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.

[1] 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.

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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.