![\"\u8fd1\u4f3c\u6d88\u606f\u4f20\u9012\uff08Approximate](\"https:\/\/etc.lmdewz.xyz\/wp-content\/uploads\/2023\/10\/AMP.png\" "\\"\u8fd1\u4f3c\u6d88\u606f\u4f20\u9012\uff08Approximate")
<\/center><\/p>\n<\/p>\n
\u7531\u4e8e$p(x_i)$\u548c$y_i$\u5747\u4e3a\u5df2\u77e5\u91cf\uff0c\u56e0\u6b64\u5728\u8fd9\u5f20\u56e0\u5b50\u56fe\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u63a8\u5bfc\u4e24\u7c7b\u6d88\u606f\uff0c\u5373\u53d8\u91cf\u8282\u70b9$x_i$\u5230\u56e0\u5b50\u8282\u70b9$p(y_a|\\bf x ,\\bf H)$\u7684\u6d88\u606f$\\mu_{i\\rightarrow a}$\u548c\u53cd\u5411\u4f20\u64ad\u7684\u6d88\u606f$\\mu_{a\\rightarrow i}$.\u4f46\u6211\u4eec\u6700\u7ec8\u9700\u8981\u7684\u5e76\u4e0d\u5b8c\u5168\u662f\u6d88\u606f\uff0c\u800c\u662f\u4fe1\u53f7\u5728\u4ee5\u4e0b\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e0b\u7684\u5747\u503c\u4e0e\u65b9\u5dee<\/p>\n
$$\n{p^t}({x_i}) = p(x_i)\\prod\\limits_{a = 1}^M {{\\mu _{a \\to i}^t}} ({x_i})\n$$<\/p>\n
\u6b64\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u88ab\u79f0\u4e3a\u8fd1\u4f3c\u7684\u8fb9\u7f18\u6982\u7387,\u4e3a\u67d0\u53d8\u91cf\u7684PDF\u5728\u56e0\u5b50\u56fe\u4e2d\u7684\u8fd1\u4f3c\u503c\u3002<\/p>\n
\u53cd\u5411\u6d88\u606f<\/h2>\n
\u9996\u5148\u8003\u8651$\\mu_{a\\rightarrow i}$\uff0c\u7531SPA\u7b97\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230<\/p>\n
$$\n\\mu_{a\\rightarrow i}^{t} = \\int p_a(y_a|\\mathbf{x},{\\bf H})\\prod_{j\\neq i}^{N}\\mu_{j\\rightarrow a}^{t}(x_{j})\\mathrm{d}\\mathbf{x}_{\\setminus i}\n$$<\/p>\n
\u7531\u4e8e\n$$\\begin{aligned}\np({y_a}|{\\bf{x}},{\\bf{H}})\n&= {\\mathcal N}\\left( {{y_a};\\sum\\limits_k {{h_{ak}}{x_k}} ,{N_0}} \\right)\\\\\n&= {\\mathcal N}\\left( {{y_a};\\sum\\limits_{k \\ne i} {{h_{ak}}{x_k} + {h_{ai}}{x_i}} ,{N_0}} \\right)\\\\\n&= {\\mathcal N}\\left( {{y_a};{h_{ai}}{x_i} + {S_{a \\to i}},{N_0}} \\right)\n\\end{aligned}$$\n\u5176\u4e2d${S_{a \\to i}} = \\sum\\limits_{k \\ne i} {{h_{ak}}{x_k}}$\uff0c\u4e3a\u4e00\u7cfb\u5217\u72ec\u7acb\u968f\u673a\u53d8\u91cf$x_k$\u7684\u6c42\u548c\u3002<\/p>\n
\u540c\u65f6\uff0c\u6d88\u606f\u53d8\u4e3a\n$$\n\\begin{aligned}\n\\mu _{a \\to i}^t &= \\int {\\mathcal N\\left( {{y_a};{h_{ai}}{x_i} + {S_{a \\to i}},{N_0}} \\right)p\\left( {{S_{a \\to i}}|{{\\bf{x}}_{\\backslash i}}} \\right)\\prod\\limits_{j \\ne i}^N {{p_{j \\to a}}} ({x_j})} {\\rm{d}}{S_{a \\to i}}{\\rm{d}}{{\\bf{x}}_{\\backslash i}}\\\\\n&= \\int {\\mathcal N\\left( {{y_a};{h_{ai}}{x_i} + {S_{a \\to i}},{N_0}} \\right)} \\left( {p\\left( {{S_{a \\to i}}|{{\\bf{x}}_{\\backslash i}}} \\right)\\prod\\limits_{j \\ne i}^N {{p_{j \\to a}}} ({x_j}){\\rm{d}}{{\\bf{x}}_{\\backslash i}}} \\right){\\rm{d}}{S_{a \\to i}}\n\\end{aligned}\n$$\n\u5728\u6b64\u5904\uff0c\u8ba4\u4e3a$x_k$\u7684PDF\u4e3a$p_{k\\to a}(x_k)=\\mu_{k\\rightarrow a}^{t}(x_{k})$\uff0c\u5176\u5e76\u4e0d\u4e00\u5b9a\u662f\u9ad8\u65af\u53d8\u91cf\uff0c\u4f46\u5176\u5b58\u5728\u5747\u503c\u53ca\u6709\u9650\u65b9\u5dee\uff0c\u5206\u522b\u4e3a${{{\\hat x}_{k \\to a}}}=\\mathbb{E}\\left(x_k\\right)$\u548c${{{\\hat v}_{k \\to a}}}=Var(x_k)$.\u8fd9\u4e24\u4e2a\u7edf\u8ba1\u91cf\u4f1a\u5728\u4e0b\u4e00\u8282\u8fdb\u884c\u63a8\u5bfc\uff0c\u6b64\u5904\u4ec5\u8ba4\u4e3a\u5176\u5b58\u5728\u3002<\/p>\n
\u5728\u5927\u7cfb\u7edf\uff08$M,N\\to \\infty, M\/N=const.$\uff09\u6761\u4ef6\u4e0b\uff0c\u5927\u6570\u5b9a\u5f8b\u6210\u7acb\uff0c${S_{a \\to i}}$\u88ab\u8fd1\u4f3c\u4e3a\u9ad8\u65af\u53d8\u91cf\u3002\u5728\u7ed9\u5b9a\u7684\u6761\u4ef6$x_k$\u4e0b\uff0c\u5176\u5747\u503c\u548c\u65b9\u5dee\u5206\u522b\u4e3a\n$$\nZ_{a \\to i}^t = \\sum\\limits_{k \\ne i}^N {{h_{ak}}{{\\hat x}_{k \\to a}^t}}\n$$\n$$\nV_{a \\to i}^t = \\sum\\limits_{k \\ne i}^N {|{h_{ak}}{|^2}{{\\hat v}_{k \\to a}^t}}\n$$\n\u5b9e\u9645\u4e0a\u6b64\u53d8\u91cf\u53ef\u4ee5\u89c6\u4e3a\u5bf9$y_a$\u5728${\\bf x}_{\\setminus i}$\u4e0b\u7684\u4f30\u8ba1\u3002\u7ee7\u7eed\u5bf9\u6d88\u606f\u8fdb\u884c\u53d8\u6362\uff1a\n$$\n\\begin{aligned}\n\\mu_{a\\rightarrow i}^{t}& \\overset{a.s.}{=}\\int {\\mathcal N}\\left(y_{a};h_{ai}x_{i}+S_{a\\rightarrow i},N_{0}\\right)\\mathcal{N}\\left(S_{a\\rightarrow i};Z_{a\\rightarrow i}^{t},V_{a\\rightarrow i}^{t}\\right)\\mathrm{d}S_{a\\rightarrow i} \\\\\n&=\\int {\\mathcal N}\\left(S_{a\\rightarrow i};y_{a}-h_{ai}x_{i},N_{0}\\right)\\mathcal{N}\\left(S_{a\\rightarrow i};Z_{a\\rightarrow i}^{t},V_{a\\rightarrow i}^{t}\\right)\\mathrm{d}S_{a\\rightarrow i} \\\\\n&\\propto {\\mathcal N}\\left(0;y_{a}-h_{ai}x_{i}-Z_{a\\rightarrow i}^{t},N_{0}+V_{a\\rightarrow i}^{t}\\right) \\\\\n&\\propto {\\mathcal N}\\Bigg(x_{i};\\frac{y_{a}-Z_{a\\rightarrow i}^{t}}{h_{ai}},\\frac{N_{0}+V_{a\\rightarrow i}^{t}}{\\left|h_{ai}\\right|^{2}}\\Bigg)\n\\end{aligned}\n$$<\/p>\n
\u6ce8\u610f\u5230\u6b64\u65f6\u6d88\u606f\u53d8\u6210\u4e86\u9ad8\u65af\u7684\uff0c\u4e14\u8ba1\u7b97\u65f6\u4e5f\u9700\u8981\u4f20\u5165\u6d88\u606f\u7684\u5747\u503c\u4e0e\u65b9\u5dee\u3002\u9ad8\u65af\u6d88\u606f\u4ec5\u5b58\u5728\u4e24\u4e2a\u5145\u5206\u7edf\u8ba1\u91cf\uff0c\u4f1a\u8ba9\u53c2\u6570\u4f20\u9012\u53d8\u5f97\u65b9\u4fbf\u3002\u5728\u6b64\uff0c\u5b9a\u4e49\u6b64\u6761\u6d88\u606f\u7684\u5747\u503c\u548c\u65b9\u5dee\u4e3a<\/p>\n
$$\n{{\\hat x}_{a \\to i}^t} = \\frac{{{y_a} - Z_{a \\to i}^t}}{{{h_{ai}}}}\n$$\n$$\n{{\\hat v}_{a \\to i}^t} = \\frac{{{N_0} + V_{a \\to i}^t}}{{{{\\left| {{h_{ai}}} \\right|}^2}}}\n$$<\/p>\n
\u6b63\u5411\u6d88\u606f<\/h2>\n
\u7531SPA\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230<\/p>\n
$$\n\\mu _{i \\to a}^{t + 1}({x_i}) \\propto p({x_i})\\prod\\limits_{b=1,b \\ne a}^M {\\mu _{b \\to i}^t} ({x_i})\n$$\n\u6ce8\u610f\u5230\uff0c\u8fd9\u6761\u6d88\u606f\u7531\u4e8e$p(x_i)$\u7684\u5b58\u5728\uff0c\u5e76\u4e0d\u5177\u5907\u9ad8\u65af\u7279\u6027\uff0c\u4f46\u5176\u4e2d\u7684\u4e00\u90e8\u5206\u53ef\u4ee5\u8fdb\u884c\u9ad8\u65af\u5408\u5e76\uff1a\n$$\n\\begin{aligned}\n\\prod\\limits_{b=1,b \\ne a}^M {\\mu _{b \\to i}^t} ({x_i}) &= \\prod\\limits_{b=1,b \\ne a}^M {{\\mathcal N}\\left( {{x_i};{{\\hat x}_{b \\to i}^t},{{\\hat v}_{b \\to i}^t}} \\right)} \\\\\n&\\propto {\\mathcal N}\\left( {{x_i};{{\\hat r}_{i \\to a}^t},{{\\hat \\Sigma }_{i \\to a}^t}} \\right)\n\\end{aligned}\n$$\n\u5176\u4e2d<\/p>\n
$$\n\\begin{aligned}\n{{\\hat r}_{i \\to a}^t} &\\overset{def}{=} {{\\hat \\Sigma }_{i \\to a}^t}\\sum\\limits_{b=1,b \\ne a}^M {\\frac{{{{\\hat x}_{b \\to i}^t}}}{{{{\\hat v}_{b \\to i}^t}}}} \\\\\n&= {{\\hat \\Sigma }_{i \\to a}^t}{\\sum\\limits_{b=1,b \\ne a}^M {\\frac{{{y_b} - Z_{b \\to i}^t}}{{{h_{b \\to i}}}}\\left( {\\frac{{{N_0} + V_{b \\to i}^t}}{{{{\\left| {{h_{bi}}} \\right|}^2}}}} \\right)} ^{ - 1}}\\\\\n&= {{\\hat \\Sigma }_{i \\to a}^t}\\sum\\limits_{b=1,b \\ne a}^M {\\frac{{h_{b \\to i}^*\\left( {{y_b} - Z_{b \\to i}^t} \\right)}}{{{N_0} + V_{b \\to i}^t}}} \\\\\n{{\\hat \\Sigma }_{i \\to a}^t} &\\overset{def}{=} {\\left( {\\sum\\limits_{b=1,b \\ne a}^M {\\frac{1}{{{{\\hat v}_{b \\to i}^t}}}} } \\right)^{ - 1}}\\\\\n&= {\\left( {\\sum\\limits_{b=1,b \\ne a}^M {\\frac{{{{\\left| {{h_{bi}}} \\right|}^2}}}{{{N_0} + V_{b \\to i}^t}}} } \\right)^{ - 1}}\n\\end{aligned}\n$$<\/p>\n
\u8fd9\u4e00\u7cfb\u5217\u9ad8\u65af\u6d88\u606f\u7684\u4e58\u79ef\u53ef\u4ee5\u89c6\u4e3a\u4ece$\\bf y_{\\setminus a}$\u6765\u7684\u6d88\u606f\u5f62\u6210\u7684\u5bf9$x_i$\u7684\u4f30\u8ba1\u3002\u4f46\u8fd9\u4e2a\u4f30\u8ba1\u5e76\u4e0d\u5168\u9762\uff0c\u56e0\u4e3a\u5176\u5e76\u672a\u5305\u542b\u975e\u9ad8\u65af\u7684\u5148\u9a8c\u4fe1\u606f$p(x_i)$\u7684\u7ea6\u675f\uff0c\u800c\u4e00\u822c\u800c\u8a00\uff0c$x_i$\u7684\u5148\u9a8c\u5206\u5e03\u53ef\u80fd\u76f8\u5f53\u590d\u6742\uff0c\u53ef\u80fd\u65e0\u6cd5\u901a\u8fc7\u8ba1\u7b97\u5f97\u5230\u4e25\u683c\u7684\u95ed\u5f0fPDF\u3002<\/p>\n
\u4f46\u662f\u4ece\u53cd\u5411\u6d88\u606f\u7684\u63a8\u5bfc\u53ef\u77e5\uff0c\u6b64\u5904\u5e76\u4e0d\u9700\u8981$\\mu _{i \\to a}^{t + 1}({x_i})$\u786e\u5207\u7684\u5206\u5e03\uff0c\u4ec5\u4ec5\u9700\u8981\u5176\u5747\u503c\u53ca\u65b9\u5dee\u3002\u56e0\u6b64\uff0c\u8ba1\u7b97\u4ee5\u4e0b\u5206\u5e03\uff1a\n$$\n\\mu_{i\\to a}^{t+1}(x_i)\\propto p(x_i)\\mathcal{N}(x_i;r_{i\\to a}^{t},\\Sigma_{i\\to a}^{t})\n$$\n\u7684\u5747\u503c\u53ca\u65b9\u5dee\u4e3a\uff1a\n$$\n\\begin{aligned}\n{{\\hat x}_{i \\to a}^{t+1}} &= \\mathbb E\\left( {{x_i}} \\right)\\\\\n&= \\int {{x_i}\\frac{1}{Z}p({x_i}){\\mathcal N}\\left( {{x_i};{{\\hat r}_{i \\to a}^t},{{\\hat \\Sigma }_{i \\to a}^t}} \\right)} {\\rm{d}}{x_i}\\\\\n&\\overset{def}{=}F(x_i;{{\\hat r}_{i \\to a}},{{\\hat \\Sigma }_{i \\to a}^t})\\\\\n{{\\hat v}_{i \\to a}^{t+1}} &= Var({x_i})\\\\\n&= \\int {{{\\left( {{x_i} - {{\\hat x}_{i \\to a}^t}} \\right)}^2}\\frac{1}{Z}p({x_i}){\\mathcal N}\\left( {{x_i};{{\\hat r}_{i \\to a}^t},{{\\hat \\Sigma }_{i \\to a}^t}} \\right)} {\\rm{d}}{x_i}\\\\\n&\\overset{def}{=}G(x_i;{{\\hat r}_{i \\to a}^t},{{\\hat \\Sigma }_{i \\to a}^t})\\\\\n\\end{aligned}\n$$\n\u5176\u4e2d$Z \\overset{def}{=} \\int {p({x_i}){\\mathcal N}\\left( {{x_i};{{\\hat r}_{i \\to a}^t},{{\\hat \\Sigma }_{i \\to a}^t}} \\right)} {\\rm{d}}{x_i}$\u4e3a\u5f52\u4e00\u5316\u53c2\u6570. \u5f53$p(x_i)$\u4e3a\u786e\u5b9a\u5206\u5e03\u65f6\uff0c\u8fd9\u4e24\u4e2a\u79ef\u5206\u53ef\u4ee5\u76f4\u63a5\u8ba1\u7b97\u6216\u8fd1\u4f3c\uff0c\u5176\u590d\u6742\u5ea6\u4e3a$\\mathcal O(1)$.<\/p>\n
\u81f3\u6b64\uff0c\u6b63\u5411\u6d88\u606f\u548c\u53cd\u5411\u6d88\u606f\u5747\u63a8\u5bfc\u5b8c\u6bd5\u3002\u5230\u8fd9\u91cc\uff0c\u5b9e\u9645\u4e0a\u662f\u5bf9LBP\u8fdb\u884c\u4e86\u8fd1\u4f3c\uff0c\u51cf\u5c11\u4e86\u4f20\u9012\u7684\u53c2\u6570\u91cf\uff0c\u5c06\u6d88\u606f\u4f20\u9012\u53d8\u6210\u4e86\u7edf\u8ba1\u91cf\u4f20\u9012<\/p>\n
\u8fdb\u4e00\u6b65\u7684\u7b80\u5316<\/h2>\n
\u4e0a\u8ff0\u6b63\u5411\u6d88\u606f\u548c\u53cd\u5411\u6d88\u606f\u7684\u8ba1\u7b97\u5747\u6d89\u53ca\u5230${\\mathcal O}(N)$\uff08\u6216${\\mathcal O}(M)$,\u4e24\u8005\u540c\u7b49\u7ea7\uff0c\u4ee5\u4e0b\u7edf\u4e00\u4e3a${\\mathcal O}(N)$\uff09\u590d\u6742\u5ea6\u7684\u8ba1\u7b97\u8fc7\u7a0b\uff0c\u540c\u65f6\uff0c\u5728\u7cfb\u7edf\u4e2d\uff0c\u6bcf\u6b21\u8fed\u4ee3\u5b58\u5728$MN$\u6761\u6b63\u5411\u548c\u53cd\u5411\u6d88\u606f\uff0c\u6545\u5b9e\u9645\u4e0a\u4ee5\u4e0a\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u7ea6\u4e3a${\\mathcal O}(N^3)$.\u8003\u8651\u5230\u5927\u7cfb\u7edf\u4e0b\u4e09\u6b21\u590d\u6742\u5ea6\u4f9d\u7136\u96be\u4ee5\u63a5\u53d7\uff0cAMP\u5bf9\u4e0a\u8ff0\u4e24\u7c7b\u6d88\u606f\u8fdb\u884c\u4e86\u8fdb\u4e00\u6b65\u7684\u7b80\u5316\u3002<\/p>\n
\u5177\u4f53\u7684\uff0c\u5b9a\u4e49\u4ee5\u4e0b\u516d\u4e2a\u4e0e\u8fb9\u65e0\u5173\u7684\u7edf\u8ba1\u91cf\uff1a\n$$\n\\begin{aligned}\nZ_a^t &= \\sum\\limits_{i = 1}^N {{h_{ai}}} \\hat x_{i \\to a}^t\\\\\nV_a^t &= \\sum\\limits_{i = 1} {|{h_{ai}}{|^2}\\hat v_{i \\to a}^t} \\\\\n\\hat \\Sigma _i^t &= {\\left( {\\sum\\limits_{a = 1}^M {\\frac{{|{h_{ai}}{|^2}}}{{N_0 + V_{a\\to i}^t}}} } \\right)^{ - 1}}\\\\\n\\hat r_i^t &= \\Sigma _i^t\\sum\\limits_{a = 1}^M {\\frac{{h_{ai}^*({y_a} - Z_{a\\to i}^t)}}{{N_0 + V_{a \\to i}^t}}} \\\\\n{{\\hat x}_i^{t+1}} &= \\int {{x_i}\\frac{1}{Z}p({x_i}){\\mathcal{N}}\\left( {{x_i};{{\\hat r}_i^t},{{\\hat \\Sigma }_i^t}} \\right)} {\\rm{d}}{x_i}\\\\\n{{\\hat v}_i^{t+1}} &= \\int {{{\\left( {{x_i} - {{\\hat x}_i}} \\right)}^2}\\frac{1}{Z}p({x_i}){\\mathcal{N}}\\left( {{x_i};{{\\hat r}_i^t},{{\\hat \\Sigma }_i^t}} \\right)} {\\rm{d}}{x_i}\n\\end{aligned}\n$$<\/p>\n
\u800c\u4e0b\u9762\u7684\u64cd\u4f5c\uff0c\u5219\u662f\u4f7f\u7528\u4e4b\u524d\u63a8\u5bfc\u7684\u6b63\u5411\u6d88\u606f\u4e0e\u53cd\u5411\u6d88\u606f\u6765\u786e\u5b9a\u4ee5\u4e0a\u516d\u4e2a\u53d8\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u9700\u8981\u5c06\u4e4b\u524d\u7684\u6b63\u5411\u548c\u53cd\u5411\u6d88\u606f\u4e0e\u8fd9\u516d\u4e2a\u53d8\u91cf\u4e4b\u95f4\u5efa\u7acb\u8054\u7cfb\uff0c\u5e76\u8bd5\u56fe\u5efa\u7acb\u6b64\u516d\u4e2a\u53d8\u91cf\u4e4b\u95f4\u7684\u8fed\u4ee3\u7b97\u6cd5\u3002<\/p>\n
\u6211\u4eec\u7ea6\u5b9a\uff0c$|h_{ij}|$\u7684\u6570\u91cf\u7ea7\u4e3a${\\mathcal O}(\\frac{1}{\\sqrt{N}})$\uff0c\u56e0\u6b64$|h_{ij}|^2$\u7684\u6570\u91cf\u7ea7\u4e3a${\\mathcal O}(\\frac{1}{{N}})$\uff1b\u540c\u65f6\u7ea6\u5b9a$x_j$\u7684\u6570\u91cf\u7ea7\u4e3a${\\mathcal O}(1)$\uff0c\u56e0\u6b64\u5176\u4f30\u8ba1\u91cf--${{\\hat x}_i}$\u548c${{\\hat v}_i}$\u6570\u91cf\u7ea7\u540c\u6837\u4e3a${\\mathcal O}(1)$\u3002\u5728\u4ee5\u4e0a\u7ea6\u5b9a\u4e0b\uff0c$y_{i}$\u7684\u6570\u91cf\u7ea7\u540c\u6837\u7684\uff0c\u4e5f\u662f${\\mathcal O}(1)$\uff0c$Z_a,V_a$\u4e5f\u540c\u6837\u5e94\u4e3a${\\mathcal O}(1)$.\u56e0\u6b64\uff0c\u6211\u4eec\u8ba4\u4e3a\u5728\u5728\u5927\u7cfb\u7edf\u6781\u9650\u4e0b\uff0c\u5355\u72ec\u5b58\u5728\u7684${\\mathcal O}(\\frac{1}{\\sqrt{N}})$\u6570\u91cf\u7ea7\u7684\u53d8\u91cf\u4ee5\u53ca$N$-\u6c42\u548c\u7684${\\mathcal O}(\\frac{1}{N})$\u6570\u91cf\u7ea7\u7684\u53d8\u91cf\uff08\u6c42\u548c\u540e\u6570\u91cf\u7ea7\u4e3a${\\mathcal O}(\\frac{1}{\\sqrt{N}})$\uff09\u53ef\u4f5c\u4e3a\u65e0\u7a77\u5c0f\u91cf\u800c\u88ab\u5ffd\u7565\u3002<\/p>\n
\u540c\u65f6\u6211\u4eec\u8ba4\u4e3a$\\mathbb E(x)=F(x;r,\\Sigma)$\u548c$Var(x)=G(x;r,\\Sigma)$ \u662f\u5173\u4e8e\u4e24\u4e2a\u53d8\u91cfLipschitz\u8fde\u7eed\u7684\u3002<\/p>\n
\u89c2\u5bdf\u4ee5\u4e0a\u53d8\u91cf\uff0c\u53ef\u4ee5\u53d1\u73b0\uff0c\u5176\u672c\u8d28\u4e0a\u5b58\u5728\u76f8\u4e92\u4f9d\u8d56\u3002\u5177\u4f53\u7684\uff0c\u53d8\u91cf\u4f9d\u8d56\u4e3a$\\Sigma_i,r_i\\to \\hat v_i,\\hat x_i \\to V_a,Z_a\\to \\Sigma_i,r_i\\to \\cdots$\uff0c\u5176\u4e2d$\\to$\u4ee3\u8868\u540e\u9762\u7684\u53d8\u91cf\u4f9d\u8d56\u4e8e\u524d\u9762\u7684\u53d8\u91cf\u3002\u56e0\u6b64\uff0c\u9996\u5148\u5bf9$r_i^t$\u548c$\\Sigma _i^t$\u8fdb\u884c\u8fd1\u4f3c\uff1a\n$$\n\\begin{aligned}\n\\hat \\Sigma _i^t - {{\\hat \\Sigma }_{i \\to a}^t} &= {\\left( {\\sum\\limits_b^M {\\frac{{{{\\left| {{h_{bi}}} \\right|}^2}}}{{{N_0} + V_{b \\to i}^t}}} } \\right)^{ - 1}} - {\\left( {\\sum\\limits_{b \\ne a}^M {\\frac{{{{\\left| {{h_{bi}}} \\right|}^2}}}{{{N_0} + V_{b \\to i}^t}}} } \\right)^{ - 1}}\\\\\n&= - \\frac{{\\frac{{{{\\left| {{h_{ai}}} \\right|}^2}}}{{{N_0} + V_{a \\to i}^t}}}}{{\\left( {\\sum\\limits_b^M {\\frac{{{{\\left| {{h_{bi}}} \\right|}^2}}}{{{N_0} + V_{b \\to i}^t}}} } \\right)\\left( {\\sum\\limits_{b \\ne a}^M {\\frac{{{{\\left| {{h_{bi}}} \\right|}^2}}}{{{N_0} + V_{b \\to i}^t}}} } \\right)}}\\\\\n&= {\\mathcal O}\\left( {\\frac{1}{N}} \\right)\\\\\n\\hat r_i^t - {{\\hat r}_{i \\to a}^t} &= {{\\hat \\Sigma }_i}\\sum\\limits_b^M {\\frac{{h_{bi}^*\\left( {{y_b} - Z_{b \\to i}^t} \\right)}}{{{N_0} + V_{b \\to i}^t}}} - {{\\hat \\Sigma }_{i \\to a}}\\sum\\limits_{b \\ne a}^M {\\frac{{h_{bi}^*\\left( {{y_b} - Z_{b \\to i}^t} \\right)}}{{{N_0} + V_{b \\to i}^t}}} \\\\\n&= {{\\hat \\Sigma }_i}\\frac{{h_{ai}^*\\left( {{y_a} - Z_{a \\to i}^t} \\right)}}{{{N_0} + V_{a \\to i}^t}} + {\\mathcal O}\\left( {\\frac{1}{N}} \\right)*\\sum\\limits_{b\\neq a}^M {\\frac{{h_{bi}^*\\left( {{y_b} - Z_{b \\to i}^t} \\right)}}{{{N_0} + V_{b \\to i}^t}}} \\\\\n&= {{\\hat \\Sigma }_i}\\frac{{h_{ai}^*\\left( {{y_a} - Z_{a \\to i}^t} \\right)}}{{{N_0} + V_{a \\to i}^t}} + {\\mathcal O}\\left( {\\frac{1}{N}} \\right)\n\\end{aligned}\n$$\n\u5373\uff0c\u6211\u4eec\u5f97\u5230\n$$\n\\begin{aligned}\n\\hat \\Sigma _i^t &\\approx {{\\hat \\Sigma }_{i \\to a}}\\\\\n\\hat r_i^t &\\approx {{\\hat r}_{i \\to a}}\n+{{\\hat \\Sigma }_i}\\frac{{h_{ai}^*\\left( {{y_a} - Z_{a \\to i}^t} \\right)}}{{{N_0} + V_{a \\to i}^t}}\n\\end{aligned}\n$$\n\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5bf9$\\hat x_{i \\to a}^{t+1}$\u548c$\\hat v_{i \\to a}^{t+1}$\u5206\u522b\u5728${{\\hat x}_i^{t+1}}$\u548c${{\\hat v}_i^{t+1}}$\u5904\u8fdb\u884c\u4e00\u9636\u6cf0\u52d2\u5c55\u5f00\u3002\u7531\u4e8eLipschitz\u8fde\u7eed\u6027\uff0c\u51fd\u6570\u51e0\u4e4e\u5904\u5904\u5b58\u5728\u4e00\u9636\u5bfc\u6570\u4e14\u5bfc\u6570\u6709\u754c\u3002\u56e0\u6b64\u53ef\u4ee5\u5f97\u5230\n$$\n\\begin{aligned}\n\\hat x_{i \\to a}^{t + 1} - \\hat x_i^{t + 1} &= -\\left( {\\hat r_i^t - \\hat r_{i \\to a}^t} \\right)\\frac{{\\partial F\\left( {{x_i};{{\\hat r}_i},{{\\hat \\Sigma }_i}} \\right)}}{{\\partial r}} - \\left( {\\hat \\Sigma _i^t - \\hat \\Sigma _{i \\to a}^t} \\right)\\frac{{\\partial F\\left( {{x_i};{{\\hat r}_i},{{\\hat \\Sigma }_i}} \\right)}}{{\\partial \\Sigma }}\\\\\n&= -\\left( {\\hat r_i^t - \\hat r_{i \\to a}^t} \\right)\\frac{{\\partial F\\left( {{x_i};{{\\hat r}_i},{{\\hat \\Sigma }_i}} \\right)}}{{\\partial r}} + {\\mathcal O}\\left( {\\frac{1}{N}} \\right)\\\\\n\\hat v_{i \\to a}^{t + 1} - \\hat v_i^{t + 1} &= -\\left( {\\hat r_i^t - \\hat r_{i \\to a}^t} \\right)\\frac{{\\partial G\\left( {{x_i};{{\\hat r}_i},{{\\hat \\Sigma }_i}} \\right)}}{{\\partial r}} - \\left( {\\hat \\Sigma _i^t - \\hat \\Sigma _{i \\to a}^t} \\right)\\frac{{\\partial G\\left( {{x_i};{{\\hat r}_i},{{\\hat \\Sigma }_i}} \\right)}}{{\\partial \\Sigma }}\\\\\n&= -\\left( {\\hat r_i^t - \\hat r_{i \\to a}^t} \\right)\\frac{{\\partial G\\left( {{x_i};{{\\hat r}_i},{{\\hat \\Sigma }_i}} \\right)}}{{\\partial r}} + {\\mathcal O}\\left( {\\frac{1}{N}} \\right)\n\\end{aligned}\n$$\n\u5e76\u4e14\u5bf9\u4e8e$F(x;r,\\Sigma)=\\mathbb E(x)=\\int {{x}\\frac{1}{Z}p({x}){\\mathcal N}\\left( {{x};{{r}},{\\Sigma}} \\right)} {\\rm{d}}{x}$\u800c\u8a00\uff0c\u5b58\u5728$\\frac{{\\partial F\\left( {x;r,\\Sigma } \\right)}}{{\\partial r}} = \\frac{{G\\left( {x;r,\\Sigma } \\right)}}{\\Sigma } = \\frac{{Var(x)}}{\\Sigma }$\uff0c\u6545\u6709\n$$\n\\begin{aligned}\n\\hat v_{i \\to a}^{t + 1} &\\approx \\hat v_i^{t + 1} - {{\\hat \\Sigma }_i}\\frac{{h_{ai}^*\\left( {{y_a} - Z_{a \\to i}^t} \\right)}}{{{N_0} + V_{a \\to i}^t}}\\frac{{\\partial G\\left( {{x_i};{{\\hat r}_i},{{\\hat \\Sigma }_i}} \\right)}}{{\\partial r}}\\\\\n\\hat x_{i \\to a}^{t + 1} &\\approx \\hat x_i^{t + 1} - \\frac{{h_{ai}^*\\left( {{y_a} - Z_{a \\to i}^t} \\right)}}{{{N_0} + V_{a \\to i}^t}}\\hat v_i^{t + 1}\n\\end{aligned}\n$$\n\u63a5\u7740\uff0c\u5c06\u4e0a\u9762\u4e8c\u5f0f\u4ee3\u5165$Z_a^t$\u548c$V_a^t$\u5f97\u5230\n$$\n\\begin{aligned}\nV_a^t &= \\sum\\limits_{i = 1}^N {|{h_{ai}}{|^2}\\hat v_{i \\to a}^t} \\\\\n&= \\sum\\limits_{i = 1}^N {\\left( {|{h_{ai}}{|^2}\\hat v_i^t - |{h_{ai}}{|^2}{{\\hat \\Sigma }_i}\\frac{{h_{ai}^*\\left( {{y_a} - Z_{a \\to i}^t} \\right)}}{{{N_0} + V_{a \\to i}^t}}\\frac{{\\partial G\\left( {{x_i};{{\\hat r}_i},{{\\hat \\Sigma }_i}} \\right)}}{{\\partial r}}} \\right)} \\\\\n&= \\sum\\limits_{i = 1}^N {\\left( {|{h_{ai}}{|^2}\\hat v_i^t - |{h_{ai}}{|^2}{{\\left( {\\sum\\limits_b^M {\\frac{{{{\\left| {{h_{bi}}} \\right|}^2}}}{{{N_0} + V_{b \\to i}^t}}} } \\right)}^{ - 1}}\\frac{{h_{ai}^*\\left( {{y_a} - Z_{a \\to i}^t} \\right)}}{{{N_0} + V_{a \\to i}^t}}\\frac{{\\partial G\\left( {{x_i};{{\\hat r}_i},{{\\hat \\Sigma }_i}} \\right)}}{{\\partial r}}} \\right)} \\\\\n&= \\sum\\limits_{i = 1}^N {|{h_{ai}}{|^2}\\hat v_i^t} + {\\mathcal O}\\left( {\\frac{1}{N}} \\right)\\\\\nZ_a^t &= \\sum\\limits_{i = 1}^N {{h_{ai}}} \\hat x_{i \\to a}^t\\\\\n&= \\sum\\limits_{i = 1}^N {{h_{ai}}} \\left( {\\hat x_i^t - \\frac{{h_{ai}^*\\left( {{y_a} - Z_{a \\to i}^{t - 1}} \\right)}}{{{N_0} + V_{a \\to i}^{t - 1}}}\\hat v_i^t} \\right)\\\\\n&= \\sum\\limits_{i = 1}^N {\\left( {{h_{ai}}\\hat x_i^t - \\frac{{|{h_{ai}}{|^2}\\left( {{y_a} - Z_{a \\to i}^{t - 1}} \\right)}}{{{N_0} + V_{a \\to i}^{t - 1}}}\\hat v_i^t} \\right)} \\\\\n&= \\sum\\limits_{i = 1}^N {{h_{ai}}\\hat x_i^t} - \\sum\\limits_{i = 1}^N {\\frac{{|{h_{ai}}{|^2}\\left( {{y_a} - Z_{a \\to i}^{t - 1}} \\right)}}{{{N_0} + V_{a \\to i}^{t - 1}}}\\hat v_i^t}\n\\end{aligned}\n$$<\/p>\n
\u540c\u65f6\uff0c\u7531\u4e8e\n$$\n\\begin{aligned}\nZ_a^t - Z_{a \\to i}^t &= {h_{ai}}\\hat x_{i \\to a}^t \\\\\n&= {h_{ai}}\\hat x_i^t - \\frac{{|{h_{ai}}{|^2}\\left( {{y_a} - Z_{a \\to i}^t} \\right)}}{{{N_0} + V_{a \\to i}^t}}\\hat v_i^t \\\\\n&= {h_{ai}}\\hat x_i^t + {\\mathcal O}\\left( {\\frac{1}{N}} \\right)\\\\\nV_a^t - V_{a \\to i}^t &= |{h_{ai}}{|^2}\\hat v_{i \\to a}^t\\\\\n&= {\\mathcal O}\\left( {\\frac{1}{N}} \\right)\n\\end{aligned}\n$$\n\u5f97\u5230\n$$\n\\begin{aligned}\nZ_a^t &= \\sum\\limits_{i = 1}^N {{h_{ai}}\\hat x_i^t} - \\sum\\limits_{i = 1}^N {\\frac{{|{h_{ai}}{|^2}\\left( {{y_a} - Z_{a \\to i}^{t - 1}} \\right)}}{{{N_0} + V_{a \\to i}^{t - 1}}}\\hat v_i^t} \\\\\n&\\approx \\sum\\limits_{i = 1}^N {{h_{ai}}\\hat x_i^t} - \\frac{{V_a^t\\left( {{y_a} - Z_a^{t - 1}} \\right)}}{{{N_0} + V_a^{t - 1}}}\n\\end{aligned}\n$$<\/p>\n
\u81f3\u6b64\uff0c\u6211\u4eec\u5f97\u5230\u4e86$Z_a^t$\u548c$V_a^t$\u7684\u4e00\u7ec4\u4e0e\u8fb9\u65e0\u5173\u7684\u8868\u8fbe\u5f0f\uff0c\u6216\u8005\u8bf4\u6211\u4eec\u5f97\u5230\u4e86\u7b2c\u4e00\u7ec4\u4ec5\u7528\u8fd9\u516d\u4e2a\u91cf\u8868\u8fbe\u7684\u53d8\u91cf\uff0c\u63a5\u4e0b\u6765\u9700\u8981\u505a\u7684\uff0c\u4fbf\u662f\u5c06\u8fd9\u516d\u4e2a\u53d8\u91cf\u4e2d\u7684\u5176\u4f59\u90e8\u5206\u90fd\u8f6c\u6362\u51fa\u4e0e\u8fb9\u65e0\u5173\u7684\u8868\u8fbe\u3002\u7531\u4e8e$\\hat x_{i \\to a}^{t+1}$\u548c$\\hat v_{i \\to a}^{t+1}$\u672c\u8eab\u4fbf\u662f\u7531$r_i^t$\u548c$\\Sigma _i^t$\u8868\u51fa\uff0c\u56e0\u6b64\u63a5\u4e0b\u6765\u53ea\u9700\u8981\u5bf9$r_i^t$\u548c$\\Sigma _i^t$\u8fdb\u884c\u8f6c\u6362\u3002\n\u5c06$Z_a^t$\u548c$V_a^t$\u4ee3\u5165$r_i^t$\u548c$\\Sigma _i^t$\u4e2d\u5f97\u5230\n$$\n\\begin{aligned}\n\\hat \\Sigma _i^t &= {\\left( {\\sum\\limits_b^M {\\frac{{{{\\left| {{h_{bi}}} \\right|}^2}}}{{{N_0} + V_{b \\to i}^t}}} } \\right)^{ - 1}}\\\\\n&= {\\left( {\\frac{{\\sum\\limits_a^M {{{\\left| {{h_{ai}}} \\right|}^2} \\prod \\limits_{b \\ne a}^M \\left( {{N_0} + V_{b}^t + O\\left( {\\frac{1}{N}} \\right)} \\right)} }}{{ \\prod \\limits_{b = 1}^M \\left( {{N_0} + V_{b}^t + O\\left( {\\frac{1}{N}} \\right)} \\right)}}} \\right)^{ - 1}}\\\\\n&= {\\left( {\\frac{{\\sum\\limits_a^M {{{\\left| {{h_{ai}}} \\right|}^2} \\prod \\limits_{b \\ne a}^M \\left( {{N_0} + V_{b}^t} \\right) + O\\left( {\\frac{1}{N}} \\right)} }}{{ \\prod \\limits_{b = 1}^M \\left( {{N_0} + V_{b}^t} \\right) + O\\left( {\\frac{1}{N}} \\right)}}} \\right)^{ - 1}}\\\\\n&\\approx {\\left( {\\frac{{\\sum\\limits_a^M {{{\\left| {{h_{ai}}} \\right|}^2} \\prod \\limits_{b \\ne a}^M \\left( {{N_0} + V_{b}^t} \\right)} }}{{ \\prod \\limits_{b = 1}^M \\left( {{N_0} + V_{b}^t} \\right)}}} \\right)^{ - 1}}\\\\\n&= {\\left( {\\sum\\limits_b^M {\\frac{{{{\\left| {{h_{bi}}} \\right|}^2}}}{{{N_0} + V_b^t}}} } \\right)^{ - 1}}\\\\\nr_i^t &= \\hat \\Sigma _i^t\\sum\\limits_b^M {\\frac{{h_{bi}^*\\left( {{y_b} - Z_{b \\to i}^t} \\right)}}{{{N_0} + V_{b \\to i}^t}}} \\\\\n&= \\hat \\Sigma _i^t\\sum\\limits_b^M {\\frac{{h_{bi}^*\\left( {{y_b} - Z_{b }^t + {h_{bi}}\\hat x_i^t}+ {\\mathcal O}\\left( {\\frac{1}{{N }}} \\right)\\right)}}{{{N_0} + V_{b \\to i}^t}}} \\\\\n&= \\hat \\Sigma _i^t\\sum\\limits_b^M {\\frac{{h_{bi}^*\\left( {{y_b} - Z_{b }^t} \\right) + |{h_{bi}}{|^2}\\hat x_i^t}}{{{N_0} + V_{b \\to i}^t}}}+{\\mathcal O}\\left( {\\frac{1}{{N }}} \\right) \\\\\n&\\approx \\hat x_i^t + \\hat \\Sigma _i^t\\sum\\limits_b^M {\\frac{{h_{bi}^*\\left( {{y_b} - Z_b^t} \\right)}}{{{N_0} + V_b^t}}} \n\\end{aligned}\n$$<\/p>\n
\u5230\u6b64\uff0cAMP\u7b97\u6cd5\u63a8\u5bfc\u5927\u81f4\u5b8c\u6210\u3002\u6b64\u65f6\uff0c\u53d8\u91cf\u8fed\u4ee3\u5747\u5728\u516d\u4e2a\u8fb9\u4e0d\u53d8\u53d8\u91cf\u4e4b\u4e2d\u8fdb\u884c\uff0c\u5c06\u8ba1\u7b97\u590d\u6742\u5ea6\u964d\u4f4e\u4e86\u4e00\u4e2a\u7ef4\u5ea6\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8fd9\u91cc\u7684\u7f29\u653e\u5047\u8bbe\u53ea\u6709\u5728\u5927\u7cfb\u7edf\u4e0b\u4f53\u73b0\u8f83\u4e3a\u660e\u663e\u3002<\/p>\n
\u7ed3\u8bed<\/h1>\n
AMP\u7684\u4e3b\u8981\u601d\u8def\u662f\u5bf9\u8fb9\u6d88\u606f\u8fdb\u884c\u8fd1\u4f3c\uff0c\u518d\u8f6c\u6362\u6210\u70b9\u6d88\u606f\u3002\u8fd1\u4f3c\u548c\u8f6c\u6362\u7684\u8fc7\u7a0b\u4e2d\u5747\u4f7f\u7528\u4e86\u4e00\u4e9b\u7cfb\u7edf\u7f29\u653e\u7684\u5047\u8bbe\uff0c\u56e0\u6b64\u6700\u7ec8\u4ea7\u751f\u4e86\u6bd4\u8f83\u5947\u5999\u7684\u7ed3\u679c\u3002<\/p>\n
\u672c\u6587\u4e8e2023.10.2\u5b8c\u6210\uff0c\u6c34\u5e73\u4e00\u822c\uff0c\u5982\u6709\u8c2c\u8bef\uff0c\u671b\u6307\u6b63\u3002<\/p>\n
\u53c2\u8003\u6587\u732e:<\/p>\n
https:\/\/arxiv.org\/abs\/2201.07487<\/a><\/p>\n https:\/\/arxiv.org\/abs\/1010.5141<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" \u66f4\u6b63 \u6700\u540e\u4e00\u4e2a$r_i^t$\u7684\u8868\u8fbe\u5f0f\u5199\u9519\u4e86\uff0c\u6539\u4e86\u8fc7\u6765\u3002\u6b22\u8fce\u5404\u4f4d\u6307\u51fa\u9519\u8bef\uff0c\u53ef\u4ee5\u5728\u8bc4\u8bba\u533a\u4ea4\u6d41\u3002 \u5f00\u59cb \u627e\u4e86\u5f88\u4e45\u90fd\u6ca1\u6709\u627e\u5230\u4e00\u4e2a\u6bd4\u8f83\u8be6\u7ec6\u7684\u5173\u4e8e\u8fd1\u4f3c\u6d88\u606f\u4f20\u9012\uff08AMP\uff09\u7684\u8be6\u5c3d\u63a8\u5bfc\uff0c\u672c\u6587\u65e8\u4ece\u56e0\u5b50\u56fe\u89d2\u5ea6\u5bf9\u5176\u8fdb\u884c\u63a8\u5bfc\u3002 \u5148\u9a8c\u77e5\u8bc6 \u5728\u8fdb\u884cAMP\u7684\u63a8\u5bfc\u4e4b\u524d\uff0c\u6211\u4eec\u9996\u5148\u9700\u8981\u4e00\u4e9b\u5148\u9a8c\u77e5\u8bc6\uff1a 1. \u6d88\u606f\uff0c\u56e0\u5b50\u56fe\u4e0e\u548c\u79ef\u7b97\u6cd5\uff08Sum Product Algorithm, SPA\uff09:\u8fd9\u4e00\u90e8\u5206\u5728\u901a\u4fe1\u4e2d\u7684\u53d8\u5206\u63a8\u7406\u6280\u672f–\u56e0\u5b50\u56fe\u548c\u6d88\u606f\u4f20\u9012\u65b9\u6cd5\u7684\u7b2c2-3\u7ae0\u6709\u8f83\u4e3a\u8be6\u7ec6\u7684\u63cf\u8ff0\uff0c\u6b64\u5904\u53ea\u505a\u7b80\u8ff0\u3002 \u9996\u5148\uff0c\u6211\u4eec\u7684\u95ee\u9898\u53ef\u4ee5\u8f6c\u5316\u4e3a\u5bf9\u67d0\u4e9b\u611f\u5174\u8da3\u53d8\u91cf\uff08\u5982$\\bf x$\u7b49\uff09\u7684\u4f30\u8ba1\uff0c\u800c\u8fd9\u4e2a\u4f30\u8ba1\u4e00\u822c\u4f1a\u5728\u53c8\u4e00\u822c\u4f1a\u5728\u5168\u5c40\u51fd\u6570\uff08$p(\\bf x, \\bf y |\\bf…… <\/p>\n \u7ee7\u7eed\u9605\u8bfb»»»<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[4,6,5],"_links":{"self":[{"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=\/wp\/v2\/posts\/94"}],"collection":[{"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=94"}],"version-history":[{"count":0,"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=\/wp\/v2\/posts\/94\/revisions"}],"wp:attachment":[{"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=94"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=94"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/etc.lmdewz.xyz\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=94"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}