Damped rank-reduction method

The estimated signal using the TSVD method, however, still contains non-negligible residual noise components, as explained in Huang et al. (2016). The problem of residual noise can be solved to some extent by further applying a thresholding operator to the singular value matrix according to the nuclear-norm minimization model (Zhou and Zhang, 2017):

$$\hat{\mathbf{S}} =\mathbf{U}_N\mathcal{T}(\boldsymbol{\Sigma}_N,\tau)\mathbf{V}_N^H,$$ (7)

where $\mathcal{T}$ denotes the thresholding operator for the singular-values, and $\tau$ denotes the threshold of the operator. Defining an optimal threshold $\tau$, however, is often challenging because a constant threshold is inadequate to handle an inhomogeneous distribution of noise energy. Huang et al. (2016) developed a more elegant way to shrinkage the singular values by deriving a damping operator:

$$\mathbf{P} = \mathbf{I} - \boldsymbol{\Gamma},$$ (8)

$$\boldsymbol{\Gamma} = \hat{\sigma}^K\left(\boldsymbol{\Sigma_N}\right)^{-K},$$ (9)

where $\mathbf{P}$ is called the damping matrix, $\mathbf{I}$ is an identity matrix, and $\boldsymbol{\Gamma}$ is referred to as the damping threshold matrix. $\hat{\sigma}$ is the $(N+1)$th singular value in the un-truncated singular matrix $\boldsymbol{\Sigma}$. $K$ in equation 9 is called the damping factor, which is used to control the strength of the damping operator. In a special case, when $K\rightarrow \infty$, $\mathbf{P}\rightarrow\mathbf{I}$, indicating that the damped rank-reduction method reverts to the traditional rank-reduction method. The mathematical details to derive the damping operator can be found in Huang et al. (2016) and Huang et al. (2017). The damping operator is used to shrinkage the singular values in the TSVD formula to reduce the residual noise:

$$\hat{\mathbf{S}} = \mathbf{U}_N\mathbf{P}\boldsymbol{\Sigma}_N\mathbf{V}_N^H.$$ (10)

Compared with the thresholding method in equation 7, where a rigid threshold $\tau$ is used to shrinkage the singular values, the damping operator applies variable thresholds to different singular values. According to equation 8, the threshold decreases as the residual noise becomes less dominant, i.e., the singular value becomes larger. The algorithm workflow for the damped rank-reduction method (DRR) is outlined in Algorithm 2.