Multi-view statistical analysis of Alzheimer's disease using DTI-derived brain imaging data and CSF measures via Generalized Eigenvalue Problem
Abstract. Eigenvalue problems are ubiquitous in computer vision,
covering a very broad spectrum of applications ranging
from estimation problems in multi-view geometry to image
segmentation. Few other linear algebra problems have a
more mature set of numerical routines available and many
computer vision libraries leverage such tools extensively.
However, the ability to call the underlying solver only as
a "black box" can often become restrictive. Many 'human
in the loop' settings in vision frequently exploit supervision
from an expert, to the extent that the user can be considered
a subroutine in the overall system. In other cases, there
is additional domain knowledge, side or even partial information
that one may want to incorporate within the formulation.
In general, regularizing a (generalized) eigenvalue
problem with such side information remains difficult. Motivated
by these needs, this paper presents an optimization
scheme to solve generalized eigenvalue problems (GEP) involving
a (nonsmooth) regularizer. We start from an alternative
formulation of GEP where the feasibility set of the
model involves the Stiefel manifold. The core of this paper
presents an end to end stochastic optimization scheme
for the resultant problem. We show how this general algorithm
enables improved statistical analysis of brain imaging
data where the regularizer is derived from other 'views' of
the disease pathology, involving clinical measurements and
other image-derived representations.
Figure: DTI image (top left) showing tensor directionality, followed by the FA image (top right) and the connectivity matrix (bottom).
Figure: Feature sensitivity. First column shows the FA image. Second column shows overlays of the weights
assigned by baseline linear kernel on this FA image. Last column shows overlays from the base R-GEP case
in Table. Green (Red) corresponds to smaller (larger) weights.
 Seong Jae Hwang, Maxwell D. Collins, Sathya N. Ravi, Vamsi K. Ithapu, Nagesh Adluru, Sterling C. Johnson, Vikas Singh,
"A Projection free method for Generalized Eigenvalue Problem with a nonsmooth Regularizer",
International Conference on Computer Vision (ICCV), 2015.
[pdf] [fixed eq (18),(20)]
SJH was supported by a University of Wisconsin CIBM fellowship (5T15LM007359-14). We acknowledge
support from NIH grants AG040396 and AG021155, NSF RI 1116584 and NSF CAREER award 1252725,
as well as UW ADRC (AG033514), UW ICTR (1UL1RR025011), Waisman Core grant (P30 HD003352-45),
UW CPCP (AI117924) and NIH grant AG027161.