Eigendecompositions and Singular Value Decompositions appear in a variety of settings in machine learning and data mining. The eigendecomposition looks like so:

$$ \mathbf{A}=\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{-1} $$

$ \mathbf{Q} $ contains the eigenvectors of $ \mathbf{A} $ and $ \mathbf{\Lambda} $ is a diagonal matrix containing the eigenvalues.

The singular value decomposition looks like:

$$ \mathbf{A} = \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^* $$

$ \mathbf{U} $ contains the eigenvectors of the covariance matrix $ \mathbf{A}\mathbf{A^T} $. $ \mathbf{V} $ contains the eigenvectors of the gram matrix $ \mathbf{A^T}\mathbf{A} $.

The **truncated** variants of these decompositions allow us to compute only a few eigenvalues(vectors) or singular values (vectors).

This is important since (i) a lot of times, the smaller eigenvalues are discarded, and (ii) you don’t want to compute the entire decomposition and retain only a few of the rows and columns of the computed matrices each time.

For core.matrix, I implemented these truncated decompositions in Kublai. Details below.