3, third floor
KAM MFF UK
Malostranske nam.25
118 00 Praha 1
Abstract
In many imaging problems such as X-ray crystallography, detectors can only record the intensity or magnitude of a diffracted wave as opposed to measuring its phase. Phase retrieval concerns the recovery of an image from such phaseless information. Although this problem is in general combinatorially hard, it is of great importance because it arises in many applications ranging from astronomical imaging to speech analysis. This talk discusses novel acquisition strategies and novel convex and non-convex algorithms which are provably exact, thereby allowing perfect phase recovery from a minimal number of noiseless and intensity-only measurements. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. This may be of special contemporary interest because phase retrieval is at the center of spectacular current research efforts collectively known under the name of coherent diffraction imaging aimed, among other things, at determining the 3D structure of large protein complexes.
