## 3 Proposed De-Noising Algorithm and Evaluation Method  
### Cross-Channel Dynamic Weighting RPCA  
Robust principal component analysis has been universally applied in many research fields [28], such as video surveillance, face recognition, latent semantic indexing, ranking and collaborative filtering, low-rank representation, target detection, signal de-noising, and classification [19; 29; 30; 31; 32; 33; 34]. The main idea of RPCA is that a data matrix \(D\) can be decomposed as a low-rank component \(L\) and a sparse component \(S\)[15], defined as  
\[min\|L\|_{*}+\lambda\|S\|_{1},\text{ subject to }D=L+S, \tag{1}\]  
where \(\|L\|_{*}\) denotes the nuclear norm (sum of singular values), and \(\|S\|_{1}\) is the \(L1\)-norm (sum of absolute values of matrix entries). In the field of arterial pulse signal de-noising and decomposition, \(D\in\mathbb{R}^{m\times n}\) is an input pulse matrix, where \(m\) and \(n\) (rows and columns of \(D\)) are the number of horizontally and vertically arranged channels in a tactile sensor, respectively. \(\lambda\) is a positive constant parameter set to \(1/\sqrt{\max(m,n)}\)[19]. Weighted robust principal component analysis, as an extension of RPCA, has been a mature and versatile method in the video frame interpolation [34], low-level vision [35], and singing voice separation fields [36]. WRPCA can be defined as follows:  
\[min\|L\|_{\omega,*}+\lambda\|S\|_{1},\text{ subject to }D=L+S, \tag{2}\]  
Figure 2: Flowchart of overall multi-channel pulse signal processing system. where \(\left\|L\right\|_{\omega,\ast}\) is the low-rank matrix with different weighted values. An efficient inexact version of the augmented Lagrange multiplier is used to solve the convex model [30]. The corresponding augmented Lagrange function is defined as  
\[Y(D,L,S,\mu)=\left\|L\right\|_{\omega,\ast}+\lambda\left\|S\right\|_{1}+Y,D-L-S+ \frac{\mu}{2}\left\|D-L-S\right\|_{F}^{2}, \tag{3}\]  
where \(Y\) is the Lagrange multiplier, \(\mu\) is a positive scaler, and \(\left\|D-L-S\right\|_{F}^{2}=D-L-S,D-L-S\). Specifically, \(\left\langle D-L-S,D-L-S\right\rangle\) is the Euclidean inner product of \(D-L-S\), and \(D\) is the observation data. The proposed cross-channel DWRPCA uses a channel-scaled factor technique to manipulate the weights of the WRPCA algorithm. We construct a convolutional neural network (CNN) to adaptively tune the channel-scaled factor according to the signal pattern of each channel. The pre-processed signals (not the extracted features) are used as the input, and the channel-scaled factor values are used as the target. Hence, the calculated CSF can dynamically adjust the weights among channels according to the signal patterns, which can optimize the de-noising effect in multi-channel signals like 3 DPI. Figure 3 illustrates the whole process of the cross-channel DWRPCA algorithm. Multi-channel pulse data matrices are normalized and transformed into \(D\prime\in\mathbb{R}^{M\times T\prime}\). The convolutional neural network constructed in this paper can predict a specific factor value for each channel. Therefore, the CSF can adaptively adjust the weight of a particular channel in a single sensor. In the context of this paper, we represent the signal input as a matrix \(D^{\prime}\in\mathbb{R}^{M\times T^{\prime}}\) (M is the number of beats in the dataset, and \(T\prime\) is the signal vector length). The details of the constructed neural network are discussed in Section 4. As the initial weighting factors for each channel in WRPCA vary, we named this method dynamic weighting RPCA. The DWRPCA method is repeated for each channel with different CSFs and extracts corresponding low-rank and sparse matrices. The low-rank matrix in a channel represents the feature components extracted from multiple cycles in that channel. Thus, the cross-channel DWRPCA algorithm is described below. In Algorithm 1, \(\rho\) is a positive numerical constant and is usually small enough to prevent \(\mu_{k}\) from increasing too fast.