An Incidence Geometry approach to Dictionary Learning

We study the Dictionary Learning (aka Sparse Coding) problem of obtaining a sparse representation of data points, by learning \emph{dictionary vectors} upon which the data points can be written as sparse linear combinations. We view this problem from a geometry perspective as the spanning set of a subspace arrangement, and focus on understanding the case when the underlying hypergraph of the subspace arrangement is specified. For this Fitted Dictionary Learning problem, we completely characterize the combinatorics of the associated subspace arrangements (i.e.\ their underlying hypergraphs). Specifically, a combinatorial rigidity-type theorem is proven for a type of geometric incidence system. The theorem characterizes the hypergraphs of subspace arrangements that generically yield (a) at least one dictionary (b) a locally unique dictionary (i.e.\ at most a finite number of isolated dictionaries) of the specified size. We are unaware of prior application of combinatorial rigidity techniques in the setting of Dictionary Learning, or even in machine learning. We also provide a systematic classification of problems related to Dictionary Learning together with various algorithms, their assumptions and performance

Geometry of Point-Hyperplane and Spherical Frameworks

In this thesis we show that the infinitesimal rigidity of point-hyperplane frameworks in Euclidean spaces is equivalent to the infinitesimal rigidity of bar-joint frameworks in spherical spaces with a set of joints (corresponding to the hyperplanes) located on a hyperplane. This is done by comparing the rigidity matrix of Euclidean point-hyperplane frameworks and the rigidity matrix of spherical frameworks. This result clearly shows how the first-order rigidity in projective spaces and Euclidean spaces are globally connected. This geometrically significant result is central to the thesis. This result leads to the equivalence of the first-order rigidity of point-hyperplane frameworks with that of bar-joint frameworks with a set of joints in a hyperplane in a Euclidean space (joint work). We also study the rigidity of point-hyperplane frameworks and characterize their rigidity in Euclidean spaces. We next highlight the relationship between point-line frameworks and slider mechanisms in the plane. Point-line frameworks are used to model various types of slider mechanisms. A combinatorial characterization of the rigidity of pinned-slider frameworks in the plane is derived directly as an immediate consequence of the analogous result for pinned bar-joint frameworks in the plane. Using fixed-normal point-line frameworks, we model a second type of slider system in which the slider directions do not change. Also, a third type of slider mechanism is introduced in which the sliders may only rotate around a fixed point but do not translate. This slider mechanism is defined using point-line frameworks with rotatory lines (no translational motion of the lines is allowed). A combinatorial characterization of the generic rigidity of these frameworks is coauthored in a joint work. Then we introduce point-hyperplane tensegrity frameworks in Euclidean spaces. We investigate the rigidity and the infinitesimal rigidity of these frameworks using tensegrity frameworks in spherical spaces. We characterize these different types of rigidity for point-hyperplane tensegrity frameworks and show how these types of rigidity are linked together. This leads to a characterization of the rigidity of a broader class of slider mechanisms in which sliders may move under variable distance constraints rather than fixed-distance constraints. Finally we investigate body-cad constraints in the plane. A combinatorial characterization of their generic infinitesimal rigidity is given. We show how angular constraints are related to non-angular constraints. This leads to a combinatorial result about the rigidity of a specific class of body-bar frameworks with point-point coincidence constraints in the space

Rigidity through a Projective Lens

In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar−joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body−hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas

The combinatorial geometry of stresses in frameworks

In this paper we formulate and prove necessary and sufficient geometric conditions for existence of generic tensegrities in the plane for arbitrary graphs. The conditions are written in terms of "meet-join" relations for the configuration spaces of fixed points and non-fixed lines through fixed points

Virtual articulation and kinematic abstraction in robotics

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 279-292).This thesis presents the theory, implementation, novel applications, and experimental validation of a general-purpose framework for applying virtual modifications to an articulated robot, or virtual articulations. These can homogenize various aspects of a robot and its task environment into a single unified model which is both qualitatively high-level and quantitatively functional. This is the first framework designed specifically for the mixed real/virtual case. It supports arbitrary topology spatial kinematics, a broad catalog of joints, on-line structure changes, interactive kinostatic simulation, and novel kinematic abstractions, where complex subsystems are simplified with virtual replacements in both space and time. Decomposition algorithms, including a novel method of hierarchical subdivision, enable scaling to large closed-chain mechanisms with 100s of joints. Novel applications are presented in two areas of current interest: operating high- DoF kinematic manipulation and inspection tasks, and analyzing reliable kinostatic locomotion strategies based on compliance and proprioception. In both areas virtual articulations homogeneously model the robot and its task environment, and abstractions structure complex models. For high-DoF operations the operator attaches virtual joints as a novel interface metaphor to define task motion and to constrain coordinated motion (by virtually closing kinematic chains); virtual links can represent task frames or serve as intermediate connections for virtual joints. For compliant locomotion, virtual articulations model relevant compliances and uncertainties, and temporal abstractions model contact state evolution.(cont.) Results are presented for experiments with two separate robotic systems in each area. For high-DoF operations, NASA/JPL's 36 DoF ATHLETE performs previously challenging coordinated manipulation/inspection moves, and a novel large-scale (100s of joints) simulated modular robot is conveniently operated using spatial abstractions. For compliant locomotion, two experiments are analyzed that each achieve high reliability in uncertain tasks using only compliance and proprioception: a novel vertical structure climbing robot that is 99.8% reliable in over 1000 motions, and a mini-humanoid that steps up an uncertain height with 90% reliability in 80 trials. In both cases virtual articulation models capture the essence of compliant/proprioceptive strategies at a higher level than basic physics, and enable quantitative analyses of the limits of tolerable uncertainty that compare well to experiment.by Marsette Arthur Vona, III.Ph.D

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum