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Register a new userUniqueness of the Gaussian Orthogonal Ensemble
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A known result in random matrix theory states the following: Given a random Wigner matrix $X$ which belongs to the Gaussian Orthogonal Ensemble (GOE), then such matrix $X$ has an invariant distribution under orthogonal conjugations. The goal of this work is to prove the converse, that is, if $X$ is a symmetric random matrix such that it is invariant under orthogonal conjugations, then such matrix $X$ belongs to the GOE. We will prove this using some elementary properties of the characteristic function of random variables.
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We show that the squared maximal height of the top path among $N$ non-intersecting Brownian bridges starting and ending at the origin is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. This result can be thought of as a discrete version of K. Johanssons result that the supremum of the Airy$_2$ process minus a parabola has the Tracy-Widom GOE distribution, and as such it provides an explanation for how this distribution arises in models belonging to the KPZ universality class with flat initial data. The result can be recast in terms of the probability that the top curve of the stationary Dyson Brownian motion hits an hyperbolic cosine barrier.
Consider a $n times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(Delta_{i,n}, 1leq ileq p)$, properly rescaled, and eventually included in any neighbourhood of the support of Wigners semi-circle law, we prove that the related counting measures $({mathcal N}_n(Delta_{i,n}), 1leq ileq p)$, where ${mathcal N}_n(Delta)$ represents the number of eigenvalues within $Delta$, are asymptotically independent as the size $n$ goes to infinity, $p$ being fixed. As a consequence, we prove that the largest and smallest eigenvalues, properly centered and rescaled, are asymptotically independent; we finally describe the fluctuations of the condition number of a matrix from the GUE.
The work concerns the Zakai equations from nonlinear filtering problems of McKean-Vlasov stochastic differential equations with correlated noises. First, we establish the Kushner-Stratonovich equations, the Zakai equations and the distribution-dependent Zakai equations. And then, the pathwise uniqueness, uniqueness in joint law and uniqueness in law of weak solutions for the distribution-dependent Zakai equations are shown. Finally, we prove a superposition principle between the distribution-dependent Zakai equations and distribution-dependent Fokker-Planck equations. As a by-product, we give some conditions under which distribution-dependent Fokker-Planck equations have unique weak solutions.
Despite the astonishing performance of deep-learning based approaches for visual tasks such as semantic segmentation, they are known to produce miscalibrated predictions, which could be harmful for critical decision-making processes. Ensemble learning has shown to not only boost the performance of individual models but also reduce their miscalibration by averaging independent predictions. In this scenario, model diversity has become a key factor, which facilitates individual models converging to different functional solutions. In this work, we introduce Orthogonal Ensemble Networks (OEN), a novel framework to explicitly enforce model diversity by means of orthogonal constraints. The proposed method is based on the hypothesis that inducing orthogonality among the constituents of the ensemble will increase the overall model diversity. We resort to a new pairwise orthogonality constraint which can be used to regularize a sequential ensemble training process, resulting on improved predictive performance and better calibrated model outputs. We benchmark the proposed framework in two challenging brain lesion segmentation tasks --brain tumor and white matter hyper-intensity segmentation in MR images. The experimental results show that our approach produces more robust and well-calibrated ensemble models and can deal with challenging tasks in the context of biomedical image segmentation.
In this note, we prove that if $g$ is uniformly continuous in $z$, uniformly with respect to $(oo,t)$ and independent of $y$, the solution to the backward stochastic differential equation (BSDE) with generator $g$ is unique.
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