[TL;DR] How I discovered sparsity and stopped worrying about it This is a journey into sparsity. The story started in 2009, in the South of France, and the nice island of Porquerolles. Here, clear blue water.
Amel Benazza-Benyahia, Jérôme Bolte, Antonin Chambolle, Caroline Chaux, Philippe Ciuciu, Albert Cohen, Patrick Louis Combettes, Claude Comtat, Christine de Mol, Alvaro R. De Pierro, Michel Defrise, Ron DeVore, Jalal Fadili, Mario Figueiredo, Massimo Fornasier, Jacques Froment, Yves Goussard, Gabor Herman, Jérome Idier, Dirk Lorenz, Russell Luke, Francois Malgouyres, Pierre Maréchal, Ali Mohammad-Djafari, Stanley Osher, Valérie Perrier, Jean-Christophe Pesquet, Gabriel Peyré, Aleksandra Pizurica, Andrew Reader, Elena Resmerita, Marc Sigelle, Gabriele Steidl, Marc Teboulle, Joel Trussell, Michael Unser, Dimitri Van de Ville, Isao Yamada
The program was wonderful. The trade-off between sea, science and sun was so easy. At that time, I had been working mostly with signals and images, processing them with sparsity-promoting decompositions: complex orthogonal/biorthogonal dual-tree wavelets with interesting covariance properties able at multichannel/multicomponent image denoising, tighly oversampled complex filter banks. We wanted to have data become sparser through a projection on some vector basis, union of bases or frames. Yet, what sparsity really meant was not fully clear to me. The count measure $\ell_0$ of course, but it is not practically usable for several reasons. [TO BE CONTINUED, THEN MATERIALS] So: SPOQ means, in Swedish, "Svenskt Pediatriskt Ortopediskt Qvalitetsregister" (Swedish Pediatric Orthopaedic Quality register).
L0 sparsity count index, L1-norm non-smooth convex proxy, L1/L2-norm ratio and Lp/Lq quasinorm/norm ratio smooth non-convex penalties.
Abstract: Underdetermined or ill-posed inverse problems require additional information for sound solutions with tractable optimization algorithms. Sparsity yields consequent heuristics to that matter, with numerous applications in signal restoration, image recovery, or machine learning. Since the
ℓ0
count measure is barely tractable, many statistical or learning approaches have invested in computable proxies, such as the ℓ1
norm. However, the latter does not exhibit the desirable property of scale invariance for sparse data. Generalizing the SOOT Euclidean/Taxicab ℓ1
/
ℓ2
norm-ratio initially introduced for blind deconvolution, we propose SPOQ, a family of smoothed scale-invariant penalty functions. It consists of a Lipschitz-differentiable surrogate for ℓp
-over-ℓ
q
quasi-norm/norm ratios with p∈]0,2[
and q≥2
. This surrogate is embedded into a novel majorize-minimize trust-region approach, generalizing the variable metric forward-backward algorithm. For naturally sparse mass-spectrometry signals, we show that SPOQ significantly outperforms ℓ0
, ℓ1
, Cauchy, Welsch, and \celo penalties on several performance measures. Guidelines on SPOQ hyperparameters tuning are also provided, suggesting simple data-driven choices.
Chants des confins (Confines' songs) is one music for each (relaive) day of confinement due to the COVID-19 coronavirus pandemia. And two lines. Two rhymes. Rarely more. I listened to each song five times in a row before posting it.
So far: Frankie Goes To Hollywood, Pink Floyd, Télépopmusik,, Henry Purcell (by Sarah Connolly and Marianne Beate Kielland), Villagers of Ioannina City, Philip Glass with Lavinia Meijer (harp), Dave Brubeck & Paul Desmond (a kind of fake on Take five) (and the Stranglers), Mano Negra, Loney Dear, Sarah Vaughan, Deep Purple (and Navarone), The Kyteman Orchestra (Colin Benders), Dominique A (et Alain Bashung), VALD, George Frideric Handel (sung by Nathalie Stutzmann & Philippe Jaroussky), Kings of convenience and Feist, Les Têtes Raides, Yom (Guillaume Humery) and the wonder rabbis, Sting (interpreted by Pat Metheny, Nils Landgren, Esbjörn Svensson, Michael Brecker, Lars Danielsson, Wolfgang Haffner), Andrea da Florentia (Ensemble Syntagma), Ilene Barnes, Marillion, Opeth, Meute (Brass techno band on Disclosure and Eliza Doolittle and a Flume Remix, Osiris, Porcupine Tree (Steven Wilson), The Stranglers (and Mariachi Mexteca feat. Hugh Cornwell), Suzanne Vega, The Nits, The Supremes, Nena, Gov't Mule, Dire Straits, Microfilm, Sylvain Chauveau, Wolves in the Throne Room, Genesis (Peter Gabriel), Barclay James Harvest, Nancy Sinatra & Lee Hazlewood, Loudblast, Archive, Calvin Russell, Laibach (The Beatles), Giovanni Battista Pergolesi (Gérard Lesne, Il seminario musicale), Yaron Herman (and Radiohead again, yes), Samaris, Bel Canto (with the beautiful voice of Anneli Drecker), Sinéad O'Connor (and The Chieftains), Youn Sun Nah, Park Jiha, Max Richter, The Waterboys, Bloc Party, Hubert-Félix-Thiéfaine, Reis-Demuth-Wiltgen, Paul Simon and Art Garfunkel, Paul Robeson, Ernest Melton, Didier Squiban, Connan Mockasin, Nina Simone, Merja Sonia, Chet Baker, Laïs, Joan Baez, Nils Frahm, Kate Bush, Megadeth, Radiohead.
Chants des confins, jour 72 : Frankie Goes To Hollywood - Welcome to the pleasure dome
Life goes on days after day, the animals are winding me up, the jungle call
Shooting stars never stop, even when they reach the top
Chants des confins, jour 70 : Télépopmusik - Breathe
Another day, Just believe
Another day, Just breathe
Chants des confins, jour 69 : Henry Purcell, Dido and Æneas, III: When I am laid in earth, or Dido's Lament (Sarah Connolly and Marianne Beate Kielland)
When I am laid, am laid in earth, May my wrongs create
Chants des confins, jour 58 : George Frideric Handel - Duet "Son nata a lagrimar", from Giulio Cesare in Egitto (Nathalie Stutzmann & Philippe Jaroussky)
Chants des confins, jour 56 : Têtes Raides - Bestiaire
La station en verticale
Pour tout vous dire c'est un fiasco
Chants des confins, jour 55 : Yom (Guillaume Humery) and the wonder rabbis - Kaddish for Superman
With love
Chants des confins, jour 54 : Sting - Fragile (Pat Metheny, Nils Landgren, Esbjörn Svensson, Michael Brecker, Lars Danielsson, Wolfgang Haffner; Jazz Baltica 2003)
On and on the rain will fall
Like tears from a star
Chants des confins, jour 53 : Andreas da Florentia/ Andrea de' Servi - Cosa crudel m'ancide (Ensemble Syntagma, Rosa e Orticha, music of the Trecento)
Se mai la vaga cosa di me fosse pietosa. Morte da me decide.
Witness Flight Lieutenant William Terrance Decker, Royal Flying Corps, returning from a patrol somewhere over France. The year is 1917. The problem is that the Lieutenant is hopelessly lost. Lieutenant Decker will soon discover that a man can be lost not only in terms of maps and miles, but also in time, and time in this case can be measured in eternities. [Lyrics from the Twilight Zone, teleplay by Richard Matheson]
[Instrumental. Title from an Greek legend about someone who buried a treasure after being defeated. The Oracle of Delphi advised researchers to look under every stone]
[Instrumental. Comment from Ernest Melton: "But it’s really a message to myself that, at the end of the day, I don’t have to be a slave to trivial stuff in my life. That’s all it is."]
(version instrumentale) Neventi vad d'ar Vretoned! [Heureuse nouvelle aux Bretons !] Ha malloz-ru d'ar C'hallaoued! [Et malédiction rouge aux Français !]
Aamulla varhain kun aurinko nousi (Early in the morning when the sun rose) Yö oli kaunis ja äänesi hellä (The night was beautiful and your voice so tender)
