Sparse Representation Algorithms

Python solvers for P0- and P1-problems.

1. P0-epsilon problem: min_z ||z||_0 s.t. ||Az - x|| < eps
2. P1-epsilon problem: min_z ||z||_1 s.t. ||Az - x|| < eps
3. Q1-epsilon problem: min_z ||Az - x||^2 + λ||z||_1

where z is a sparse representation of the input x, and A is a dictionary matrix (either fixed or learned).

Documentation is available at https://sparse-representation.readthedocs.io

The theoretical foundations of the implemented functional are in edX course Sparse Representations in Signal and Image Processing.

Algorithms

Coherence of a matrix

• Mutual Coherence
• Babbel Function
• Spark of a matrix

Solving P0- and P1-epsilon problems with a fixed dictionary

• Greedy algorithms, approximating the P0-problem:
  • Orthogonal Matching Pursuit (OMP)
  • Least Squares OMP (LS-OMP)
  • Matching Pursuit (MP)
  • Weak Matching Pursuit (WMP)
  • Thresholding Algorithm (THR)
• Relaxation algorithms, approximating the P0-problem:
  • Basis Pursuit (L1-relaxation)
  • Basis Pursuit + ADMM
  • Iterative Shrinkage Algorithm (ISTA, Fast ISTA)

Dictionary learning

• BasisPursuit dictionary learning (similar to MOD)
• Learned Iterative Shrinkage-Thresholding Algorithm (LISTA)

sparse.nn module contains PyTorch implementation of the listed above algorithms.

Quick start example

Fixed dictionary

Reproduce with edX/finproj/project2_all.py

To illustrate an example with a fixed dictionary, we

1. simulate an image x of size n x n, constructed with bars;
2. add noise and corrupt the image -> x_noisy;
3. generate a fixed dictionary matrix A of size n^2 x m (m >> n^2) with random bars as atoms (columns);
4. reconstruct x' from x_noisy by seeking the sparsest vector z such that Az ≈ x.

Learned dictionary

Start a Visdom server and run the examples with

$ python -m visdom.server
$ python sparse/examples/basis_pursuit_pytorch.py

Then navigate to localhost:8097 to see the training progress.

The "output sparsity" is the sparsity of the embedding vector from which the original image is reconstructed.

Demo

More examples are on http://85.217.171.57:8097. Choose environments with MatchingPursuit.

Installation

$ git clone https://github.com/dizcza/sparse-representation.git
$ cd sparse-representation
$ pip install -e .[extra]

Extra requirements install PyTorch for sparse.nn module.