The **Ψ** Subproblem

With fixed **Y** and *λ*, the optimization problem to obtain **Ψ** is given by

$$\Psi = \underset{\mathbf{Y}}{\text{arg min}} \ \|\mathbf{Y} - \mathcal{S}\_{\lambda}(\mathbf{Y}^T \mathbf{X})\|\_{F}^{2} + \frac{\eta\_{3}}{\eta\_{1}} \|\boldsymbol{\Phi}\mathbf{Y}^T - \mathbf{I}\|\_{F}^{2}.\tag{18}$$

#### **Algorithm 1:** Sparse coding algorithm.

#### **Input and Initialization:**

Training data **<sup>X</sup>** ∈ R*N*×*L*, iteration number *<sup>r</sup>*, initial value *<sup>λ</sup>* = 0.

#### **Output:**

Sparse coefficients **Y**, and threshold values *λ*
