Fisher's Linear Discriminant Analysis (LDA) Algorithm

Fisher's Linear Discriminant Analysis (LDA) is a classic supervised dimensionality reduction and feature extraction method proposed by statistician R.A. Fisher, widely used in pattern recognition and machine learning. Its core principle involves finding optimal projection directions by maximizing the ratio between inter-class scatter and intra-class scatter, thereby clustering samples from the same class together while separating samples from different classes as much as possible.

The algorithm logic consists of the following key steps: Compute within-class scatter matrix: Calculate the deviation between each class's samples and their mean values, reflecting the compactness of data within the same class (implementation typically involves summing covariance matrices for each class). Compute between-class scatter matrix: Measure the differences between mean values of different classes, representing class separability (calculated using the differences between class means weighted by class sizes). Solve generalized eigenvalue problem: Find projection vectors that maximize the discriminant criterion through matrix operations, typically converted to solving eigenvectors of the matrix product (between-class scatter matrix) × (inverse of within-class scatter matrix). Dimensionality reduction projection: Select the top k eigenvectors corresponding to the largest eigenvalues to form a transformation matrix, projecting original data into a lower-dimensional space.

Fisher LDA's significant advantage lies in its ability to preserve class discriminant information, making it particularly suitable for small-sample datasets. However, it's important to note that the method assumes data follows Gaussian distribution with equal covariance matrices across classes – practical applications must verify whether these prerequisites are satisfied. Code implementation typically requires matrix inversion and eigenvalue decomposition operations, which can be efficiently handled using linear algebra libraries like NumPy or MATLAB.