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Journal Articles IEEE Signal Processing Letters Year : 2012

On LARS/homotopy equivalence conditions for over-determined LASSO

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Abstract

We revisit the positive cone condition given by Efron et al. [1] for the over-determined least absolute shrinkage and selection operator (LASSO). It is a sufficient condition ensuring that the number of nonzero entries in the solution vector keeps increasing when the penalty parameter decreases, based on which the least angle regression (LARS) [1] and homotopy [2] algorithms yield the same iterates. We show that the positive cone condition is equivalent to the diagonal dominance of the Gram matrix inverse, leading to a simpler way to check the positive cone condition in practice. Moreover, we elaborate on a connection between the positive cone condition and the mutual coherence condition given by Donoho and Tsaig [3], ensuring the exact recovery of any k-sparse representation using both LARS and homotopy.
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Dates and versions

hal-00738715 , version 1 (05-10-2012)

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Junbo Duan, Charles Soussen, David Brie, Jérôme Idier, Yu-Ping Wang. On LARS/homotopy equivalence conditions for over-determined LASSO. IEEE Signal Processing Letters, 2012, 19 (12), pp.894-897. ⟨10.1109/LSP.2012.2221712⟩. ⟨hal-00738715⟩
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