| Summary: | Sparse dictionary learning (SDL) is a classic representation learning method and has been widely used in data analysis. Recently, the <inline-formula> <math display="inline"> <semantics> <msub> <mo>ℓ</mo> <mi>m</mi> </msub> </semantics> </math> </inline-formula>-norm (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>≥</mo> <mn>3</mn> <mo>,</mo> <mi>m</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </semantics> </math> </inline-formula>) maximization has been proposed to solve SDL, which reshapes the problem to an optimization problem with orthogonality constraints. In this paper, we first propose an <inline-formula> <math display="inline"> <semantics> <msub> <mo>ℓ</mo> <mi>m</mi> </msub> </semantics> </math> </inline-formula>-norm maximization model for solving dual principal component pursuit (DPCP) based on the similarities between DPCP and SDL. Then, we propose a smooth unconstrained exact penalty model and show its equivalence with the <inline-formula> <math display="inline"> <semantics> <msub> <mo>ℓ</mo> <mi>m</mi> </msub> </semantics> </math> </inline-formula>-norm maximization model. Based on our penalty model, we develop an efficient first-order algorithm for solving our penalty model (PenNMF) and show its global convergence. Extensive experiments illustrate the high efficiency of PenNMF when compared with the other state-of-the-art algorithms on solving the <inline-formula> <math display="inline"> <semantics> <msub> <mo>ℓ</mo> <mi>m</mi> </msub> </semantics> </math> </inline-formula>-norm maximization with orthogonality constraints.
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