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SUMMARY:Multi-parameter regularisation for solving inverse problems of unm
 ixing problems: theoretical and practical aspects - Valeriya Naumova (Simu
 la\, Norway)
DTSTART:20180619T133000Z
DTEND:20180619T143000Z
UID:TALK107533@talks.cam.ac.uk
CONTACT:Carola-Bibiane Schoenlieb
DESCRIPTION:Motivated by real-life applications in signal processing and i
 mage analysis\, where the quantity of interest is generated by several sou
 rces to be accurately modelled and separated\, as well as by recent advanc
 es in sparse regularisation theory and optimisation\, we present a theoret
 ical and algorithmic framework for optimal support recovery in inverse pro
 blems of unmixing type by means of multi-penalty regularisation. While mul
 ti-penalty regularisation is not a novel technique [1]\, we aim at providi
 ng precise reconstruction guarantees and methods for adaptive regularisati
 on parameter choice.\n \n\n\nWe consider and analyse a regularisation fun
 ctional composed of a data-fidelity term\, where signal and noise are addi
 tively mixed\, a non-smooth\, convex\, sparsity promoting term\, and a con
 vex penalty term to model the noise. We prove not only that the well-esta
 blished theory for sparse recovery in the single parameter case can be tra
 nslated to the multi-penalty settings\, but we also demonstrate the enhanc
 ed properties of multi-penalty regularisation in terms of support identifi
 cation compared to sole $\\ell^1$-minimisation. We further propose an ite
 rative alternating algorithm based on simple iterative thresholding steps 
 to perform the minimisation of the extended multi-penalty functional\, con
 taining  non-smooth and non-convex sparsity promoting term.\nExtending th
 e notion of Lasso path\, we additionally propose an efficient procedure fo
 r an adaptive parameter choice in multi-penalty regularisation\, focusing 
 on the recovery of the correct support of the solution. The approach essen
 tially enables a fast construction of a tiling over the parameter space in
  such a way that each tile corresponds to a different sparsity pattern of 
 the solution. Finally\, we briefly discuss the applicability of multi-pena
 lty for recovery of low-rank matrices with approximately sparse singular v
 ectors from a limited number of measurements.\n \n\nTo exemplify the robu
 stness and effectiveness of the multi-penalty framework\, we provide an ex
 tensive numerical analysis of our method and compare it with state-of-the-
 art single-penalty algorithms for compressed sensing problems.\n \n\nThis
  is joint work with Markus Grasmair [3\, 4]\, Norwegian University of Scie
 nce and Technology\; Timo Klock [4]\, Simula Research Laboratory\; Steffen
  Peter [2] and Johannes Maly\, Technical University of Munich.\n\n\n\nRefe
 rences:\n\n1. Y. Meyer\, Oscillating patterns in image processing and nonl
 inear evolution equations: the fifteenth Dean Jacqueline B. Lewis memorial
  lectures\, University Lecture Series\, vol. 22\, 2001.\n2. V. Naumova and
  S. Peter. Minimization of multi-penalty functionals by alternating iterat
 ive thresholding and optimal parameter choices. Inverse Problems\, 30:1250
 03\, 1–34\, 2014.\n3. M. Grasmair and V. Naumova. Conditions on optimal 
 support recovery in unmixing problems by means of  multi-penalty regulariz
 ation. Inverse Problems\, 32(10):104007\, 2016.\n4. M. Grasmair\, T. Klock
 \, and V. Naumova. Multiple parameter learning with regularization path al
 gorithms\, submitted\, 2017.\n5. M. Fornasier\, J. Maly\, and V. Naumova. 
 A-T-LAS: A Multi-Penalty Approach to Compressed Sensing of Low-Rank Matric
 es with Sparse Decompositions\, submitted 2018.
LOCATION:MR 14
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