Hierarchical relations are prevalent and indispensable for organizing human knowledge captured by a knowledge graph (KG). The key property of hierarchical relations is that they induce a partial ordering over the entities, which needs to be modeled in order to allow for hierarchical reasoning.

GivenatrainingdatasetD =(X,y)offeaturesandcorresponding labels from {1, ..., T} classes,D is partitioned recursively to two subsets, according to classes, at each tree level until reaching leaf nodes with data from only one class. More concretely, initially, feature vectors for all samples are obtained (using a NN), then a class prototype is generated by averaging the feature vectors belonging to the same class for all classes.

We show that the matrix perspective function, which is jointly convex in the Cartesian product of a standard Euclidean vector space and a conformal space of symmetric matrices, has a proximity operator in an almost closed form.

Zhang et al. have recently shown that geodesic convex concave Riemannian problems always admit saddle-point solutions.

We consider the problem of estimating the Fréchet and conditional Fréchet mean from data taking values in separable metric spaces. Unlike Euclidean spaces, where well-established methods are available, there is no practical estimator that works universally for all metric spaces. Therefore, we introduce a computable estimator for the Fréchet mean based on random quantization techniques and establish its universal consistency across any separable metric spaces. Additionally, we propose another estimator for the conditional Fréchet mean, leveraging data-driven partitioning and quantization, and demonstrate its universal consistency when the output space is any Banach space.

Luigi Mangione trial date dispute emerges as defense calls Manhattan DA's proposed July 1 start date "unrealistic" due to conflicting federal trial in Brian Thompson case.