arXiv subjects

Bala Krishnamoorthy

Publications and source records attributed to Bala Krishnamoorthy.

At least 37 records · Page 2Linked to original sources

arXiv2023

Robust Feasibility of Systems of Quadratic Equations Using Topological Degree Theory

We consider the problem of measuring the margin of robust feasibility of solutions to a system of nonlinear equations. We study the special case of a system of quadratic equations, which shows up in many practical applications such as the power grid and other infrastructure networks. This problem is a generalization of quadratically constrained quadratic programming (QCQP), which is NP-Hard in the general setting. We develop approaches based on topological degree theory to estimate bounds on the robustness margin of such systems. Our methods use tools from convex analysis and optimization theory to cast the problems of checking the conditions for robust feasibility as a nonlinear optimization problem. We then develop inner bound and outer bound procedures for this optimization problem, which could be solved efficiently to derive lower and upper bounds, respectively, for the margin of robust feasibility. We evaluate our approach numerically on standard instances taken from the MATPOWER and NESTA databases of AC power flow equations that describe the steady state of the power grid. The results demonstrate that our approach can produce tight lower and upper bounds on the margin of robust feasibility for such instances.

math.OC↗

arXiv2023

Persistent Homology to Study Cold Hardiness of Grape Cultivars

Persistent homology is a branch of computational algebraic topology that studies shapes and extracts features over multiple scales. In this paper, we present an unsupervised approach that uses persistent homology to study divergent behavior in agricultural point cloud data. More specifically, we build persistence diagrams from multidimensional point clouds, and use those diagrams as the basis to compare and contrast different subgroups of the population. We apply the framework to study the cold hardiness behavior of 5 leading grape cultivars, with real data from over 20 growing seasons. Our results demonstrate that persistent homology is able to effectively elucidate divergent behavior among the different cultivars; identify cultivars that exhibit variable behavior across seasons; and identify seasonal correlations.

math.AT↗

arXiv2022

SFCDecomp: Multicriteria Optimized Tool Path Planning in 3D Printing using Space-Filling Curve Based Domain Decomposition

We explore efficient optimization of toolpaths based on multiple criteria for large instances of 3D printing problems. We first show that the minimum turn cost 3D printing problem is NP-hard, even when the region is a simple polygon. We develop SFCDecomp, a space filling curve based decomposition framework to solve large instances of 3D printing problems efficiently by solving these optimization subproblems independently. For the Buddha model, our framework builds toolpaths over a total of 799,716 nodes across 169 layers, and for the Bunny model it builds toolpaths over 812,733 nodes across 360 layers. Building on SFCDecomp, we develop a multicriteria optimization approach for toolpath planning. We demonstrate the utility of our framework by maximizing or minimizing tool path edge overlap between adjacent layers, while jointly minimizing turn costs. Strength testing of a tensile test specimen printed with tool paths that maximize or minimize adjacent layer edge overlaps reveal significant differences in tensile strength between the two classes of prints.

cs.GR↗

arXiv2021

Stitch Fix for Mapper and Topological Gains

The mapper construction is a powerful tool from topological data analysis that is designed for the analysis and visualization of multivariate data. In this paper, we investigate a method for stitching a pair of univariate mappers together into a bivariate mapper, and study topological notions of information gains, referred to as topological gains, during such a process. We further provide implementations that visualize such topological gains for mapper graphs.

cs.CG↗

arXiv2021

Pheno-Mapper: An Interactive Toolbox for the Visual Exploration of Phenomics Data

High-throughput technologies to collect field data have made observations possible at scale in several branches of life sciences. The data collected can range from the molecular level (genotypes) to physiological (phenotypic traits) and environmental observations (e.g., weather, soil conditions). These vast swathes of data, collectively referred to as phenomics data, represent a treasure trove of key scientific knowledge on the dynamics of the underlying biological system. However, extracting information and insights from these complex datasets remains a significant challenge owing to their multidimensionality and lack of prior knowledge about their complex structure. In this paper, we present Pheno-Mapper, an interactive toolbox for the exploratory analysis and visualization of large-scale phenomics data. Our approach uses the mapper framework to perform a topological analysis of the data, and subsequently render visual representations with built-in data analysis and machine learning capabilities. We demonstrate the utility of this new tool on real-world plant (e.g., maize) phenomics datasets. In comparison to existing approaches, the main advantage of Pheno-Mapper is that it provides rich, interactive capabilities in the exploratory analysis of phenomics data, and it integrates visual analytics with data analysis and machine learning in an easily extensible way. In particular, Pheno-Mapper allows the interactive selection of subpopulations guided by a topological summary of the data and applies data mining and machine learning to these selected subpopulations for in-depth exploration.

cs.CG↗

arXiv2021

Euler Transformation of Polyhedral Complexes

We propose an Euler transformation that transforms a given $d$-dimensional cell complex $K$ for $d=2,3$ into a new $d$-complex $\hat{K}$ in which every vertex is part of a uniform even number of edges. Hence every vertex in the graph $\hat{G}$ that is the $1$-skeleton of $\hat{K}$ has an even degree, which makes $\hat{G}$ Eulerian, i.e., it is guaranteed to contain an Eulerian tour. Meshes whose edges admit Eulerian tours are crucial in coverage problems arising in several applications including 3D printing and robotics. For $2$-complexes in $\mathbb{R}^2$ ($d=2$) under mild assumptions (that no two adjacent edges of a $2$-cell in $K$ are boundary edges), we show that the Euler transformed $2$-complex $\hat{K}$ has a geometric realization in $\mathbb{R}^2$, and that each vertex in its $1$-skeleton has degree $4$. We bound the numbers of vertices, edges, and $2$-cells in $\hat{K}$ as small scalar multiples of the corresponding numbers in $K$. We prove corresponding results for $3$-complexes in $\mathbb{R}^3$ under an additional assumption that the degree of a vertex in each $3$-cell containing it is $3$. In this setting, every vertex in $\hat{G}$ is shown to have a degree of $6$. We also present bounds on parameters measuring geometric quality (aspect ratios, minimum edge length, and maximum angle) of $\hat{K}$ in terms of the corresponding parameters of $K$ (for $d=2, 3$). Finally, we illustrate a direct application of the proposed Euler transformation in additive manufacturing.

cs.CG↗

arXiv2021

Representations of Energy Landscapes by Sublevelset Persistent Homology: An Example With n-Alkanes

Encoding the complex features of an energy landscape is a challenging task, and often chemists pursue the most salient features (minima and barriers) along a highly reduced space, i.e. 2- or 3-dimensions. Even though disconnectivity graphs or merge trees summarize the connectivity of the local minima of an energy landscape via the lowest-barrier pathways, there is more information to be gained by also considering the topology of each connected component at different energy thresholds (or sublevelsets). We propose sublevelset persistent homology as an appropriate tool for this purpose. Our computations on the configuration phase space of n-alkanes from butane to octane allow us to conjecture, and then prove, a complete characterization of the sublevelset persistent homology of the alkane $C_m H_{2m+2}$ potential energy landscapes, for all $m$, and in all homological dimensions. We further compare both the analytical configurational potential energy landscapes and sampled data from molecular dynamics simulation, using the united and all-atom descriptions of the intramolecular interactions. In turn, this supports the application of distance metrics to quantify sampling fidelity and lays the foundation for future work regarding new metrics that quantify differences between the topological features of high-dimensional energy landscapes.

physics.chem-ph↗

arXiv2021

The Maximum Distance Problem and Minimal Spanning Trees

Given a compact $E \subset \mathbb{R}^n$ and $s > 0$, the maximum distance problem seeks a compact and connected subset of $\mathbb{R}^n$ of smallest one dimensional Hausdorff measure whose $s$-neighborhood covers $E$. For $E\subset \mathbb{R}^2$, we prove that minimizing over minimum spanning trees that connect the centers of balls of radius $s$, which cover $E$, solves the maximum distance problem. The main difficulty in proving this result is overcome by the proof of Lemma 3.5, which states that one is able to cover the $s$-neighborhood of a Lipschitz curve $Γ$ in $\mathbb{R}^2$ with a finite number of balls of radius $s$, and connect their centers with another Lipschitz curve $Γ_\ast$, where $\mathcal{H}^1(Γ_\ast)$ is arbitrarily close to $\mathcal{H}^1(Γ)$. We also present an open source package for computational exploration of the maximum distance problem using minimum spanning trees, available at https://github.com/mtdaydream/MDP_MST.

math.CA↗

arXiv2020

Continuous Toolpath Planning in Additive Manufacturing

We develop a framework that creates a new polygonal mesh representation of the sparse infill domain of a layer-by-layer 3D printing job. We guarantee the existence of a single, continuous tool path covering each connected piece of the domain in every layer. We present a tool path algorithm that traverses each such continuous tool path with no crossovers. The key construction at the heart of our framework is an Euler transformation which converts a 2-dimensional cell complex K into a new 2-complex K^ such that every vertex in the 1-skeleton G^ of K^ has even degree. Hence G^ is Eulerian, and a Eulerian tour can be followed to print all edges in a continuous fashion. We start with a mesh K of the union of polygons obtained by projecting all layers to the plane. We compute its Euler transformation K^. In the slicing step, we clip K^ at each layer using its polygon to obtain a complex that may not necessarily be Euler. We then patch this complex by adding edges such that any odd-degree nodes created by slicing are transformed to have even degrees again. We print extra support edges in place of any segments left out to ensure there are no edges without support in the next layer. These support edges maintain the Euler nature of the complex. Finally we describe a tree-based search algorithm that builds the continuous tool path by traversing "concentric" cycles in the Euler complex. Our algorithm produces a tool path that avoids material collisions and crossovers, and can be printed in a continuous fashion irrespective of complex geometry or topology of the domain (e.g., holes). We implement our test our framework on several 3D objects. Apart from standard geometric shapes, we demonstrate the framework on the Stanford bunny.

cs.CG↗

arXiv2019

Hyppo-X: A Scalable Exploratory Framework for Analyzing Complex Phenomics Data

Phenomics is an emerging branch of modern biology that uses high throughput phenotyping tools to capture multiple environmental and phenotypic traits, often at massive spatial and temporal scales. The resulting high dimensional data represent a treasure trove of information for providing an in-depth understanding of how multiple factors interact and contribute to the overall growth and behavior of different genotypes. However, computational tools that can parse through such complex data and aid in extracting plausible hypotheses are currently lacking. In this paper, we present Hyppo-X, a new algorithmic approach to visually explore complex phenomics data and in the process characterize the role of environment on phenotypic traits. We model the problem as one of unsupervised structure discovery, and use emerging principles from algebraic topology and graph theory for discovering higher-order structures of complex phenomics data. We present an open source software which has interactive visualization capabilities to facilitate data navigation and hypothesis formulation. We test and evaluate Hyppo-X on two real-world plant (maize) data sets. Our results demonstrate the ability of our approach to delineate divergent subpopulation-level behavior. Notably, our approach shows how environmental factors could influence phenotypic behavior, and how that effect varies across different genotypes and different time scales. To the best of our knowledge, this effort provides one of the first approaches to systematically formalize the problem of hypothesis extraction for phenomics data. Considering the infancy of the phenomics field, tools that help users explore complex data and extract plausible hypotheses in a data-guided manner will be critical to future advancements in the use of such data.

q-bio.QM↗

arXiv2018

Median Shapes

We introduce and begin to explore the mean and median of finite sets of shapes represented as integral currents. The median can be computed efficiently in practice, and we focus most of our theoretical and computational attention on medians. We consider questions on the existence and regularity of medians. While the median might not exist in all cases, we show that a mass-regularized median is guaranteed to exist. When the input shapes are modeled by integral currents with shared boundaries in codimension $1$, we show that the median is guaranteed to exist, and is contained in the \emph{envelope} of the input currents. On the other hand, we show that medians can be \emph{wild} in this setting, and smooth inputs can generate non-smooth medians. For higher codimensions, we show that \emph{books} are minimizing for a finite set of $1$-currents in $\Bbb{R}^3$ with shared boundaries. As part of this proof, we present a new result in graph theory---that \emph{cozy} graphs are \emph{comfortable}---which should be of independent interest. Further, we show that regular points on the median have book-like tangent cones in this case. From the point of view of computation, we study the median shape in the settings of a finite simplicial complex. When the input shapes are represented by chains of the simplicial complex, we show that the problem of finding the median shape can be formulated as an integer linear program. This optimization problem can be solved as a linear program in practice, thus allowing one to compute median shapes efficiently. We provide open source code implementing our methods, which could also be used by anyone to experiment with ideas of their own. The software could be accessed at https://github.com/tbtraltaa/medianshape.

math.DG↗

arXiv2018

Interesting Paths in the Mapper

The Mapper produces a compact summary of high dimensional data as a simplicial complex. We study the problem of quantifying the interestingness of subpopulations in a Mapper, which appear as long paths, flares, or loops. First, we create a weighted directed graph G using the 1-skeleton of the Mapper. We use the average values at the vertices of a target function to direct edges (from low to high). The difference between the average values at vertices (high-low) is set as the edge's weight. Covariation of the remaining h functions (independent variables) is captured by a h-bit binary signature assigned to the edge. An interesting path in G is a directed path whose edges all have the same signature. We define the interestingness score of such a path as a sum of its edge weights multiplied by a nonlinear function of their ranks in the path. Second, we study three optimization problems on this graph G. In the problem Max-IP, we seek an interesting path in G with the maximum interestingness score. We show that Max-IP is NP-complete. For the special case when G is a directed acyclic graph (DAG), we show that Max-IP can be solved in polynomial time - in O(mnd_i) where d_i is the maximum indegree of a vertex in G. In the more general problem IP, the goal is to find a collection of edge-disjoint interesting paths such that the overall sum of their interestingness scores is maximized. We also study a variant of IP termed k-IP, where the goal is to identify a collection of edge-disjoint interesting paths each with k edges, and their total interestingness score is maximized. While k-IP can be solved in polynomial time for k <= 2, we show k-IP is NP-complete for k >= 3 even when G is a DAG. We develop polynomial time heuristics for IP and k-IP on DAGs.

cs.CG↗

arXiv2016

Thinner is not Always Better: Cascade Knapsack Problems

In the context of branch-and-bound (B&B) for integer programming (IP) problems, a direction along which the polyhedron of the IP has minimum width is termed a thin direction. We demonstrate that a thin direction need not always be a good direction to branch on for solving the problem efficiently. Further, the integer width, which is the number of B&B nodes created when branching on the direction, may also not be an accurate indicator of good branching directions.

math.OC↗

arXiv2016

Flat Norm Decomposition of Integral Currents

Currents represent generalized surfaces studied in geometric measure theory. They range from relatively tame integral currents representing oriented compact manifolds with boundary and integer multiplicities, to arbitrary elements of the dual space of differential forms. The flat norm provides a natural distance in the space of currents, and works by decomposing a $d$-dimensional current into $d$- and (the boundary of) $(d+1)$-dimensional pieces in an optimal way. Given an integral current, can we expect its flat norm decomposition to be integral as well? This is not known in general, except in the case of $d$-currents that are boundaries of $(d+1)$-currents in $\mathbb{R}^{d+1}$ (following results from a corresponding problem on the $L^1$ total variation ($L^1$TV) of functionals). On the other hand, for a discretized flat norm on a finite simplicial complex, the analogous statement holds even when the inputs are not boundaries. This simplicial version relies on the total unimodularity of the boundary matrix of the simplicial complex -- a result distinct from the $L^1$TV approach. We develop an analysis framework that extends the result in the simplicial setting to one for $d$-currents in $\mathbb{R}^{d+1}$, provided a suitable triangulation result holds. In $\mathbb{R}^2$, we use a triangulation result of Shewchuk (bounding both the size and location of small angles), and apply the framework to show that the discrete result implies the continuous result for $1$-currents in $\mathbb{R}^2$.

math.DG↗

arXiv2015

A Knapsack-Like Code Using Recurrence Sequence Representations

We had recently shown that every positive integer can be represented uniquely using a recurrence sequence, when certain restrictions on the digit strings are satisfied. We present the details of how such representations can be used to build a knapsack-like public key cryptosystem. We also present new disguising methods, and provide arguments for the security of the code against known methods of attack.

math.NT↗

arXiv2014

Topological Features In Cancer Gene Expression Data

We present a new method for exploring cancer gene expression data based on tools from algebraic topology. Our method selects a small relevant subset from tens of thousands of genes while simultaneously identifying nontrivial higher order topological features, i.e., holes, in the data. We first circumvent the problem of high dimensionality by dualizing the data, i.e., by studying genes as points in the sample space. Then we select a small subset of the genes as landmarks to construct topological structures that capture persistent, i.e., topologically significant, features of the data set in its first homology group. Furthermore, we demonstrate that many members of these loops have been implicated for cancer biogenesis in scientific literature. We illustrate our method on five different data sets belonging to brain, breast, leukemia, and ovarian cancers.

q-bio.GN↗

arXiv2013

Simplicial Flat Norm with Scale

We study the multiscale simplicial flat norm (MSFN) problem, which computes flat norm at various scales of sets defined as oriented subcomplexes of finite simplicial complexes in arbitrary dimensions. We show that the multiscale simplicial flat norm is NP-complete when homology is defined over integers. We cast the multiscale simplicial flat norm as an instance of integer linear optimization. Following recent results on related problems, the multiscale simplicial flat norm integer program can be solved in polynomial time by solving its linear programming relaxation, when the simplicial complex satisfies a simple topological condition (absence of relative torsion). Our most significant contribution is the simplicial deformation theorem, which states that one may approximate a general current with a simplicial current while bounding the expansion of its mass. We present explicit bounds on the quality of this approximation, which indicate that the simplicial current gets closer to the original current as we make the simplicial complex finer. The multiscale simplicial flat norm opens up the possibilities of using flat norm to denoise or extract scale information of large data sets in arbitrary dimensions. On the other hand, it allows one to employ the large body of algorithmic results on simplicial complexes to address more general problems related to currents.

math.DG↗

arXiv2013

Non Total-Unimodularity Neutralized Simplicial Complexes

Given a simplicial complex K with weights on its simplices and a chain on it, the Optimal Homologous Chain Problem (OHCP) is to find a chain with minimal weight that is homologous (over the integers) to the given chain. The OHCP is NP-complete, but if the boundary matrix of K is totally unimodular (TU), it becomes solvable in polynomial time when modeled as a linear program (LP). We define a condition on the simplicial complex called non total-unimodularity neutralized, or NTU neutralized, which ensures that even when the boundary matrix is not TU, the OHCP LP must contain an integral optimal vertex for every input chain. This condition is a property of K, and is independent of the input chain and the weights on the simplices. This condition is strictly weaker than the boundary matrix being TU. More interestingly, the polytope of the OHCP LP may not be integral under this condition. Still, an integral optimal vertex exists for every right-hand side, i.e., for every input chain. Hence a much larger class of OHCP instances can be solved in polynomial time than previously considered possible. As a special case, we show that 2-complexes with trivial first homology group are guaranteed to be NTU neutralized.

math.AT↗