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A* orthogonal matching pursuit - or is it

The delightful thing about the A* routing algorithm is that it is provably the optimal algorithm for the purpose, in the sense that it's the algorithm that visits the fewest possible path nodes given the information made available. See the original paper for proof. Despite its simplicity, it is apparently still used in a lot of industrial routing algorithms today, and it can be adapted to help solve other sorts of problem.

A colleague pointed out a paper about "A*OMP" - an algorithm that performs a kind of OMP (Orthogonal Matching Pursuit) with a tree search added to try out different paths towards fitting a good sparse representation. "Aha," I thought, "if they can use A* then they can get some nice recovery properties inherited from the A* search."

However, in reading the paper I find two issues with the A*OMP algorithm which make me reluctant to use the name "A*" for it:

  1. The heuristics used are not "consistent" (this means you can't guarantee they are always less-than-or-equal to the true distance remaining). This lack of consistency means the proof of A*'s optimality doesn't apply. (Remember, A*'s "optimality" is about the number of nodes inspected before finding the best path.) (EDIT: a colleague pointed out that it's actually worse than this - if the heuristic isn't consistent then it's not just sub-optimal search, it may fail to inspect the best path.)
  2. Since A*OMP restricts the number of paths it adds (to the top "B" atoms having largest cross-product with the residual) there are no guarantees that it will even inspect the true basis.

These issues are independent of each other. If you leave out the pragmatic restriction on the number of search paths (to get round the second issue), the first issue still applies. OMP itself is greedy rather than exact, so this doesn't make A*OMP worse than OMP, but to my mind it's "not as good as A*".

In practice, the authors' A*OMP algorithm might indeed get good results, and the experiments shown seem to do so. So my quibbles may be mere quibbles. But the name "A*" led me to expect guarantees that just aren't there (e.g. guarantees of being better than OMP). It's quite easy to construct a toy problem for which A*OMP will not get you nearer the true solution than OMP will.

It's not obvious how to come up with a consistent heuristic. For a given problem, if we knew there was an exact solution (i.e. zero residual was possible within the sparsity constraints) then we could use the residual energy, but since we can't know that in general then the residual energy may overestimate the distance to be "travelled" to the goal.

One minor thing: their "equivalent path pruning" in section 4.2.3 is a bit overkill - I know a simpler way to avoid visiting duplicate paths. I'll leave that as an exercise for the reader :)

Friday 13th July 2012 | science | Permalink

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