Title: Rapid Near-Neighbor Interaction of High-dimensional Data via Hierarchical Clustering,
Abstract: Calculation of near-neighbor interactions among high dimensional, irregularly
distributed data points is a fundamental task to many graph-based or
kernel-based machine learning algorithms and applications. Such calculations,
involving large, sparse interaction matrices, expose the limitation of
conventional data-and-computation reordering techniques for improving space and
time locality on modern computer memory hierarchies. We introduce a novel
method for obtaining a matrix permutation that renders a desirable sparsity
profile. The method is distinguished by the guiding principle to obtain a
profile that is block-sparse with dense blocks. Our profile model and measure
capture the essential properties affecting space and time locality, and permit
variation in sparsity profile without imposing a restriction to a fixed
pattern. The second distinction lies in an efficient algorithm for obtaining a
desirable profile, via exploring and exploiting multi-scale cluster structure
hidden in but intrinsic to the data. The algorithm accomplishes its task with
key components for lower-dimensional embedding with data-specific principal
feature axes, hierarchical data clustering, multi-level matrix compression
storage, and multi-level interaction computations. We provide experimental
results from case studies with two important data analysis algorithms. The
resulting performance is remarkably comparable to the BLAS performance for the
best-case interaction governed by a regularly banded matrix with the same
sparsity.