## 5 Theoretical Analysis  
In this section, we prove that the equivalent sensing matrix \(A\) follows the RIP property. We first propose some definitions and corollaries.  
**Definition 2.** (_Sub-Gaussian [41]) A random variable \(\wp\) is called sub-Gaussian if there exists a constant \(c>0\) such that_  
\[E(\exp(\wp\,t^{\prime}))\leq\exp({c^{2}{t^{\prime}}^{2}}/{2}) \tag{19}\]  
_holds for all \(t^{\prime}\in\mathbb{R}\). We use the notation \(\wp\,\sim\,Sub(c^{2})\) to denote that\({}_{\wp}\) satisfies Equation (19)._  
**Theorem 1.** ([41]) Suppose \(\wp_{\partial}=\left[\wp_{\partial 1},\wp_{\partial 2},\dots,\wp_{\partial M }\right]^{\prime}\), where each \(\wp_{\partial i}\) is i.i.d. \(\wp_{\partial i}\,\sim\,Sub(c^{2})\) and \(E(\wp_{\partial i}^{2})=\sigma^{2}\). Then_  
\[E(\|\wp_{\partial i}\|_{2}^{2})=M\sigma^{2} \tag{20}\]  
_Furthermore, for any \(\alpha^{\prime}\in(0,1)\), \(\beta^{\prime}\in[c^{2}/\sigma^{2},\beta_{\max}]\), there exists a constant \(c^{*}\) such that_  
\[P(\|\wp\|_{2}^{2}\leq\alpha^{\prime}M\sigma^{2})\leq\exp(-M(1-\alpha^{\prime} )^{2}/c^{*}) \tag{21}\]  
_and_  
\[P(\|\wp\|_{2}^{2}\leq\beta^{\prime}M\sigma^{2})\leq\exp(-M(\beta^{\prime}-1)^ {2}/c^{*}) \tag{22}\]  
**Lemma 1.**_Fix \(\delta\in(0,1)\), \(A=\Psi\Phi\) satisfies \((1-\delta_{K})\|x\|_{2}^{2}\leq\|\Phi x\|_{2}^{2}\leq(1+\delta_{K})\|x\|_{2}^{2}\) for all \(N\)-dimensional \(K\)-sparse signal \(x\)._  
**Proof.** As mentioned above, diffusion operator \(O\) generated by \(\Lambda\), while \(O\) has the same eigenvalues as \(\Lambda\) (less than 1). The modified diffusion wavelets are the concatenation of the scale functions and wavelet functions. Hence, we believe that the entries of the representative basis \(\Psi\) are randomly sequenced, which is denoted by \(\wp_{1},\wp_{2},\wp_{3},\dots\wp_{N}\). Moreover, since the non-zero entries for each row of measurement matrix \(\Phi\) are mutually independent, the same probability is chosen for the projection nodes. Therefore, each row of \(A\) independently selects elements at random from \(\Psi\). Alternatively, we suppose that \(A\) is generated by independent and identically distributed (i.i.d.)random variables \(\wp_{\partial 1},\wp_{\partial 2},\wp_{\partial 3},\dots\wp_{\partial M}\).  
The next step is normalization, \(A=\sqrt{\frac{N}{M}}[\Theta_{1},\Theta_{2},\dots,\Theta_{M}]^{T}\); in addition, we obtain Equations (23) and (24), which follow a similar idea as [18]:  
\[E(\Theta^{T}(i,j)^{2})=(\sqrt{\frac{N}{M}})^{2}\,E(\Theta(j,i)^{  
Here, we bring Equations (23) and (24) into Equation (25), and Equation (25) can be expressed as Equation (26):  
\[E(\|Y\|_{2}^{2})=\|x\|_{2}^{2}. \tag{26}\]  
Alternatively, we set \(a^{\prime}=1-\delta\) and \(\beta^{\prime}=1+\delta\). Hence, Equations (21) and (22) can be expressed as follows:  
\[P(\|Y\|_{2}^{2}\leq(1-\delta)E(\|Y\|_{2}^{2}))\leq\exp(-M\delta^{2}/c^{*}) \tag{27}\]  
\[P(\|\Phi x\|_{2}^{2}\leq(1-\delta)\|x\|_{2}^{2})\leq\exp(-M\delta^{2}/c^{*}) \tag{28}\]  
and  
\[P(\|\Phi x\|_{2}^{2}\geq(1+\delta)\|x\|_{2}^{2})\leq\exp(-M\delta^{2}/c^{*}). \tag{29}\]  
However, there are \((N,K)\) possible \(K\)-dimensional subspaces of \(A\) relying on [PERSON]'s approximation. So, the inequality is given as \((N,K)\leq(eN/K)^{K}\). Thus, the probability of signal \(x\) needs to obey the conditions of Equation (30):  
\[(\mathrm{e}N/K)^{K}\cdot 2\mathrm{e}^{(-M\delta^{2}/c^{*})}=2\mathrm{e}^{(-M \delta^{2}/c^{*}+K\log(N/K)+1)}, \tag{30}\]  
such that  
\[(1-\delta_{K})\|x\|_{2}^{2}\leq\|\Phi x\|_{2}^{2}\leq(1+\delta_{K})\|x\|_{2} ^{2}. \tag{31}\]  
Finally, \(M=O(K\log(N/K))\) is selected to follow the RIP property with the probability approximation to 1, which completes the proof.