Jonathan Richard Shewchuk

Jonathan Richard Shewchuk

The restricted Delaunay triangulation of a closed surface $Σ$ and a finite point set $V \subset Σ$ is a subcomplex of the Delaunay tetrahedralization of $V$ whose triangles approximate $Σ$. It is well known that if $V$ is a sufficiently dense sample of a smooth $Σ$, then the union of the restricted Delaunay triangles is homeomorphic to $Σ$. We show that an $ε$-sample with $ε\leq 0.3245$ suffices. By comparison, Dey proves it for a $0.18$-sample; our improved sampling bound reduces the number of sample points required by a factor of $3.25$. More importantly, we improve a related sampling bound of Cheng et al. for Delaunay surface meshing, reducing the number of sample points required by a factor of $21$. The first step of our homeomorphism proof is particularly interesting: we show that for a $0.44$-sample, the restricted Voronoi cell of each site $v \in V$ is homeomorphic to a disk, and the orthogonal projection of the cell onto $T_vΣ$ (the plane tangent to $Σ$ at $v$) is star-shaped.

Jonathan Richard Shewchuk, Sagnik Bhattacharya

We characterize the geometry and topology of the set of all weight vectors for which a linear neural network computes the same linear transformation $W$. This set of weight vectors is called the fiber of $W$ (under the matrix multiplication map), and it is embedded in the Euclidean weight space of all possible weight vectors. The fiber is an algebraic variety that is not necessarily a manifold. We describe a natural way to stratify the fiber--that is, to partition the algebraic variety into a finite set of manifolds of varying dimensions called strata. We call this set of strata the rank stratification. We derive the dimensions of these strata and the relationships by which they adjoin each other. Although the strata are disjoint, their closures are not. Our strata satisfy the frontier condition: if a stratum intersects the closure of another stratum, then the former stratum is a subset of the closure of the latter stratum. Each stratum is a manifold of class $C^\infty$ embedded in weight space, so it has a well-defined tangent space and normal space at every point (weight vector). We show how to determine the subspaces tangent to and normal to a specified stratum at a specified point on the stratum, and we construct elegant bases for those subspaces. To help achieve these goals, we first derive what we call a Fundamental Theorem of Linear Neural Networks, analogous to what Strang calls the Fundamental Theorem of Linear Algebra. We show how to decompose each layer of a linear neural network into a set of subspaces that show how information flows through the neural network. Each stratum of the fiber represents a different pattern by which information flows (or fails to flow) through the neural network. The topology of a stratum depends solely on this decomposition. So does its geometry, up to a linear transformation in weight space.