John Cobb

Paul Breiding, John Cobb, Aviva K. Englander et al.

Many hypersurfaces in algebraic geometry, such as discriminants, arise as the projection of another variety. The real complement of such a hypersurface partitions its ambient space into open regions. In this paper, we propose a new method for computing these regions. Existing methods for computing regions require the explicit equation of the hypersurface as input. However, computing this equation by elimination can be computationally demanding or even infeasible. Our approach instead derives from univariate interpolation by computing the intersection of the hypersurface with a line. Such an intersection can be done using so-called pseudo-witness sets without computing a defining equation for the hypersurface - we perform elimination without actually eliminating. We implement our approach in a forthcoming Julia package and demonstrate, on several examples, that the resulting algorithm accurately recovers all regions of the real complement of a hypersurface.

John Cobb, Thomas Gebhart

Many inference tasks on knowledge graphs, including relation prediction, operate on knowledge graph embeddings -- vector representations of the vertices (entities) and edges (relations) that preserve task-relevant structure encoded within the underlying combinatorial object. Such knowledge graph embeddings can be modeled as an approximate global section of a cellular sheaf, an algebraic structure over the graph. Using the diffusion dynamics encoded by the corresponding sheaf Laplacian, we optimally propagate known embeddings of a subgraph to inductively represent new entities introduced into the knowledge graph at inference time. We implement this algorithm via an efficient iterative scheme and show that on a number of large-scale knowledge graph embedding benchmarks, our method is competitive with -- and in some scenarios outperforms -- more complex models derived explicitly for inductive knowledge graph reasoning tasks.