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Geometric methods for solving open-world off-road navigation tasks, by learning occupancy and metric maps, provide good generalization but can be brittle in outdoor environments that violate their assumptions (e.g., tall grass).…

In this work we propose to combine the advantages of learningbased and combinatorial formalisms for 3D shape matching. While learningbased methods lead to state-of-the-art matching performance, they do not ensure geometric consistency, so…

Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion…

Let $\mathrm{PG}(3,q)$ be the projective space of dimension three over the finite field with $q$ elements. Consider a twisted cubic in $\mathrm{PG}(3,q)$. The structure of the point-plane incidence matrix in $\mathrm{PG}(3,q)$ with respect…

We endeavour to estimate numerous multi-dimensional means of various probability distributions on a common space based on independent samples. Our approach involves forming estimators through convex combinations of empirical means derived…

This paper is about models for a vector of probabilities whose elements must have a multiplicative structure and sum to 1 at the same time; in certain applications, as basket analysis, these models may be seen as a constrained version of…

We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an…

We consider the bit complexity of computing Chow forms and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational…

One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erd\H{o}s: Every set of points $E$ in a projective plane determines at least $|E|$ lines, unless all the points are contained in a…

One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this…

The purpose of this survey is to present analytic versions of the injectivity theorem and their applications. The proof of our injectivity theorems is based on a combination of the L^2-method for the dbar-equation and the theory of harmonic…

This note reports on some experiments, using a handful of standard automated reasoning tools, for exploring Steinitz-Rademacher polyhedra, which are models of a certain first-order theory of incidence structures. This theory and its models,…

We show a one-to-one correspondence between arrangements of d lines in the projective plane, and lines in P^{d-2}. We apply this correspondence to classify (3,q)-nets over the complex numbers for all q<=6. When q=6, we have twelve possible…

Oriented matroids (often called order types) are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a…

In this study, we propose a projection estimation method for large-dimensional matrix factor models with cross-sectionally spiked eigenvalues. By projecting the observation matrix onto the row or column factor space, we simplify factor…

Diffusion-based policies have shown remarkable capability in executing complex robotic manipulation tasks but lack explicit characterization of geometry and semantics, which often limits their ability to generalize to unseen objects and…

In this paper we will do the following: (1) show how to geometrically define multiplication, using only basic plane geometry, independently of area and any notion of similar triangles; (2) prove all the properties of multiplication using…

The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in…

The Expectation--Maximization (EM) algorithm is a simple meta-algorithm that has been used for many years as a methodology for statistical inference when there are missing measurements in the observed data or when the data is composed of…

We propose an iterative solution method for the 3D high-frequency Helmholtz equation that exploits a contour integral formulation of spectral projectors. In this framework, the solution in certain invariant subspaces is approximated by…