This paper introduces Generalized Principal Component Analysis (GPCA), an algebraic-geometric approach to subspace segmentation. GPCA represents the union of \( n \) subspaces of unknown and possibly different dimensions as the zero set of a set of homogeneous polynomials of degree \( n \). The coefficients of these polynomials can be estimated linearly from sample data points, and the normal vectors to each subspace can be obtained by evaluating the derivatives of these polynomials at points on the subspaces. The main contributions include: 1. **Algebraic-Geometric Representation**: GPCA represents the union of subspaces using homogeneous polynomials, which allows for a unified approach to subspace segmentation. 2. **Linear Estimation**: The coefficients of the polynomials can be estimated linearly from sample data, simplifying the estimation process. 3. **Derivative-Based Subspace Bases**: The normal vectors to each subspace can be obtained by differentiating the polynomials at points on the subspaces, providing a direct method to compute subspace bases. 4. **Recursive Point Selection**: Points on each subspace can be recursively selected using polynomial division, dealing with moderate noise in the data. 5. **Global Non-Iterative Algorithm**: GPCA provides a global, non-iterative solution to subspace segmentation, avoiding the need for initialization and iterative refinement. The paper also discusses extensions of GPCA to handle low-dimensional subspaces in high-dimensional spaces and unknown numbers of subspaces. Experimental results show that GPCA outperforms existing algebraic algorithms and improves the performance of iterative techniques in various computer vision applications, such as face clustering, temporal video segmentation, and 3D motion segmentation.This paper introduces Generalized Principal Component Analysis (GPCA), an algebraic-geometric approach to subspace segmentation. GPCA represents the union of \( n \) subspaces of unknown and possibly different dimensions as the zero set of a set of homogeneous polynomials of degree \( n \). The coefficients of these polynomials can be estimated linearly from sample data points, and the normal vectors to each subspace can be obtained by evaluating the derivatives of these polynomials at points on the subspaces. The main contributions include: 1. **Algebraic-Geometric Representation**: GPCA represents the union of subspaces using homogeneous polynomials, which allows for a unified approach to subspace segmentation. 2. **Linear Estimation**: The coefficients of the polynomials can be estimated linearly from sample data, simplifying the estimation process. 3. **Derivative-Based Subspace Bases**: The normal vectors to each subspace can be obtained by differentiating the polynomials at points on the subspaces, providing a direct method to compute subspace bases. 4. **Recursive Point Selection**: Points on each subspace can be recursively selected using polynomial division, dealing with moderate noise in the data. 5. **Global Non-Iterative Algorithm**: GPCA provides a global, non-iterative solution to subspace segmentation, avoiding the need for initialization and iterative refinement. The paper also discusses extensions of GPCA to handle low-dimensional subspaces in high-dimensional spaces and unknown numbers of subspaces. Experimental results show that GPCA outperforms existing algebraic algorithms and improves the performance of iterative techniques in various computer vision applications, such as face clustering, temporal video segmentation, and 3D motion segmentation.