Sparse dictionary learning (SDL) is a classic representation learning method and has been widely used in data analysis. Recently, the ℓ m -norm ( m ≥ 3 , m ∈ N ) 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 ℓ m -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 ℓ m -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 ℓ m -norm maximization with orthogonality constraints.
In recent years, there has been an increasing interest in applying inverse data envelopment analysis (DEA) to a wide range of disciplines, and most applications have adopted radial-based inverse DEA models. However, results given by existing radial based inverse DEA models can be unreliable as they neglect slacks while evaluating decision-making units’ (DMUs) overall efficiency level, whereas classic radial DEA models measure the efficiency level through not only radial efficiency index but also slacks. This paper points out these disadvantages with a counterexample, where current inverse DEA models give results that outputs shall increase when inputs decrease. We show that these unreasonable results are the consequence of existing inverse DEA models’ failure in preserving DMU’s efficiency level. To rectify this problem, we propose a revised model for the situation where the investigated DMU has no slacks. Compared to existing radial inverse DEA models, our revised model can preserve radial efficiency index as well as eliminating all slacks, thus fulfilling the requirement of efficiency level invariant. Numerical examples are provided to illustrate the validity and limitations of the revised model.
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