Sparsification of Large Ultrametric Matrices: Insights into the Microbial Tree of Life

Ultrametric matrices have a rich structure that is not apparent from their definition. Notably, the subclass of strictly ultrametric matrices are covariance matrices of certain weighted rooted binary trees. In applications, these matrices can be large and dense, making them difficult to store and handle. In this manuscript, we exploit the underlying tree structure of these matrices to sparsify them via a similarity transformation based on Haar-like wavelets. We show that, with overwhelmingly high probability, only an asymptotically negligible fraction of the off-diagonal entries in random but large strictly ultrametric matrices remain non-zero after the transformation; and develop a fast algorithm to compress such matrices directly from their tree representation. We also identify the subclass of matrices diagonalized by the wavelets and supply a sufficient condition to approximate the spectrum of strictly ultrametric matrices outside this subclass. Our methods give computational access to a covariance model of the microbiologists' Tree of Life, which was previously inaccessible due to its size, and motivate defining a new but wavelet-based phylogenetic $\beta$-diversity metric. Applying this metric to a metagenomic dataset demonstrates that it can provide novel insight into noisy high-dimensional samples and localize speciation events that may be most important in determining relationships between environmental factors and microbial composition.