## 2 RASMR for Big Data Analysis under GLM
### Solving the Score-Matching Equation with Splitting Points  
In this section, we investigate the number of solutions for the score-matching Equation (3) and explain why more stable and accurate solutions can be obtained for Equation (3) by further splitting the data blocks, which is our motivation to propose the RASMR algorithms in later sections. According to Equations (2) and (4), the \(k\)th representative \((\tilde{\mathbf{X}}_{k},\tilde{y}_{k})\) can be obtained as weighted averages of the \(k\)th data block \(\mathcal{D}_{k}=\{(\mathbf{X}_{i},y_{i}),i\in I_{k}\}\). The weights of \(y_{i}\)'s and \(\mathbf{X}_{i}\)'s are \(v(\eta_{i})\eta_{i}\) and \(v(\eta_{i})[y_{i}-G(\eta_{i})]\), respectively. To stabilize the constructed representatives, we propose to keep the weights with the same sign (that is, all positive or all negative) in each individual data block. That is, besides splitting the data block according to \(\text{sgn}(\eta_{i})\), as suggested in Remark 3.1 of [43], we suggest further splitting the data blocks by \(\text{sgn}(y_{i}-G(\eta_{i}))\). Now, we investigate the number of solutions to Equation (3) and how to further split the data block to ensure a unique solution. Following [43], we denote \(S(\eta)=v(\eta)[\tilde{y}_{k}-G(\eta)]\eta\). According to the proof for Theorem 3.1 in [43], Equation (3) can be rewritten as follows:  
\[S(\tilde{\eta}_{k})=\frac{1}{n_{k}}\sum_{i\in I_{k}}S(\eta_{i}). \tag{5}\]  
We further denote \(\eta_{k}^{\wedge}=\min_{i\in I_{k}}\{\eta_{i}\}\) and \(\eta_{k}^{\vee}=\max_{i\in I_{k}}\{\eta_{i}\}\). If both \(v(\eta)\) and \(G(\eta)\) are continuous, then there exists a solution \(\eta_{*}\in[\eta_{k}^{\wedge},\eta_{k}^{\vee}]\) that solves Equation (5) (see Theorem 3.1 in [43]). Examples of \(v(\eta)\) and \(G(\eta)\) for commonly used GLMs can be found in Table 1 of [43], which are all continuous. If \(S(\eta)\) is strictly monotone on \([\eta_{k}^{\wedge},\eta_{k}^{\vee}]\), then there exists a unique \(\eta_{*}\in[\eta_{k}^{\wedge},\eta_{k}^{\vee}]\) that solves (3) (see Theorem A1 in Appendix A). If \(v(\eta)\) is a constant, such as for the normal model with an identity link, the Bernoulli model with a logit link, the Poisson model with a log link, the gamma model with a reciprocal link, and the inverse Gaussian model with an inverse-square link, without any loss of generality, we rewrite \(S(\eta)=[\tilde{y}_{k}-G(\eta)]\eta\). Then, its first derivative \(S^{\prime}(\eta)=\tilde{y}_{k}-[G(\eta)+G^{\prime}(\eta)\eta]\), with its key component \(T(\eta)\stackrel{{\triangle}}{{=}}G(\eta)+G^{\prime}(\eta)\eta\). In this case, we have the following: \(\tilde{y}_{k}=\frac{\sum_{i\in I_{k}}\eta_{i}y_{i}}{\sum_{i\in I_{k}}\eta_{i}}= \frac{n_{k}^{-1}\sum_{i\in I_{k}}\eta_{i}y_{i}}{\tilde{y}_{k}}\), where \(\tilde{y}_{k}=n_{k}^{-1}\sum_{i\in I_{k}}\eta_{i}\) corresponds to the mean representative (MR). We denote the following: \(\tilde{G}_{k}=\frac{\sum_{i\in I_{k}}\eta_{i}G(\eta_{i})}{\sum_{i\in I_{k}}\eta_{i}}= \frac{n_{k}^{-1}\sum_{i\in I_{k}}\eta_{i}G(\eta_{i})}{\eta_{k}}\). Then, we have the following:  
\[\tilde{S}\stackrel{{\triangle}}{{=}}\frac{1}{n_{k}}\sum_{i\in I_{ k}}S(\eta_{i})=\frac{1}{n_{k}}\sum_{i\in I_{k}}[\tilde{y}_{k}-G(\eta_{i})]\eta_{i}= \tilde{\eta}_{k}(\tilde{y}_{k}-\tilde{G}_{k}). \tag{6}\]  
For the normal model with an identity link, that is, the usual linear model, there can be up to two solutions to (3) (see Theorem A2 in Appendix A). For Bernoulli models (see Table 1 of [43]), \(y_{i}\in\{0,1\}\) and \(G(\eta_{i})\in(0,1)\) for all \(i\). Then, \(y_{i}<G(\eta_{i})\) always implies \(y_{i}=0\), and \(y_{i}>G(\eta_{i})\) always implies \(y_{i}=1\). Suppose (i) either \(\eta_{i}>0\) for all \(i\in I_{k}\) or \(\eta_{i}<0\) for all \(i\in I_{k}\); and (ii) either \(y_{i}>G(\eta_{i})\) for all \(i\in I_{k}\) or \(y_{i}<G(\eta_{i})\) for all \(i\in I_{k}\). Then, \(\tilde{y}_{k}\) is either \(0\) or \(1\).