94. Mathematical Colloquium
Emmanuel Candes
Stanford
ROBUST PRINCIPAL COMPONENT ANALYSIS
Wednesday February 4, 2015, 14:00room S5, 2nd floor
Malostranské nám. 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, 14:00room S3, 3rd floor
Malostranské nám. 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.
About the speaker
Profesor Emmanuel Candes je zajiste jednim z nejznamejsich matematiku soucasnosti. Ve sve osobe spojuje idealne matematiku cistou i aplikovanou, dokonce ve smyslu matematiky industrialni. Jeho prace je bez nadsazky strhujicim prikladem sire a relevance soucasne matematiky.
Prof. Emmanuel Candes studoval na Ecole Polytechnique, Universite Paris VI a IX a posleze na Stanford University, kde ziskal Ph.D. v roce 1998. Vykonal staze na nekolika prednich akademickych pracovistich jak v USA tak Evrope a stal se profesorem na Caltechu a od roku 2009 je profesorem matematiky a statistiky na Stanfordove univerzite, kde je od roku 2012 drzitelem Barnumovy-Simonsovy profesury.
Candesova cinnost zasahuje do nekolika oblasti: teoreticke informatiky, matematicke optimalizace, teorie informace, scientific computing, vysoce dimenzionalni statistiky s aplikacemi v inverznich problemech a zpracovani obrazu. Je jednim ze spolutvurcu dnes velmi intenzivne studovane oblasti compressed sensing (spolu s D. L. Donoho a T. Tao). Je autorem pres 90 puvodnich praci a mj. 3 US patentu. Za svou praci byl vyznamenan radou oceneni (jiz od studentskych let), z nichz jmenujeme alespoň: J. H. Wilkinson Prize (SIAM), A. T. Waterman Medaile (NSF), Polyova cena (SIAM), Collatzova cena (ICIAM), Lagrangeova cena (SIAM) a v letosnim roce cena G. D. Birkhoffa udelovana spolecne AMS a SIAM. Candes je clenem jak Narodni akademie tak Americke akademie umeni a vedy.
Prof. Emmanuel Candes je skvelym prednasejicim, napr. jeho plenarni prednaska na ICM 2014 v Soulu byla toho presvedcivym dokladem. Je nasi velkou cti, ze prof. Emmanuel Candes prednese 94. kolokvium a nasledne jeste jednu prednasku. Mame se na co tesit!
