Meeting 6 (02.04)

Goal: discussion of the first prototype implementation of our auto-encoder and the learned atoms on image patches.

Related notebooks: formulation, image.

Achieved

  • Implementation of a prototype in Python.
    • Optimizing for Z last leads to a lower global objective function (two Z terms, only one D).
    • The solution does not depend too much on the initialization, even if the global problem is not convex.
  • See the formulation notebook for a construction of the algorithm form the base (linear least square fitting) to the version with encoder (no graph yet). Tests on synthetic examples are included.
  • See the image notebook for an application to image data along with observations. We try here to recover Gabor filters.
  • Results from Xavier: J’ai testé les 2 contraintes sur le dictionary: (1) ||d_j|| = 1 et (2) ||d_j|| <= 1. L’algorithme est 2x plus rapide avec (2) ! Je pense que cela vient simplement du fait que la contrainte (2) est convexe, contrairement a (1).

Discussion

  • \(E\) seems to approach \(D^T\) when it is less constrained, i.e. \(m>>n\). This was due to an implementation error which constrained the L1 norm of the whole matrix instead of independently constraining each column.
  • Used atoms norm are oddly not 1 (which should minimize the objective function). Caused by the aforementioned bug.
  • Centric atoms seem to appear when the dictionary is more constrained.
  • Measure how \(E\) is similar to \(D^T\).

Notes

  • FISTA is not guaranteed to be monotone.

Michaël Defferrard

I am currently pursuing master studies in Information Technologies at EPFL. My master project, conducted at the LTS2 Signal Processing laboratory led by Prof. Pierre Vandergheynst, is about audio classification with structured deep learning. I previously devised an image inpainting algorithm. It used a non-local patch graph representation of the image and a structure detector which leverages the graph representation and influences the fill-order of the exemplar-based algorithm. I've been a Research Assistant in the lab, where I did investigate Super Resolution methods for Mass Spectrometry. I develop PyUNLocBoX, a convex optimization toolbox in Python.

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