Trainlets: cropped wavelet decomposition for high-dimensional learning

It's being a lonng time: element 120 from the aperiodic table of wavelets is the trainlet, from Jeremias Sulam, Student Member, Boaz Ophir, Michael Zibulevsky, and Michael Elad, Trainlets: Dictionary Learning in High Dimensions:

Abstract: Sparse representations has shown to be a very powerful model for real world signals, and has enabled the development of applications with notable performance. Combined with the ability to learn a dictionary from signal examples, sparsity-inspired algorithms are often achieving state-of-the-art results in a wide variety of tasks. Yet, these methods have traditionally been restricted to small dimensions mainly due to the computational constraints that the dictionary learning problem entails. In the context of image processing, this implies handling small image patches. In this work we show how to efficiently handle bigger dimensions and go beyond the small patches in sparsity-based signal and image processing methods. We build our approach based on a new cropped wavelet decomposition, which enables a multi-scale analysis with virtually no border effects. We then employ this as the base dictionary within a double sparsity model to enable the training of adaptive dictionaries. To cope with the increase of training data, while at the same time improving the training performance, we present an Online Sparse Dictionary Learning (OSDL) algorithm to train this model effectively, enabling it to handle millions of examples. This work shows that dictionary learning can be up-scaled to tackle a new level of signal dimensions, obtaining large adaptable atoms that we call trainlets.

They offer a base dictionary used within a double sparsity model to enable the training of adaptive dictionaries. The associated package is here, from Michael Elad software page.  The  cropped wavelet decomposition enables a multi-scale analysis with virtually no border effects. An entry  to trainlets has added to WITS, the aperiodic table of wavelets.

But things always ends up with a song! Two of my favorite train songs, by  Porcupine tree (Trains) and the Nits (The train).

The turn of a friendly card - Train numbering trick

Coming back from Orléans (France) on March 2010 for the Second conference  "Mathematics and Image processing" (Deuxième colloque "Méthodes mathématiques pour l’image") organized at MAPMO. Having presented stuff on Statistical estimators based on Stein's principle for  M-band wavelets and filter banks (abstract, slides, codes). Waiting in a train, staring again (and pixing) at the eight-seat compartment  numbering once discussed here. 

Since Igor Carron posted this Magic trick for summer vacations on Nuit Blanche (constest won by Laurent Jacques), I propose the following one again. Pictures attest the realness of the data, in contrast to Igor thought experiment (vicious tackle). Seat numbering is split in odd and even (unlike in six-seat cars), face-to-face, as:

.1 .3 .7 .5

.2 .8 .4 .6

The 3 (mod 8) sum is obvious. Easy enough for front-to-front booking by ancient computers. So why not the simple child Gauss-like arrangement?

.1 .3 .5 .7

.8 .6 .4 .2

Suspect some kind of compressive coding of seat booking? Contributions welcome. 

One good reason to listen on Nits again - The train. TGIF; my train of thoughts is leaving (le train de mes pensées s'égare).

Or Alan Parson's project - Turn of the friendly card:

Un train peut en cacher des nombres

Il y a eu des lettres, mais peu de chiffres encore. Une petite énigme sur l'arithmétique ferrovière, sur des modèles de wagons qui ont peut-être disparu aujourd'hui.

J'ai croisé dans le passé plusieurs trains aux compartiments munis de deux rangées de quatre sièges. D'un côté marche, les numéros impairs, de l'autre arrière, les numéros pairs, chaque compartiment portant un chiffre de dizaine différent : les sièges 1. dans le premier, les 2. dans le second. La subtilité réside dans l'agencement des places, par exemple, face-à-face :

31 33 37 35
32 38 34 36

On remarque un apparent désordre. Étourderie, peu probable, la forme est toujours la même :

.1 .3 .7 .5
.2 .8 .4 .6

J'imagine une astuce léguée par un polytechnicien farceur en poste à la SNCF, comme un bit de parité permettant de gérer la réservation de places face-à-face sur des logiciels primitifs. Et de fait, la somme donne toujours 3 en faisant une preuve par 8 : 1+2 = 3, 7+4 = 3+8, etc., ce qui peut être géré assez simplement par addition sur un octet (8 bits). Mais il y avait plus simple, comme aurait dit Bertrand Renard, avec un ordre inverse qui selon la légende fut le premier coup d'éclat de Gauss en très jeune âge :

.1 .3 .5 .7
.8 .6 .4 .2

Malgré les grèves, le train nous cache encore de belles choses. J'ai écrit en 2002 à la SNCF, qui me remercie de mon intérêt pour leur service. Sans réponse, je suspecte un code ésotérique. Contributions bienvenues...