The Singular Value Decomposition and Applications

The Singular Value Decomposition (SVD) is a natural matrix factorization that offers one a quantitative means of discerning what is, and what is not, of importance in the underlying data or model. We build this factorization from ingredients we assembled in our study of the eigen-decomposition of symmetric matrices in Chapter 6. We apply this factorization, in its guise as Principal Component Analysis (PCA), to data reduction in the context of sorting spikes that reach a single recording electrode from multiple sources. PCA is a technique for choosing coordinates in which the data exhibits maximal variance. We observe that a simple Hebbian learning rule achieves the same outcome. We then demonstrate how the SVD may be used to reduce the dimension of dynamical models. We show that a 400-dimensional quasi-active cable may be accurately simulated with as few as five variables.