ℓ₀-norm Minimization for Basis Selection

Part of Advances in Neural Information Processing Systems 17 (NIPS 2004)

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David Wipf, Bhaskar Rao


Finding the sparsest, or minimum ℓ0-norm, representation of a signal given an overcomplete dictionary of basis vectors is an important prob- lem in many application domains. Unfortunately, the required optimiza- tion problem is often intractable because there is a combinatorial increase in the number of local minima as the number of candidate basis vectors increases. This deficiency has prompted most researchers to instead min- imize surrogate measures, such as the ℓ1-norm, that lead to more tractable computational methods. The downside of this procedure is that we have now introduced a mismatch between our ultimate goal and our objective function. In this paper, we demonstrate a sparse Bayesian learning-based method of minimizing the ℓ0-norm while reducing the number of trou- blesome local minima. Moreover, we derive necessary conditions for local minima to occur via this approach and empirically demonstrate that there are typically many fewer for general problems of interest.