4.2.1. Subspace Separation Principle
Assume matrix X is decomposed into the sum of two other matrices X=B+Γ, where X∈RN×K(K<N) stands for the observed data, B∈RN×K represents the true signal and Γ∈RN×K denotes the noise matrix. SSP attempts to partition the range space of X into two subspaces, where one subspace is spanned by the source signal and the other subspace is spanned by noise. One possible way to do this is via singular value decomposition (SVD). Specifically, the SVD of X takes the form:(10) X=UΣVT=∑i=1KσkukvkT where the singular values are arranged in a descending order σ1≥σ2≥...≥σK≥0. In the situation where the noise level is low and the signal matrix is not ill-conditioned, the significant singular values (singular values with larger values) correspond to the signal subspace, and the remaining negligible singular values correspond to the noise subspace. Under the assumption of keeping (L<K) singular values as the signal singular values, the SVD of Equation (10) can be decomposed into two components, corresponding to the signal (XL) and noise component (XR), respectively. (11) X=ULΣLVLT︸XL+URΣRVRT︸XR
The first term in Equation (11), i.e., XL, is called the L-rank Eckart–Young–Mirsky (EYM) approximation of X and represents the higher signal-to-noise ratio (SNR) representation of X. Matrix ΣL is a diagonal matrix, and it contains the first L singular values corresponding to the signal component; and UL and VL correspond to the left and right singular vectors, respectively. Similarly, ΣR is a diagonal matrix containing the last K−L singular values corresponding to the noise part, and UR and VR correspond to the left and right noise singular vectors, respectively. Hence, the space of the observed measurements is approximately decomposed into two separate subspaces: signal and noise subspace, respectively. If we still further denote X as the product of the two matrices A∈RN×M and S∈RM×K, i.e., X=AS+Γ as shown in Equation (1), it is shown in [32] that UR represents a robust approximation of the left null space of A in the case L=M.