arXiv subjects

Meera Sitharam

Publications and source records attributed to Meera Sitharam.

At least 37 records · Page 2Linked to original sources

arXiv2018

Corner-Sharing Tetrahedra for Modeling Micro-Structure

State-of-the-art representations of volumetric multi-scale shape and structure can be classified into three broad categories: continuous, continuous-from-discrete, and discrete representations. We propose modeling micro-structure with a class of discrete Corner-Sharing Tetrahedra (CoSTs). CoSTs can represent bar-joint, tensegrity, line-incidence, and similar constraint systems that capture local physical constraints and global multi-scale properties for design and analysis. The paper develops a palette of simple geometry processing operations on CoSTs including graph manipulation, hierarchical refinement, randomization, and generating associated continuous representations.

cs.CG↗

arXiv2016

Combinatorial rigidity of Incidence systems and Application to Dictionary learning

Given a hypergraph $H$ with $m$ hyperedges and a set $Q$ of $m$ \emph{pinning subspaces}, i.e.\ globally fixed subspaces in Euclidean space $\mathbb{R}^d$, a \emph{pinned subspace-incidence system} is the pair $(H, Q)$, with the constraint that each pinning subspace in $Q$ is contained in the subspace spanned by the point realizations in $\mathbb{R}^d$ of vertices of the corresponding hyperedge of $H$. This paper provides a combinatorial characterization of pinned subspace-incidence systems that are \emph{minimally rigid}, i.e.\ those systems that are guaranteed to generically yield a locally unique realization. Pinned subspace-incidence systems have applications in the \emph{Dictionary Learning (aka sparse coding)} problem, i.e.\ the problem of obtaining a sparse representation of a given set of data vectors by learning \emph{dictionary vectors} upon which the data vectors can be written as sparse linear combinations. Viewing the dictionary vectors from a geometry perspective as the spanning set of a subspace arrangement, the result gives a tight bound on the number of dictionary vectors for sufficiently randomly chosen data vectors, and gives a way of constructing a dictionary that meets the bound. For less stringent restrictions on data, but a natural modification of the dictionary learning problem, a further dictionary learning algorithm is provided. Although there are recent rigidity based approaches for low rank matrix completion, we are unaware of prior application of combinatorial rigidity techniques in the setting of Dictionary Learning. We also provide a systematic classification of problems related to dictionary learning together with various algorithms, their assumptions and performance.

cs.CG↗

arXiv2016

Symmetry in Sphere-based Assembly Configuration Spaces

Many remarkably robust, rapid and spontaneous self-assembly phenomena in nature can be modeled geometrically starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly processes. Since spheres within bunches could be identical and bunches could be identical as well, the underlying symmetry groups could be of large order that grows with the number of participating spheres and bunches. Thus, understanding symmetries and associated isomorphism classes of microstates correspond to various types of macrostates can significantly reduce the complexity of computing entropy and free energy, as well as paths and kinetics, in high dimensional configuration spaces. In addition, a precise understanding of symmetries is crucial for giving provable guarantees of algorithmic accuracy and efficiency in such computations. In particular, this may aid in predicting crucial assembly-driving interactions. This is a primarily expository paper that develops a novel, original framework for dealing with symmetries in configuration spaces of assembling spheres with the following goals. (1) We give new, formal definitions of various concepts relevant to sphere-based assembly that occur in previous work, and in turn, formal definitions of their relevant symmetry groups leading to the main theorem concerning their symmetries. These previously developed concepts include, for example, (a) assembly configuration spaces, (b) stratification of assembly configuration space into regions defined by active constraint graphs, (c) paths through the configurational regions, and (d) coarse assembly pathways. (2) We demonstrate the new symmetry concepts to compute sizes and numbers of orbits in two example settings appearing in previous work. (3) We give formal statements of a variety of open problems and challenges using the new conceptual definitions.

math.CO↗

arXiv2015

Optimal Decomposition and Recombination of Isostatic Geometric Constraint Systems for Designing Layered Materials

Optimal recursive decomposition (or DR-planning) is crucial for analyzing, designing, solving or finding realizations of geometric constraint sytems. While the optimal DR-planning problem is NP-hard even for general 2D bar-joint constraint systems, we describe an O(n^3) algorithm for a broad class of constraint systems that are isostatic or underconstrained. The algorithm achieves optimality by using the new notion of a canonical DR-plan that also meets various desirable, previously studied criteria. In addition, we leverage recent results on Cayley configuration spaces to show that the indecomposable systems---that are solved at the nodes of the optimal DR-plan by recombining solutions to child systems---can be minimally modified to become decomposable and have a small DR-plan, leading to efficient realization algorithms. We show formal connections to well-known problems such as completion of underconstrained systems. Well suited to these methods are classes of constraint systems that can be used to efficiently model, design and analyze quasi-uniform (aperiodic) and self-similar, layered material structures. We formally illustrate by modeling silica bilayers as body-hyperpin systems and cross-linking microfibrils as pinned line-incidence systems. A software implementation of our algorithms and videos demonstrating the software are publicly available online (visit http://cise.ufl.edu/~tbaker/drp/index.html.)

cs.CG↗

arXiv2015

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.

cs.CG↗

arXiv2015

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.

cs.LG↗

arXiv2015

Combinatorial rigidity and independence of generalized pinned subspace-incidence constraint systems

Given a hypergraph $H$ with $m$ hyperedges and a set $X$ of $m$ \emph{pins}, i.e.\ globally fixed subspaces in Euclidean space $\mathbb{R}^d$, a \emph{pinned subspace-incidence system} is the pair $(H, X)$, with the constraint that each pin in $X$ lies on the subspace spanned by the point realizations in $\mathbb{R}^d$ of vertices of the corresponding hyperedge of $H$. We are interested in combinatorial characterization of pinned subspace-incidence systems that are \emph{minimally rigid}, i.e.\ those systems that are guaranteed to generically yield a locally unique realization. As is customary, this is accompanied by a characterization of generic independence as well as rigidity. In a previous paper \cite{sitharam2014incidence}, we used pinned subspace-incidence systems towards solving the \emph{fitted dictionary learning} problem, i.e.\ dictionary learning with specified underlying hypergraph, and gave a combinatorial characterization of minimal rigidity for a more restricted version of pinned subspace-incidence system, with $H$ being a uniform hypergraph and pins in $X$ being 1-dimension subspaces. Moreover in a recent paper \cite{Baker2015}, the special case of pinned line incidence systems was used to model biomaterials such as cellulose and collagen fibrils in cell walls. In this paper, we extend the combinatorial characterization to general pinned subspace-incidence systems, with $H$ being a non-uniform hypergraph and pins in $X$ being subspaces with arbitrary dimension. As there are generally many data points per subspace in a dictionary learning problem, which can only be modeled with pins of dimension larger than $1$, such an extension enables application to a much larger class of fitted dictionary learning problems.

cs.CG↗

arXiv2014

Best of Both Worlds: Uniform sampling in Cartesian and Cayley Molecular Assembly Configuration Space

EASAL (efficient atlasing and sampling of assembly landscapes) is a recently reported geometric method for representing, visualizing, sampling and computing integrals over the potential energy landscape tailored for small molecular assemblies. EASAL's efficiency arises from the fact that small assembly landscapes permit the use of so-called Cayley parameters (inter-atomic distances) for geometric representation and sampling of the assembly configuration space regions; this results in their isolation, convexification, customized sampling and systematic traversal using a comprehensive topological roadmap, ensuring reasonable coverage of crucial but narrow regions of low effective dimension. However, this alone is inadequate for accurate computation of configurational entropy and other integrals, required for estimation of both free energy and kinetics - where it is essential to obtain uniform sampling in appropriate cartesian or moduli space parameterization. Standard adjustment of Cayley sampling via the Jacobian of the map between the two parameterizations is fraught with challenges stemming from an illconditioned Jacobian. This paper formalizes and analyzes these challenges to provide modifications to EASAL that secure the advantages of Cayley sampling while ensuring certain minimum distance and coverage relationships between sampled configurations - in Cartesian space. The modified EASAL's performance is compared with the basic EASAL and the data are presented for Human and Rat Islet Amylin Polypeptide (HiAPP, PDB-2KJ7 and RiAPP PDB-2KB8) dimerization (the two differ in only 6 out of 37 residues, but the former aggregates into fibrils, while the latter does not).

physics.chem-ph↗

arXiv2014

Cayley Analysis of Mechanism Configuration Spaces using CayMos: Software Functionalities and Architecture

For a common class of 2D mechanisms called 1-dof tree decomposable linkages, we present a software CayMos which uses new theoretical results to implement efficient algorithmic solutions for: (a) meaningfully representing and visualizing the connected components in the Euclidean realization space; (b) finding a path of continuous motion between two realizations in the same connected component, with or without restricting the realization type (sometimes called orientation type); (c) finding two ``closest'' realizations in different connected components.

cs.CG↗

arXiv2013

Nucleation-free $3D$ 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 $2D$.

cs.CG↗

arXiv2013

Maxwell-independence: a new rank estimate for 3D rigidity matroids

The problem of combinatorially determining the rank of the 3-dimensional bar-joint {\em rigidity matroid} of a graph is an important open problem in combinatorial rigidity theory. Maxwell's condition states that the edges of a graph $G=(V, E)$ are {\em independent} in its $d$-dimensional generic rigidity matroid only if $(a)$ the number of edges $|E|$ $\le$ $d|V| - {d+1\choose 2}$, and $(b)$ this holds for every induced subgraph with at least $d$ vertices. We call such graphs {\em Maxwell-independent} in $d$ dimensions. Laman's theorem shows that the converse holds for $d=2$ and thus every maximal Maxwell-independent set of $G$ has size equal to the rank of the 2-dimensional generic rigidity matroid. While this is false for $d=3$, we show that every maximal, Maxwell-independent set of a graph $G$ has size at least the rank of the 3-dimensional generic rigidity matroid of $G$. This answers a question posed by Tibór Jordán at the 2008 rigidity workshop at BIRS \cite{bib:birs}. Along the way, we construct subgraphs (1) that yield alternative formulae for a rank upper bound for Maxwell-independent graphs and (2) that contain a maximal (true) independent set. We extend this bound to special classes of non-Maxwell-independent graphs. One further consequence is a simpler proof of correctness for existing algorithms that give rank bounds.

cs.CG↗

arXiv2012

Cayley Configuration Spaces of 1-dof Tree-decomposable Linkages, Part II: Combinatorial Characterization of Complexity

We continue to study Cayley configuration spaces of 1-dof linkages in 2D begun in Part I of this paper, i.e. the set of attainable lengths for a non-edge. In Part II, we focus on the algebraic complexity of describing endpoints of the intervals in the set, i.e., the Cayley complexity. Specifically, We focus on Cayley configuration spaces of a natural class of 1-dof linkages, called 1-dof tree-decomposable linkages. The underlying graphs G satisfy the following: for some base non-edge f, G \cup f is quadratic-radically solvable (QRS), meaning that G \cup f is minimally rigid, and given lengths \bar{l} of all edges, the corresponding linkage (G \cup f, \bar{l}) can be simply realized by ruler and compass starting from f. It is clear that the Cayley complexity only depends on the graph G and possibly the non-edge f. Here we ask whether the Cayley complexity depends on the choice of a base non-edge f. We answer this question in the negative, thereby showing that low Cayley complexity is a property of the graph G (independent of the non-edge f). Then, we give a simple characterization of graphs with low Cayley complexity, leading to an efficient algorithmic characterization, i.e. an efficient algorithm for recognizing such graphs. Next, we show a surprising result that (graph) planarity is equivalent to low Cayley complexity for a natural subclass of 1-dof triangle-decomposable graphs. While this is a finite forbidden minor graph characterization of low Cayley complexity, we provide counterexamples showing impossibility of such finite forbidden minor characterizations when the above subclass is enlarged.

cs.CG↗

arXiv2011

Learning Hierarchical Sparse Representations using Iterative Dictionary Learning and Dimension Reduction

This paper introduces an elemental building block which combines Dictionary Learning and Dimension Reduction (DRDL). We show how this foundational element can be used to iteratively construct a Hierarchical Sparse Representation (HSR) of a sensory stream. We compare our approach to existing models showing the generality of our simple prescription. We then perform preliminary experiments using this framework, illustrating with the example of an object recognition task using standard datasets. This work introduces the very first steps towards an integrated framework for designing and analyzing various computational tasks from learning to attention to action. The ultimate goal is building a mathematically rigorous, integrated theory of intelligence.

cs.LG↗

arXiv2010

Body-and-cad Geometric Constraint Systems

Motivated by constraint-based CAD software, we develop the foundation for the rigidity theory of a very general model: the body-and-cad structure, composed of rigid bodies in 3D constrained by pairwise coincidence, angular and distance constraints. We identify 21 relevant geometric constraints and develop the corresponding infinitesimal rigidity theory for these structures. The classical body-and-bar rigidity model can be viewed as a body-and-cad structure that uses only one constraint from this new class. As a consequence, we identify a new, necessary, but not sufficient, counting condition for minimal rigidity of body-and-cad structures: nested sparsity. This is a slight generalization of the well-known sparsity condition of Maxwell.

cs.CG↗

arXiv2009

Tree Orbits under Permutation Group Action: Algorithm, Enumeration and Application to Viral Assembly

This paper uses combinatorics and group theory to answer questions about the assembly of icosahedral viral shells. Although the geometric structure of the capsid (shell) is fairly well understood in terms of its constituent subunits, the assembly process is not. For the purpose of this paper, the capsid is modeled by a polyhedron whose facets represent the monomers. The assembly process is modeled by a rooted tree, the leaves representing the facets of the polyhedron, the root representing the assembled polyhedron, and the internal vertices representing intermediate stages of assembly (subsets of facets). Besides its virological motivation, the enumeration of orbits of trees under the action of a finite group is of independent mathematical interest. If $G$ is a finite group acting on a finite set $X$, then there is a natural induced action of $G$ on the set $\mathcal{T}_X$ of trees whose leaves are bijectively labeled by the elements of $X$. If $G$ acts simply on $X$, then $|X| := |X_n| = n \cdot |G|$, where $n$ is the number of $G$-orbits in $X$. The basic combinatorial results in this paper are (1) a formula for the number of orbits of each size in the action of $G$ on $\mathcal{T}_{X_n}$, for every $n$, and (2) a simple algorithm to find the stabilizer of a tree $τ\in \mathcal{T}_X$ in $G$ that runs in linear time and does not need memory in addition to its input tree.

math.CO↗

arXiv2009

Characterizing graphs with convex and connected configuration spaces

We define and study exact, efficient representations of realization spaces Euclidean Distance Constraint Systems (EDCS), which includes Linkages and Frameworks. Each representation corresponds to a choice of Cayley parameters and yields a different parametrized configuration space. Significantly, we give purely graph-theoretic, forbidden minor characterizations that capture (i) the class of graphs that always admit efficient configuration spaces and (ii) the possible choices of representation parameters that yield efficient configuration spaces for a given graph. In addition, our results are tight: we show counterexamples to obvious extensions. This is the first step in a systematic and graded program of combinatorial characterizations of efficient configuration spaces. We discuss several future theoretical and applied research directions. Some of our proofs employ an unusual interplay of (a) classical analytic results related to positive semi-definiteness of Euclidean distance matrices, with (b) recent forbidden minor characterizations and algorithms related to the notion of d-realizability of EDCS. We further introduce a novel type of restricted edge contraction or reduction to a graph minor, a "trick" that we anticipate will be useful in other situations.

cs.CG↗

arXiv2008

Characterizing 1-Dof Henneberg-I graphs with efficient configuration spaces

We define and study exact, efficient representations of realization spaces of a natural class of underconstrained 2D Euclidean Distance Constraint Systems(EDCS) or Frameworks based on 1-dof Henneberg-I graphs. Each representation corresponds to a choice of parameters and yields a different parametrized configuration space. Our notion of efficiency is based on the algebraic complexities of sampling the configuration space and of obtaining a realization from the sample (parametrized) configuration. Significantly, we give purely combinatorial characterizations that capture (i) the class of graphs that have efficient configuration spaces and (ii) the possible choices of representation parameters that yield efficient configuration spaces for a given graph. Our results automatically yield an efficient algorithm for sampling realizations, without missing extreme or boundary realizations. In addition, our results formally show that our definition of efficient configuration space is robust and that our characterizations are tight. We choose the class of 1-dof Henneberg-I graphs in order to take the next step in a systematic and graded program of combinatorial characterizations of efficient configuration spaces. In particular, the results presented here are the first characterizations that go beyond graphs that have connected and convex configuration spaces.

cs.CG↗

arXiv2005

Real Mutually Unbiased Bases

We tabulate bounds on the optimal number of mutually unbiased bases in R^d. For most dimensions d, it can be shown with relatively simple methods that either there are no real orthonormal bases that are mutually unbiased or the optimal number is at most either 2 or 3. We discuss the limitations of these methods when applied to all dimensions, shedding some light on the difficulty of obtaining tight bounds for the remaining dimensions that have the form d=16n^2, where n can be any number. We additionally give a simpler, alternative proof that there can be at most d/2+1 real mutually unbiased bases in dimension d instead of invoking the known results on extremal Euclidean line sets by Cameron and Seidel, Delsarte, and Calderbank et al.

quant-ph↗