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Minkovskian Gradient for Sparse Optimization
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DOI：10.1109/JSTSP.2013.2241014.png)
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Abstract
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Information geometry is used to elucidate convex optimization problems under L-1 constraint. A convex function induces a Riemannian metric and two dually coupled affine connections in the manifold of parameters of interest. A generalized Pythagorean theorem and projection theorem hold in such a manifold. An extended LARS algorithm, applicable to both under-determined and over-determined cases, is studied and properties of its solution path are given. The algorithm is shown to be a Minkovskian gradient-descent method, which moves in the steepest direction of a target function under the Minkovskian L-1 norm. Two dually coupled affine coordinate systems are useful for analyzing the solution path.
Keywords:
Extended LARS
information geometry
L1-constraint
sparse convex optimization
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