First thoughts on Orthogonal Matching Pursuit

I am working on implementing the Orthogonal Matching Pursuit (OMP) algorithm for the scikit. It is an elegant algorithm (that almost writes itself in Numpy!) to compute a greedy approximation to the solution of a sparse coding problem:

\$\$ \text{argmin} \big|\big|\gamma\big|\big|_0 \text{ subject to }\big|\big|x-D\gamma\big|\big|_2\^2 \leq \epsilon\$\$

or (in a different parametrization)

\$\$ \text{argmin} \big|\big|x - D\gamma\big|\big|_2\^2\text{ subject to }\big|\big|\gamma\big|\big|_0 \leq m\$\$

The second formulation is interesting in that it gives one of the few algorithms for sparse coding that can control the actual number of non-zero entries in the solution. Some dictionary learning methods need this (I’m thinking of K-SVD).

Both problems are solved by the same algorithm, with a different stopping condition. The gist of it is to include at each iteration, the atom with the highest correlation to the current residual. However, as opposed to regular Matching Pursuit, here, after choosing the atom, the input signal is orthogonally projected to the space spanned by the chosen atoms. This involves the solution of a least squares problem at each step. However, because the problem is almost the same at each iteration, only with one more column added to the matrix, this can be easily solved by maintaining a QR or Cholesky decomposition of the dictionary matrix that is updated at each step.

Rubinstein et al. [1] came up with a clever method to optimize the calculations, based on the fact that usually in practice we never have to find a sparse coding for a single signal, but usually for a batch. They called this method Batch OMP, and it is based on a straightforward modification of the Cholesky update algorithm, taking advantage of precomputing the Gram matrix [latex] G=D’D[/latex].

Based on my experiments, their batch update is the fastest, even though it lags behind if invoked with too small a batch. As soon as I make sure the implementation is robust and ready for use, I will make some benchmarks.

Update: Here’s a little proof that it works!
[Stem plot for sparse signals recovered by OMP][]

Update 2: Here’s a little benchmark:
[Orthogonal Matching Pursuit benchmark][]
[1] http://www.cs.technion.ac.il/\~ronrubin/Publications/KSVD-OMP-v2.pdf

[Stem plot for sparse signals recovered by OMP]: http://localhost:8001/wp-content/uploads/2011/06/omp.png [Orthogonal Matching Pursuit benchmark]: http://localhost:8001/wp-content/uploads/2011/06/omp_bench.png

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