  Bayesian multidimensional scaling (BMDS) is a probabilistic dimension
reduction tool that allows one to model and visualize data consisting of
dissimilarities between pairs of objects. Although BMDS has proven useful
within, e.g., Bayesian phylogenetic inference, its likelihood and gradient
calculations require a burdensome order of $N^2$ floating-point operations,
where $N$ is the number of data points. Thus, BMDS becomes impractical as $N$
grows large. We propose and compare two sparse versions of BMDS (sBMDS) that
apply log-likelihood and gradient computations to subsets of the observed
dissimilarity matrix data. Landmark sBMDS (L-sBMDS) extracts columns, while
banded sBMDS (B-sBMDS) extracts diagonals of the data. These sparse variants
let one specify a time complexity between $N^2$ and $N$. Under simplified
settings, we prove posterior consistency for subsampled distance matrices.
Through simulations, we examine the accuracy and computational efficiency
across all models using both the Metropolis-Hastings and Hamiltonian Monte
Carlo algorithms. We observe approximately 3-fold, 10-fold and 40-fold speedups
with negligible loss of accuracy, when applying the sBMDS likelihoods and
gradients to 500, 1,000 and 5,000 data points with 50 bands (landmarks); these
speedups only increase with the size of data considered. Finally, we apply the
sBMDS variants to the phylogeographic modeling of multiple influenza subtypes
to better understand how these strains spread through global air transportation
networks.
