## 2 UWB Signal Processing Steps
### Data Cleansing and Clutter Rejection
#### 2.1.3 RPCA via Inexact ALM  
The above-defined classical PCA aims at the exact recovery problem from corrupted low-rank data owing to small errors and noise, but it cannot effectively deal with incomplete or missing real-world data suffering greater corruption. Therefore, the robust principle component analysis (RPCA) has been proposed for corrupted low-rank data recovery [18; 19; 20; 21; 22; 23].  
Recently, [PERSON] et al. [24] have shown that under rather broad conditions the answer is affirmative: as long as the error matrix \(E\) is sufficiently sparse (relative to the rank of \(A\)), one can exactly recover the low-rank matrix \(A\) from \(X=A+E\) by solving the following convex optimization problem:  
\[\underset{A,\,E}{min}\,||A||_{*}\,\,+\,\,\lambda\,||E||_{1}\,\,\,,\,\,\text{ subject to }X=A+E, \tag{8}\]  
where \(\|\cdot\|_{*}\) is the nuclear norm of a matrix, and thus the sum of its singular values, \(\|\cdot\|_{1}\) is the sum of the absolute values of matrix entries, and \(\lambda\) is a positive weighting parameter on the sparse error.  
Different algorithms have been recently developed to solve the RPCA problem. They all model the background with a low rank subspace, while the moving foreground objects constitute the correlated sparse outliers.  
As stated in the first section of this paper, the proposed model follows the techniques developed by Augmented Lagrange Multiplier. Given a radar-scan data matrix \(X\) whose dimensions are M \(\times\) N, and a weight \(\lambda\) on sparse error term, if we identify:  
\[\left\{\begin{array}{c}X=\left(\text{A},\text{E}\right),\\ f(X)=\|\ A\ \|_{*}+\lambda\|\ E\ \|_{1}\\ h(X)=\text{X}-\text{A}-\text{E}\end{array}\right., \tag{9}\]  
where \(A\) and \(E\) are the low-rank and sparse matrices respectively. The general augmented Lagrangian function will then be defined as follows:  
\[L\left(X,\,Y,\,\,\mu\right)=\,f\left(X\right)+\left\langle\,Y,\,h\left(X \right)\right\rangle+\,\frac{\mu}{2}\,||h\left(X\right)||_{F}^{2}\,\,, \tag{10}\]  
where \(\mu\) is a positive scalar. It has been proven in [25] that if \(\mu\) is increasingly changing in each iteration and \(f\) and h are continuously differentiable functions, then under rather general conditions the ALM method converges Q-linearly toward the optimal solution.  
In [11] [PERSON] et al. explained how the use of RPCA algorithm for online processing is challenged by the nuclear norm defined in Equation 8. In order to find the minimum nuclear norm, a Singular Value Decomposition (SVD) has to be performed across all samples, though SVD is the operation that consumes the most running time. In [26], the inexact ALM (IALM) approach was defined and converged to global minimum by computing only the first few, largest singular values.  
The inexact ALM method will then be defined by Algorithm 1:The algorithm takes \(X\) (an \(MxN\) radar-Scans matrix) and \(\lambda\) (a positive weighting parameter on the sparse error) as inputs and returns \(A_{k}\), \(E_{k}\): the estimated background and foreground respectively.  
In [14] the value of \(\lambda\) is said to be calculated as \(\frac{1}{\sqrt{max(M,N)}}\), but our extensive experiments proved that if \(M\)\(>\)\(>\)\(N\) then \(\lambda=\frac{1}{\sqrt{M/N}}\) (or vice-versa) will be used; otherwise, the first holds.  
From Theorem 1 and Theorem 2 elaborated in [26], we can get the following conclusion: for geometrically growing \(\mu_{k}\), IALM converges Q-linearly, and the faster \(\mu_{k}\) grows, the faster IALM converges. However, when \(\rho\) is too large such that the condition \(\underset{k\rightarrow+\infty}{lim}\mu_{k}\left(E_{k+1}-E_{k}\right)\ =0\) is violated, IALM may fail to converge to the optimal solution. Thus, if seeking both optimality and fast convergence, the values of \(\mu_{0}\) and \(\rho\) must be chosen cautiously.