Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach

Joint with Federico López, Beatrice Pozzetti, Michael Strube, and Anna Wienhard

Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learn-ing applications. We propose the systematic use of symmetric spaces in representation learning, a class encompassing many of the previously used embedding targets. This enables us to introduce anew method, the use of Finsler metrics integrated in a Riemannian optimization scheme, that better adapts to dissimilar structures in the graph. We develop a tool to analyze the embeddings and infer structural properties of the data sets. For implementation, we choose Siegel spaces, a versatile family of symmetric spaces. Our approach out-performs competitive baselines for graph reconstruction tasks on various synthetic and real-world datasets. We further demonstrate its applicability on two downstream tasks, recommender systems and node classification.

Ray-marching Thurston Geometries

Joint work with Remi Coulon, Sabetta Matsumoto and Henry Segerman.

We describe algorithms that produce accurate real-time interactive in-space views of the eight Thurston geometries using ray-marching. We give a theoretical framework for our algorithms, independent of the geometry involved. In addition to scenes within a geometry X, we also consider scenes within quotient manifolds and orbifolds X/Γ. We adapt the Phong lighting model to non-euclidean geometries. The most difficult part of this is the calculation of light intensity, which relates to the area density of geodesic spheres. We also give extensive practical details for each geometry.

Algebraic Number Starscapes

Joint with Edmund Harris and Kate Stange

We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by these images, called algebraic starscapes, we describe the geometry of the map from the coefficient space of polynomials to the root space, focussing on the quadratic and cubic cases. The geometry describes and explains notable features of the illustrations, and motivates a geometric-minded recasting of fundamental results in the Diophantine approximation of the complex plane. The images provide a case-study in the symbiosis of illustration and research, and an entry-point to geometry and number theory for a wider audience. The paper is written to provide an accessible introduction to the study of homogeneous geometry and Diophantine approximation. We investigate the homogeneous geometry of root and coefficient spaces under the natural PSL(2;ℂ) action, especially in degrees 2 and 3. We rediscover the quadratic and cubic root formulas as isometries, and determine when the map sending certain families of polynomials to their complex roots (our starscape images) are embeddings. We consider complex Diophantine approximation by quadratic irrationals, in terms of hyperbolic distance and the discriminant as a measure of arithmetic height. We recover the quadratic case of results of Bugeaud and Evertse, and give some geometric explanation for the dichotomy they discovered (Bugeaud, Y. and Evertse, J.-H., Approximation of complex algebraic numbers by algebraic numbers of bounded degree, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), no. 2, 333-368). Our statements go a little further in distinguishing approximability in terms of whether the target or approximations lie on rational geodesics. The paper comes with accompanying software, and finishes with a wide variety of open problems.

Families of Geometries

Geometric Transition is a continuously varying family of homogeneous spaces which abruptly changes isomorphism type. The ideas of transitional geometry trace all the way back to the original work of Bolyai, who in his construction of hyperbolic geometry described a limiting procedure to recover the standard theorems and relations of Euclidean geometry/trigonometry. Beginning wtih the work of Klein, the usual description of a transition of geometries is as a conjugacy limit inside of some ambient geometry, usually projective space. The choice of such an ambient space affects both the results (which geometries limit to which others), and the methods of proof (which geometric techniques can be used to show a particular transition is actually continuous.) This preprint recounts work from my thesis, constructing an alternative formalism for discussing geometric transitions intrinsically, without need for an ambient space.

Currently being expanded on from thesis work.

Chabauty Compactification of the Orthogonal Groups

The space of closed subsets of a compact metric space $X$ is itself naturally a compact metric space under the Hausdorff metric. With care, this construction generalizes to broader contexts; in particular the Lie subgroups of a given (not necessarily compact) Lie group $G$ are topologized by the Chabauty topology. This Chabauty space of $G$ allows us to reason topologically about the relation between different isomorphism types of subgroups of $G$, and is fundamental to the study of limits of subgeometries / subgroups in geometric topology. The collection of orthogonal groups in $\mathsf{GL}(n,\mathbb{R})$ is a union of components (in fact, conjugacy classes) determined by signature. The conjugacy limits of the orthogonal groups lie in the Chabauty closure of this space, whose topology we describe here. In particular, we investigate in detail a slice of this space, the Chabauty compactification of the space of orthogonal groups with respect to diagonal quadratic forms. Perhaps surprisingly the compactification is a manifold, succintly described as the (maximal) de Concini Procesi wonderful compactification of the coordinate hyperplane arrangement in $\mathbb{R}\mathsf{P}^{n-1}$. The original conjugacy classes of orthogonal groups form top dimensional permutohedral cells in this manifold, and the lower dimensional faces / their combinatorics parameterize all limits / their relationships.

Currrently being expanded from Thesis

Riemannian Limits of the Product Geometries

Hyperbolic geometry degenrates to Euclidean within projective geometry, as first noticed by Klein. More generally, up to finite index all eight Thurston geometries can be constructed as subgeometries of $\mathbb{R}\mathsf{P}^3$ and it is natural to ask when does Thurston geometry X have a conjugacy limit containing Thurston geometry Y? In this preprint we fill in the final two open cases in this $8\times 8$ grid: showing that the product geometries $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$ do not limit to $\mathsf{Nil}$.

The Heisenbgerg Plane

A manifold $M$ admits a geometric structure locally modeled on a homogeneous space $(G,X)$ if it can be built out of little chunks of $X$ glued together isometrically by maps in $G$. Classical examples are Euclidean and Hyperbolic structures, but more exotic strutctures such as Convex Real Projecti ve, Anti de Sitter, Half Pipe, and Heisenberg structures are also important in geometric topology. This paper studies the geometry given by the projective action of the Heisenberg group on the plane. The closed orbifolds admitting Heisenberg structures are those with vanishing Euler characteristic and singularities of order at most two, and the corresponding deformation spaces are computed. Heisenberg geometry is of interest as a transitional geometry between any two of the constant-curvature geometries $\mathbb{S}^2,\mathbb{E}^2,\mathbb{H}^2$, and regenerations of Heisenberg tori into these geometries are completely described.

Nil Geometry

Joint work with Remi Coulon, Sabetta Matsumoto and Henry Segerman.

This paper provides an account of the intrinsic geometry of Nil. In particular, we focus on the the location of conjugate points in the Riemannian metric, and their effect on internal view. We also discuss the intrinsic geometry of closed Nil manifolds built as a torus bundle over the circle with Dehn twist monodromy. This paper has been submitted to the proceedings of the Bridges 2020 conference.

Sol Geometry

Joint work with Remi Coulon, Sabetta Matsumoto and Henry Segerman.

This paper provides an expository account of the intrinsic geometry of Sol. In particular, we focus how the spiraling nature of geodesics strongly affects vision, even when looking at simple idealized objects such as the $xy$ plane. We also consider discuss the intrinsic geometry of closed Sol manifolds built as a torus bundle over the circle with anosov monodromy. This paper has been submitted to the proceedings of the Bridges 2020 conference.

PhD Thesis

This thesis details the results of four interrelated projects. The first of these presents a new proof of the theorem of Cooper, Danciger and Wienhard classifying the limits under conjugacy of the orthogonal groups in GL(n; R). The second provides a detailed investigation into Heisenberg geometry, which is the maximally degenerate such limit in dimension two. The remaining two projects concern understanding geometric transitions which do not occur naturally as limits under conjugacy in some ambient geometry. The third project describes a new degeneration of complex hyperbolic space, formed by degenerating the complex numbers as a real algebra, into the algebra R+R. Inspired by this example, the final project attempts to build the beginnings of a framework for studying transitions between geometries abstractly. As a first application of this, we generalize the previous result and describe a collection of new geometric transitions, defined by constructing analogs of familiar geometries (projective geometry, hyperbolic geometry, etc) over real algebras, and then allowing this algebra to vary.

In Progress

Embedding Knowledge Graphs in Symmetric Positive Definite Manifolds

Joint with Federico López, Beatrice Pozzetti, Michael Strube, and Anna Wienhard

Geometric Dynamics, Gauge Theory and Sub Riemannian Geometry of Deformable Systems in Classical Mechanics

Joint with Brian Day and Sabetta Matsumoto.

A paper laying out precisely a variety of mathematical tools that are useful in the physics of deformable systems, from swimming to robotics.

Optimal Strokes for Stokesian and Geometric Swimmers

Joint with Brian Day and Sabetta Matsumoto.

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Using sub-Riemannian geometry to derive the system of differential equations describing the optimal motion of certain idealized microscopic organisims in Stokes flow, as well as a means of gradient descent constructed to respect this sub-Riemannian structure to investigate these paths numerically.