MATLAB Implementation of Fuzzy Support Vector Machine with Code Descriptions

Fuzzy Support Vector Machine (FSVM) is an extension of traditional Support Vector Machines that introduces the concept of membership degrees to handle uncertainty and noise in data. Unlike standard SVM which treats all samples equally, FSVM assigns different membership values to different samples, thereby reducing the impact of noise or outliers on the classification boundary. The core implementation of linear fuzzy support vector machine in MATLAB focuses on two key aspects: first, how to assign appropriate membership degrees to training samples, and second, how to integrate these membership degrees into the SVM optimization problem. Common membership assignment methods include distance-based approaches (where samples closer to class centers receive higher membership) and density-based methods. The optimization objective of linear FSVM incorporates membership weights, allowing samples with lower membership degrees to have reduced influence on the classification hyperplane constraints. Important implementation considerations include: - Normalization of membership degrees (e.g., mapping to the 0-1 range) - Modifying the SVM loss function term by using membership degrees as penalty coefficients for samples - Utilizing quadratic programming solvers (such as MATLAB's quadprog function) to handle the weighted optimization problem Although this implementation does not use kernel functions, it provides significant reference value for understanding FSVM's core mechanism—particularly how fuzzy processing enhances model robustness. Future extension directions include integration with nonlinear kernel functions and dynamic membership adjustment strategies. Key MATLAB implementation details: - Use pdist2 function to calculate distances for membership assignment - Implement membership normalization using rescale or min-max scaling functions - Modify the standard SVM dual problem by incorporating membership weights in the constraint conditions - Solve the quadratic programming problem using quadprog with properly defined H, f, A, and b matrices - The decision function follows w'*x + b format with membership-weighted support vectors