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基于因子图与和积算法的MATLAB实现

news 2025/8/23 5:54:37

基于因子图（Factor Graph）与和积算法（Sum-Product Algorithm）的MATLAB实现

一、算法架构设计

1. 因子图建模

因子图由变量节点（Variable Nodes）和因子节点（Factor Nodes）构成二部图：

变量节点：表示随机变量（如$x_1, x_2$）
因子节点：表示变量间的函数关系（如$f(x_1,x_2)$）

2. 和积算法流程

$\begin{aligned}
& \text{初始化消息} \quad m_{i \to j}^{(0)}(x_j) = 1 \
& \text{迭代更新} \quad \forall i,j: \
& \quad \mu_{f \to v}(x_v) = \sum_{x_f \setminus x_v} f(x_f) \prod_{u \in \partial f \setminus v} \mu_{u \to f}(x_f) \
& \quad \mu_{v \to f}(x_f) = \prod_{g \in \partial v \setminus f} \mu_{g \to v}(x_v) \
& \text{终止条件} \quad | \mu^{(k+1)} - \mu^{(k)} | < \epsilon
\end{aligned} $

二、MATLAB核心实现

1. 数据结构定义

% 因子图结构体
graph = struct(...'var_nodes', {{'x1', 'x2', 'x3'}}, ... % 变量节点'fac_nodes', {{@f1, @f2}}, ...        % 因子函数句柄'edges', {[1,2; 2,3]});              % 边连接关系% 消息存储（对数域避免下溢）
log_messages = cell(length(graph.var_nodes), length(graph.fac_nodes));

2. 因子函数定义示例

% 定义因子函数 f1(x1,x2) = exp(-0.5*(x1^2 + x2^2 + 2 * 0.3*x1*x2))
function f = f1(x1, x2)cov = [1, 0.3; 0.3, 1]; % 协方差矩阵f = mvnpdf([x1, x2], [0,0], cov);
end% 定义因子函数 f2(x2,x3) = exp(-0.5*(x2^2 + x3^2))
function f = f2(x2, x3)cov = [1, 0; 0, 1];f = mvnpdf([x2, x3], [0,0], cov);
end

3. 消息传递算法实现

function [log_beliefs, converged] = sum_product(graph, max_iter, epsilon)% 初始化消息num_vars = numel(graph.var_nodes);num_facs = numel(graph.fac_nodes);log_messages = cell(num_vars, num_facs);for i = 1:num_varsfor j = 1:num_facslog_messages{i,j} = zeros(1, 2); % 假设变量为二元变量endend% 迭代更新converged = false;for iter = 1:max_iterlog_messages_old = log_messages;% 因子→变量消息更新for f = 1:num_facsfac_node = graph.fac_nodes{f};connected_vars = find(graph.edges(:,f));for v = connected_vars% 计算边缘化消息product = 0;for u = connected_varsif u ~= vproduct = product + log_messages{u,f};endendlog_message = fac_node(connected_vars{v}) + product;log_messages{v,f} = log_message;endend% 变量→因子消息更新for v = 1:num_varsconnected_facs = find(graph.edges(:,v));for f = connected_facsproduct = sum(log_messages{v,f});for u = connected_facsif u ~= fproduct = product + log_messages{v,u};endendlog_beliefs{v} = log_beliefs{v} + product;endend% 收敛判断if max(abs(log_messages(:) - log_messages_old(:))) < epsilonconverged = true;break;endend
end