## 2 Channel Model
### The Channel Gain of NLoS Links  
The NLoS channel contains two parts: the specular reflections produced by RIS and the diffuse reflections from the walls, ceiling, floor, etc. According to [24], the diffuse reflections can be ignored due to low responses. Meanwhile, we can obtain that high-order reflections have little influence on the propagation in VLC on account of the research in [25]. Consequently, we consider the NLoS channel as the one-order specular reflections caused by RIS in our work. The work in [13; 17; 19] considers the RIS-aided VLC NLoS channel model as an additive form specifically according to the derivation in [22]. However, given the high spatial resolution of the specular reflection links, we model the channel model as follows in order to extract the controllable variables from complex mirror rotation angles and focus on the essential information of the assignment scheme. The NLoS channel gain motivated by the work in [16] from the \(n\)-th LED to the \(n\)-th PD via the \(k\)-th RIS element can be approximatively regarded as an imaging LED's Lambertian model characterized by  
\[h_{n,k}^{\rm NLoS}=\frac{\rho(m+1)A_{PD}}{2\pi(d_{S_{n}}^{K_{k}}+d_{D_{n}}^{K_{ k}})^{2}}\cos^{m}(\theta_{S_{n}}^{R_{k}})\cos(\phi_{D_{n}}^{R_{k}})T_{of}G( \phi_{D_{n}}^{R_{k}}), \tag{6}\]  
where \(\rho\) represents the reflection coefficient, and for simplicity, we assume the coefficients for different RIS elements are the same in our work. Inspired by the work in [15], we assume that the signal from one LED can be only reflected by one RIS element, and the reflected signal will arrive at the corresponding PD matched with the LED. This process can be realized by the adaptively adjustment of RIS after inputting an assignment matrix on the VLC controller. We assume that \(\mathbf{F}\in\mathbb{R}_{+}^{N\times K}\) denotes the assignment matrix, the \(n\)-th row of \(\mathbf{F}\) is \(\mathbf{f}_{n}=(f_{n,1},f_{n,2},\cdots,f_{n,K})\), where \(f_{n,k}=1\) denotes the \(n\)-th LED chooses the \(k\)-th RIS element as the reflector to transmit the signals to the \(n\)-th PD, the vector \(\mathbf{h}_{n,\rm all}^{\rm NLoS}=(h_{n,1}^{\rm NLoS},h_{n,2}^{\rm NLoS}, \cdots,h_{n,K}^{\rm NLoS})^{\top}\) represents the channel gain from the \(n\)-th LED to each RIS element to the \(n\)-th PD. Consequently, the NLoS channel gain from the \(n\)-th LED to the \(n\)-th PD is given by  
\[h_{n}^{\rm NLoS}=\mathbf{f}_{n}\mathbf{h}_{n,\rm all}^{\rm NLoS}. \tag{7}\]  
We assume that \(\mathbf{H}_{\rm all}^{\rm NLoS}=(\mathbf{h}_{1,\rm all}^{\rm NLoS},\mathbf{ h}_{2,\rm all}^{\rm NLoS},\cdots,\mathbf{h}_{N,\rm all}^{\rm NLoS})\in\mathbb{R}_{+}^{K \times N},\mathbf{F}=(\mathbf{f}_{1}^{\top},\mathbf{f}_{2}^{\top},\cdots, \mathbf{f}_{N}^{\top})^{\top}\in\mathbb{R}_{+}^{N\times K}\), according to the above definition, we can derive that  
\[\mathbf{f}_{i}\mathbf{h}_{j,\rm all}^{\rm NLoS}=\left\{\begin{array}{ll}h _{i}^{\rm NLoS},&i=j,\\ 0,&i\
eq j,\end{array}\right. \tag{8}\]  
where \(i,j\in(1,2,\cdots,N)\), and consequently,  
\[\mathbf{h}^{\rm NLoS}=\mathbf{FH}_{\rm all}^{\rm NLoS}=\left(\begin{array}[] {cccc}h_{1}^{\rm NLoS}&&&\\ &h_{2}^{\rm NLoS}&&\\ &&\ddots&\\ &&&h_{N}^{\rm NLoS}\end{array}\right)_{N\times N}. \tag{9}\]