On Flattenability of Graphs

We consider a generalization of the concept of $d$-flattenability of graphs - introduced for the $l_2$ norm by Belk and Connelly - to general $l_p$ norms, with integer $P$, $1 \le p < \infty$, though many of our results work for $l_\infty$ as well. The following results are shown for graphs $G$, using notions of genericity, rigidity, and generic $d$-dimensional rigidity matroid introduced by Kitson for frameworks in general $l_p$ norms, as well as the cones of vectors of pairwise $l_p^p$ distances of a finite point configuration in $d$-dimensional, $l_p$ space: (i) $d$-flattenability of a graph $G$ is equivalent to the convexity of $d$-dimensional, inherent Cayley configurations spaces for $G$, a concept introduced by the first author; (ii) $d$-flattenability and convexity of Cayley configuration spaces over specified non-edges of a $d$-dimensional framework are not generic properties of frameworks (in arbitrary dimension); (iii) $d$-flattenability of $G$ is equivalent to all of $G$'s generic frameworks being $d$-flattenable; (iv) existence of one generic $d$-flattenable framework for $G$ is equivalent to the independence of the edges of $G$, a generic property of frameworks; (v) the rank of $G$ equals the dimension of the projection of the $d$-dimensional stratum of the $l_p^p$ distance cone. We give stronger results for specific norms for $d = 2$: we show that (vi) 2-flattenable graphs for the $l_1$-norm (and $l_\infty$-norm) are a larger class than 2-flattenable graphs for Euclidean $l_2$-norm case and finally (vii) prove further results towards characterizing 2-flattenability in the $l_1$-norm. A number of conjectures and open problems are posed

Nucleation-free 3D3D rigidity

When all non-edge distances of a graph realized in $\mathbb{R}^{d}$ as a {\em bar-and-joint framework} are generically {\em implied} by the bar (edge) lengths, the graph is said to be {\em rigid} in $\mathbb{R}^{d}$. For $d=3$, characterizing rigid graphs, determining implied non-edges and {\em dependent} edge sets remains an elusive, long-standing open problem. One obstacle is to determine when implied non-edges can exist without non-trivial rigid induced subgraphs, i.e., {\em nucleations}, and how to deal with them. In this paper, we give general inductive construction schemes and proof techniques to generate {\em nucleation-free graphs} (i.e., graphs without any nucleation) with implied non-edges. As a consequence, we obtain (a) dependent graphs in $3D$ that have no nucleation; and (b) $3D$ nucleation-free {\em rigidity circuits}, i.e., minimally dependent edge sets in $d=3$. It additionally follows that true rigidity is strictly stronger than a tractable approximation to rigidity given by Sitharam and Zhou \cite{sitharam:zhou:tractableADG:2004}, based on an inductive combinatorial characterization. As an independently interesting byproduct, we obtain a new inductive construction for independent graphs in $3D$. Currently, very few such inductive constructions are known, in contrast to $2D$

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