- Multifidelity Covariance Estimation through Regression on the Manifold of Symmetric Constructive Particular Matrices(arXiv)
Writer : Aimee Maurais, Terrence Alsup, Benjamin Peherstorfer, Youssef Marzouk
Summary : We introduce a multifidelity estimator of covariance matrices formulated as the answer to a regression drawback on the manifold of symmetric constructive particular matrices. The estimator is constructive particular by development, and the Mahalanobis distance minimized to acquire it possesses properties which allow sensible computation. We present that our manifold regression multifidelity (MRMF) covariance estimator is a most probability estimator below a sure error mannequin on manifold tangent house. Extra broadly, we present that our Riemannian regression framework encompasses present multifidelity covariance estimators constructed from management variates. We exhibit through numerical examples that our estimator can present important decreases, as much as one order of magnitude, in squared estimation error relative to each single-fidelity and different multifidelity covariance estimators. Moreover, preservation of constructive definiteness ensures that our estimator is suitable with downstream duties, akin to information assimilation and metric studying, through which this property is crucial.
2. Modeling Graphs Past Hyperbolic: Graph Neural Networks in Symmetric Constructive Particular Matrices(arXiv)
Writer : Wei Zhao, Federico Lopez, J. Maxwell Riestenberg, Michael Strube, Diaaeldin Taha, Steve Trettel
Summary : Current analysis has proven that alignment between the construction of graph information and the geometry of an embedding house is essential for studying high-quality representations of the info. The uniform geometry of Euclidean and hyperbolic areas permits for representing graphs with uniform geometric and topological options, akin to grids and hierarchies, with minimal distortion. Nevertheless, real-world graph information is characterised by a number of sorts of geometric and topological options, necessitating extra refined geometric embedding areas. On this work, we make the most of the Riemannian symmetric house of symmetric constructive particular matrices (SPD) to assemble graph neural networks that may robustly deal with complicated graphs. To do that, we develop an revolutionary library that leverages the SPD gyrocalculus instruments cite{lopez2021gyroSPD} to implement the constructing blocks of 5 widespread graph neural networks in SPD. Experimental outcomes exhibit that our graph neural networks in SPD considerably outperform their counterparts in Euclidean and hyperbolic areas, in addition to the Cartesian product thereof, on complicated graphs for node and graph classification duties. We launch the library and datasets at url{https://github.com/andyweizhao/SPD4GNNs}.
