Spectral isolation of naturally reductive metrics on simple Lie groups

We show that within the class of left-invariant naturally reductive metrics $\mathcal{M}_{\operatorname{Nat}}(G)$ on a compact simple Lie group $G$, every metric is spectrally isolated. We also observe that any collection of isospectral compact symmetric spaces is finite; this follows from a somewhat stronger statement involving only a finite part of the spectrum.Comment: 19 pages, new title and abstract, revised introduction, new result demonstrating that any collection of isospectral compact symmetric spaces must be finite, to appear Math Z. (published online Dec. 2009

Gallot-Tanno theorem for pseudo-Riemannian metrics and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist

We generalize for pseudo-Riemannian metrics a classical result of Gallot and Tanno and use it to reprove a recent result of Alekseevsky, Cortes, Galaev and Leistner that decomposable cones over complete closed pseudo-Riemannian manifolds do not exist.Comment: 6 pages, no figure

Uncertainty quantification in medical image synthesis

Machine learning approaches to medical image synthesis have shown outstanding performance, but often do not convey uncertainty information. In this chapter, we survey uncertainty quantification methods in medical image synthesis and advocate the use of uncertainty for improving clinicians’ trust in machine learning solutions. First, we describe basic concepts in uncertainty quantification and discuss its potential benefits in downstream applications. We then review computational strategies that facilitate inference, and identify the main technical and clinical challenges. We provide a first comprehensive review to inform how to quantify, communicate and use uncertainty in medical synthesis applications

Uncertainty in multitask learning: joint representations for probabilistic MR-only radiotherapy planning

Multi-task neural network architectures provide a mechanism that jointly integrates information from distinct sources. It is ideal in the context of MR-only radiotherapy planning as it can jointly regress a synthetic CT (synCT) scan and segment organs-at-risk (OAR) from MRI. We propose a probabilistic multi-task network that estimates: 1) intrinsic uncertainty through a heteroscedastic noise model for spatially-adaptive task loss weighting and 2) parameter uncertainty through approximate Bayesian inference. This allows sampling of multiple segmentations and synCTs that share their network representation. We test our model on prostate cancer scans and show that it produces more accurate and consistent synCTs with a better estimation in the variance of the errors, state of the art results in OAR segmentation and a methodology for quality assurance in radiotherapy treatment planning.Comment: Early-accept at MICCAI 2018, 8 pages, 4 figure

Energy properness and Sasakian-Einstein metrics

In this paper, we show that the existence of Sasakian-Einstein metrics is closely related to the properness of corresponding energy functionals. Under the condition that admitting no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of Sasakian-Einstein metric implies a Moser-Trudinger type inequality. At the end of this paper, we also obtain a Miyaoka-Yau type inequality in Sasakian geometry.Comment: 27 page

On the degrees of freedom of a semi-Riemannian metric

A semi-Riemannian metric in a n-manifold has n(n-1)/2 degrees of freedom, i.e. as many as the number of components of a differential 2-form. We prove that any semi-Riemannian metric can be obtained as a deformation of a constant curvature metric, this deformation being parametrized by a 2-for

Bi-Legendrian manifolds and paracontact geometry

We study the interplays between paracontact geometry and the theory of bi-Legendrian manifolds. We interpret the bi-Legendrian connection of a bi-Legendrian manifold M as the paracontact connection of a canonical paracontact structure induced on M and then we discuss many consequences of this result both for bi-Legendrian and for paracontact manifolds. Finally new classes of examples of paracontact manifolds are presented.Comment: to appear in Int. J. Geom. Meth. Mod. Phy

2009-2010 Drake Memorial Library Annual Report

The 2009-2010 annual report of Drake Memorial Library of The College at Brockport, as compiled by Mary Jo Orzech, Bob Cushman, Pam O\u27Sullivan and Jennifer Smathers with contributions from the Drake Faculty and Staff

Stochastic Filter Groups for Multi-Task CNNs: Learning Specialist and Generalist Convolution Kernels

The performance of multi-task learning in Convolutional Neural Networks (CNNs) hinges on the design of feature sharing between tasks within the architecture. The number of possible sharing patterns are combinatorial in the depth of the network and the number of tasks, and thus hand-crafting an architecture, purely based on the human intuitions of task relationships can be time-consuming and suboptimal. In this paper, we present a probabilistic approach to learning task-specific and shared representations in CNNs for multi-task learning. Specifically, we propose "stochastic filter groups'' (SFG), a mechanism to assign convolution kernels in each layer to "specialist'' or "generalist'' groups, which are specific to or shared across different tasks, respectively. The SFG modules determine the connectivity between layers and the structures of task-specific and shared representations in the network. We employ variational inference to learn the posterior distribution over the possible grouping of kernels and network parameters. Experiments demonstrate that the proposed method generalises across multiple tasks and shows improved performance over baseline methods.Comment: Accepted for oral presentation at ICCV 201