Non-negative sparse priors

Sparseness priors, which impose that most of the weights are small or zero, are very effective in constraining regression problems. The prototypical sparseness prior is the Laplacian prior (aka L1-prior), which imposes a penalty on the absolute value of individual weights. Regression problems (and GLMs) with Laplacian priors can be easily solved by Maximum a Posteriori methods; the negative log-posterior is convex in the model parameters.

It’s also possible to assume that model weights are non-negative, in addition to being sparse. Such non-negative sparse priors are less frequently used that straight sparseness priors, but they can be useful from time to time. You might have a model where model parameters cannot logically be negative, on physical grounds. Or you might want to express a database of images as a sum of features.

Perhaps surprisingly, it’s pretty easy to solve for such problems. Patrik Hoyer shows in his non-negative sparse coding paper how to solve for model weights using a simple iterative method here. He has software available which implements non-negative sparse coding, non-negative matrix factorization, and associated algorithms.

YALL1 (Your ALgorithms for L1) is a Matlab package which can solve many problems related to L1. This includes what it calls the L1/L2+ problem, \arg \min ||x||_1 + (1/2\rho)||Ax - b||_2^2 such that x \ge 0.

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