## 3 The Proposed SSM-NTF
### The Stability Analysis of the Proposed Model  
In sparse representation problem, a given noiseless signal \(\mathbf{x}\), can be formulated formulated as  
\[(P_{0}):\quad\min_{\mathbf{x}}\|\mathbf{y}\|_{0}\quad s.t.\ \mathbf{x}=\mathbf{ \Phi y} \tag{14}\]  
where \(\mathbf{\Phi}\) is the sparse representation dictionary and \(\mathbf{y}\) is the sparse coefficients. While \(\mathbf{x}=\mathbf{\Phi y}\) is an underdetermined linear system, the problem \((P_{0})\) has the unique solution \(\mathbf{y}_{0}\) as soon as it satisfies the uniqueness property which is formulated as  
\[\|\mathbf{\hat{y}}\|_{0}<\frac{1}{2}(1+1/\mu) \tag{15}\]  
where \(\mu\) is the mutual-coherence of \(\mathbf{\Phi}\)[9]. However, the signals are usually acquired with noise, then the problem \((P_{0})\) should be relaxed to the problem \((P_{\epsilon})\) which is expressed as  
\[(P_{\epsilon}):\quad\min_{\mathbf{x}}\|\mathbf{y}\|_{0}\quad s.t.\ \|\mathbf{x}- \mathbf{\Phi y}\|_{F}^{2}\leq\epsilon \tag{16}\]  
where \(\epsilon\) is an error-tolerant which exists due to the noise. The problem \((P_{\epsilon})\) will no longer maintain the uniqueness of solution as \(\mathbf{x}=\mathbf{\Phi y}+\epsilon\) is an inequality system. Thus, the notion of Uniqueness Property (15) is replaced by the notion of stability which claims that all the alternative solutions reside very close to the ideal solution. Under the stable guarantee, we can yet ensure that the recovery results of our methods produce meaningful solutions. Assume that \(\mathbf{y}_{0}\) is the ideal solution to the problem \((P_{\epsilon})\) and \(\mathbf{\hat{y}}\) is the candidate one, the traditional sparse model has a stability claim of the form [9]  
\[\|\mathbf{\hat{y}}-\mathbf{y}_{0}\|_{2}^{2}\leq\frac{4\epsilon^{2}}{1-(2s_{0} -1)\mu}, \tag{17}\]  
where \(\mu\) is the mutual coherence which is formulated as \(\mu=\max_{i\
eq j}|<\boldsymbol{\phi}_{i},\boldsymbol{\phi}_{j}>|,i,j=1,2, \cdots,M\). Apparently, the error bound of Equation (17) can only be determined with given sparsity \(s_{0}\) and the mutual coherence \(\mu\). However, the mutual coherence of an unknown dictionary is very difficult to calculate which lead to a result that we can not ensure the stability in the dictionary learning case. In contrast, we derive a similar stability claim of our proposed SSM-NTF model. Defining \(\mathbf{d}=\mathbf{\hat{y}}-\mathbf{y}_{0}\) with \(\mathbf{y}_{0}\) as the ideal solution to the model, we have that \(\|\mathbf{\Phi\hat{y}}-\mathbf{\Phi y}_{0}\|_{2}=\|\mathbf{\Phi d}\|_{2}\leq 2\epsilon\). From the previous subsection, we have know that the frame \(\mathbf{\Phi}\) satisfies the RIP with the corresponding parameter \(\delta(\mathbf{\Phi})\). Thus, using this property and exploiting the lower-bound part in Equation (1), we get  
\[(1-\hat{\delta}(\mathbf{\Phi}))\|\mathbf{d}\|_{2}^{2}\leq\|\mathbf{\Phi d}\|_ {2}^{2}\leq 4\epsilon^{2} \tag{18}\]  
where \(\hat{\delta}(\mathbf{\Phi})=\frac{\Gamma(\mathbf{\Phi})-1}{\Gamma(\mathbf{ \Phi})+1}=\frac{\frac{B}{2}-1}{\frac{B}{A}+1}\). Thus, we get a stability claim of the form  
\[\|\mathbf{d}\|_{2}^{2}=\|\mathbf{\hat{y}}-\mathbf{y}_{0}\|_{2}^{2}\leq\frac{4 \epsilon^{2}}{1-\frac{\frac{B}{A}-1}{\frac{B}{A}+1}} \tag{19}\]  
Obviously, the error bound of the SSM-NTF is determined by \(\frac{B}{A}\), the ratio of the upper bound to the lower bound of the frame, rather than the specific values of \(A\) and \(B\). Thus, for the convenience of numerical experiments, we usually set \(A\) to a fixed value. A main advantage of standard orthogonal transformations is that they maintain the energy of the signals in the transform domain as its frame bounds \(A\) and \(B\) are equal to \(1\). However, the standard orthogonal basis is non-redundant that limits its performance in sparse representation.