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Knowledge bases often consist of facts which are harvested from a variety of sources, many of which are noisy and some of which conflict, resulting in a level of uncertainty for each triple. Knowledge bases are also often incomplete,…
We study embedding a subset $K$ of the unit sphere to the Hamming cube $\{-1,+1\}^m$. We characterize the tradeoff between distortion and sample complexity $m$ in terms of the Gaussian width $\omega(K)$ of the set. For subspaces and several…
Although shape correspondence is a central problem in geometry processing, most methods for this task apply only to two-dimensional surfaces. The neglected task of volumetric correspondence--a natural extension relevant to shapes extracted…
Many innovative applications require establishing correspondences among 3D geometric objects. However, the countless possible deformations of smooth surfaces make shape matching a challenging task. Finding an embedding to represent the…
Learning the embedding space, where semantically similar objects are located close together and dissimilar objects far apart, is a cornerstone of many computer vision applications. Existing approaches usually learn a single metric in the…
Transformer models learn to encode and decode an input text, and produce contextual token embeddings as a side-effect. The mapping from language into the embedding space maps words expressing similar concepts onto points that are close in…
The study of neural representations, both in biological and artificial systems, is increasingly revealing the importance of geometric and topological structures. Inspired by this, we introduce Event2Vec, a novel framework for learning…
Associated to any finite metric space are a large number of objects and quantities which provide some degree of structural or geometric information about the space. In this paper we show that in the setting of subsets of weighted Hamming…
We present MMbeddings, a probabilistic embedding approach that reinterprets categorical embeddings through the lens of nonlinear mixed models, effectively bridging classical statistical theory with modern deep learning. By treating…
We survey connections between the theory of bi-Lipschitz embeddings and the Sparsest Cut Problem in combinatorial optimization. The story of the Sparsest Cut Problem is a striking example of the deep interplay between analysis, geometry,…
Hyperbolic space is a geometry that is known to be well-suited for representation learning of data with an underlying hierarchical structure. In this paper, we present a novel hyperbolic distribution called \textit{pseudo-hyperbolic…
Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with…
Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case…
Geospatial Knowledge Graphs (GeoKGs) model geoentities (e.g., places and natural features) and spatial relationships in an interconnected manner, providing strong knowledge support for geographic applications, including data retrieval,…
In light of the inherent entailment relations between images and text, hyperbolic point vector embeddings, leveraging the hierarchical modeling advantages of hyperbolic space, have been utilized for visual semantic representation learning.…
Phylogenetic models have polynomial parametrization maps. For symmetric group-based models, Matsen studied the polynomial inequalities that characterize the joint probabilities in the image of these parametrizations. We employ this…
Recent years brought advancements in using neural networks for representation learning of various language or visual phenomena. New methods freed data scientists from hand-crafting features for common tasks. Similarly, problems that require…
We study a variant of the median problem for a collection of point sets in high dimensions. This generalizes the geometric median as well as the (probabilistic) smallest enclosing ball (pSEB) problems. Our main objective and motivation is…
We propose a learning framework to find the representation of a robot's kinematic structure and motion embedding spaces using graph neural networks (GNN). Finding a compact and low-dimensional embedding space for complex phenomena is a key…
Representing words by vectors, or embeddings, enables computational reasoning and is foundational to automating natural language tasks. For example, if word embeddings of similar words contain similar values, word similarity can be readily…