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In the majority of recent work on sparse estimation algorithms, performance has been evaluated using ideal or quasi-ideal dictionaries (e.g., random Gaussian or Fourier) characterized by unit L2 norm, incoherent columns or features. But these types of dictionaries represent only a subset of the dictionaries that are actually used in practice (largely restricted to idealized compressive sensing applications). In contrast, herein sparse estimation is considered in the context of structured dictionaries possibly exhibiting high coherence between arbitrary groups of columns and/or rows. Sparse penalized regression models are analyzed with the purpose of finding, to the extent possible, regimes of dictionary invariant performance. In particular, a class of non-convex, Bayesian-inspired estimators with dictionary-dependent sparsity penalties is shown to have a number of desirable invariance properties leading to provable advantages over more conventional penalties such as the L1 norm, especially in areas where existing theoretical recovery guarantees no longer hold. This can translate into improved performance in applications such model selection with correlated features, source localization, and compressive sensing with constrained measurement directions. Moreover, the underlying methodology naturally extends to related rank minimization problems.
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