General Bayesian Inference over the Stiefel Manifold via the Givens Transform

We introduce the Givens Transform, a novel transform between the space of orthonormal matrices and $\mathbb{R}^D$. The Givens Transform allows for the application of any general Bayesian inference algorithm to probabilistic models containing constrained unit-vectors or orthonormal matrix parameters. This includes a variety of matrix factorizations and dimensionality reduction models such as Probabilistic PCA (PPCA), Exponential Family PPCA (BXPCA), and Canonical Correlation Analysis (CCA). While previous Bayesian approaches to these models relied on separate sampling update rules for constrained and unconstrained parameters, the Givens Transform enables the treatment of unit-vectors and orthonormal matrices agnostically as unconstrained parameters. Thus any Bayesian inference algorithm can be used on these models without modification. This opens the door to not just sampling algorithms, but Variational Inference (VI) as well. We illustrate with several examples and supplied code, how the Givens Transform allows end-users to easily build complex models in their favorite Bayesian modeling framework such as Stan, Edward, or PyMC3, a task that was previously intractable due to technical constraints.

Geodesic-based Salient Object Detection

Saliency detection has been an intuitive way to provide useful cues for object detection and segmentation, as desired for many vision and graphics applications. In this paper, we provided a robust method for salient object detection and segmentation. Other than using various pixel-level contrast definitions, we exploited global image structures and proposed a new geodesic method dedicated for salient object detection. In the proposed approach, a new geodesic scheme, namely geodesic tunneling is proposed to tackle with textures and local chaotic structures. With our new geodesic approach, a geodesic saliency map is estimated in correspondence to spatial structures in an image. Experimental evaluation on a salient object benchmark dataset validated that our algorithm consistently outperformed a number of the state-of-art saliency methods, yielding higher precision and better recall rates. With the robust saliency estimation, we also present an unsupervised hierarchical salient object cut scheme simply using adaptive saliency thresholding, which attained the highest score in our F-measure test. We also applied our geodesic cut scheme to a number of image editing tasks as demonstrated in additional experiments.