Interpolation on the manifold of k component GMMs
Hyunwoo J. Kim, Nagesh Adluru, Monami Banerjee, Baba C. Vemuri, Vikas Singh,
Interpolation on the manifold of k component Gaussian Mixture Models (GMMs), In International Conference on Computer Vision (ICCV), December 2015.
Abstract
Probability density functions (PDFs) are fundamental "objects" in
mathematics with numerous applications in computer vision, machine
learning and medical imaging. The feasibility of basic operations
such as computing the distance between two PDFs and estimating a mean
of a set of PDFs is a direct function of the representation we choose
to work with.
In this paper, we study the Gaussian mixture model (GMM)
representation of the PDFs motivated by its numerous attractive
features. (1) GMMs are arguably more interpretable than, say, square
root parameterizations (2) the model complexity can be explicitly
controlled by the number of components and (3) they are already widely
used in many applications.
The main contributions of this paper are numerical algorithms to
enable basic operations on such objects that strictly respect their
underlying geometry. For instance, when operating with a set of k
component GMMs, a first order expectation is that the result of simple
operations like interpolation and averaging should provide an object
that is also a k component GMM. The literature provides very
little guidance on enforcing such requirements systematically. It
turns out that these tasks are important internal modules for analysis
and processing of a field of ensemble average propagators (EAPs),
common in diffusion weighted magnetic resonance imaging. We provide
proof of principle experiments showing how the proposed algorithms for
interpolation can facilitate statistical analysis of such data,
essential to many neuroimaging studies. Separately, we also derive
interesting connections of our algorithm with functional spaces of Gaussians, that may be of
independent interest.
Acknowledgments
This work was supported in part by NIH grants AG040396 (VS), NS066340 (BCV), IIS1525431 (BCV), NSF CAREER award 1252725 (VS), 1UL1RR025011 (NA), and P30 HD00335245 (NA). Partial support was also provided by the Center for Predictive Computational Phenotyping (CPCP) at UWMadison (AI117924).
