Basistracking

Basistracking refers to methods for maintaining and updating a basis of a subspace to reflect evolving data. In linear algebra, a basis is a set of vectors that spans a subspace; basistracking seeks a current basis that captures the dominant directions of a data stream or time-varying dataset. The goal is to produce a low-dimensional representation that adapts as new observations arrive, often under constraints such as orthogonality or unit length.

Common approaches include online algorithms that update the basis with each new sample. Incremental or online singular value decomposition (SVD) and projection-PCA techniques recompute components efficiently from the previous basis. Oja’s rule and related online neural-network–inspired updates provide simple iterative procedures for updating principal components. More recent work formulates basistracking as optimization on the Grassmannian, the space of subspaces, enabling principled updates that minimize projection error while tracking changes. Robust and sparse variants address outliers and noise.

Applications of basistracking span several domains. In video analysis, it is used for background modeling and subtraction by maintaining a subspace model of the scene. In audio and communications, it supports adaptive beamforming and source separation. It also appears in sensor networks and real-time data analytics where the data distribution evolves over time, requiring continual adaptation of the low-dimensional representation.

Challenges include balancing responsiveness to change with stability, handling abrupt shifts or concept drift, robustness to outliers, and computational efficiency on streaming data. Evaluation typically uses subspace error measures such as principal angles, projection or reconstruction error, and running time. Related topics include subspace tracking and online principal component analysis.