Showing papers by "Chris Peterson published in 2017"

Persistence images: a stable vector representation of persistent homology

Abstract: Many data sets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a data set. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.

Sparse Locally Linear Embedding

Abstract: The Locally Linear Embedding (LLE) algorithm has proven useful for determining structure preserving, dimension reducing mappings of data on manifolds. We propose a modification to the LLE optimization problem that serves to minimize the number of neighbors required for the representation of each data point. The algorithm is shown to be robust over wide ranges of the sparsity parameter producing an average number of nearest neighbors that is consistent with the best performing parameter selection for LLE. Given the number of non-zero weights may be substantially reduced in comparison to LLE, Sparse LLE can be applied to larger data sets. We provide three numerical examples including a color image, the standard swiss roll, and a gene expression data set to illustrate the behavior of the method in comparison to LLE. The resulting algorithm produces comparatively sparse representations that preserve the neighborhood geometry of the data in the spirit of LLE.

Visualizing data sets on the Grassmannian using self-organizing mappings

Abstract: We extend the self-organizing mapping algorithm to the problem of visualizing data on Grassmann manifolds. In this setting, a collection of k points in n-dimensions is represented by a k-dimensional subspace, e.g., via the singular value or QR-decompositions. Data assembled in this way is challenging to visualize given abstract points on the Grassmannian do not reside in Euclidean space. The extension of the SOM algorithm to this geometric setting only requires that distances between two points can be measured and that any given point can be moved towards a presented pattern. The similarity between two points on the Grassmannian is measured in terms of the principal angles between subspaces, e.g., the chordal distance. Further, we employ a formula for moving one subspace towards another along the shortest path, i.e., the geodesic between two points on the Grassmannian. This enables a faithful implementation of the SOM approach for visualizing data consisting of k-dimensional subspaces of n-dimensional Euclidean space. We illustrate the resulting algorithm on a hyperspectral imaging application.

Stratifying High-Dimensional Data Based on Proximity to the Convex Hull Boundary

Abstract: The convex hull of a set of points, $C$, serves to expose extremal properties of $C$ and can help identify elements in $C$ of high interest. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be difficult to determine. One solution is to expand the list of high interest candidates to points lying near the boundary of the convex hull. We propose a quadratic program for the purpose of stratifying points in a data cloud based on proximity to the boundary of the convex hull. For each data point, a quadratic program is solved to determine an associated weight vector. We show that the weight vector encodes geometric information concerning the point's relationship to the boundary of the convex hull. The computation of the weight vectors can be carried out in parallel, and for a fixed number of points and fixed neighborhood size, the overall computational complexity of the algorithm grows linearly with dimension. As a consequence, meaningful computations can be complete...

Numerical irreducible decomposition over a number field

Abstract: Let 𝒢 = {g1,…,gn} be a set of elements in the polynomial ring R = ℂ[z1,…,zN], let I ⊂ R denote the ideal generated by the elements of 𝒢, and let I denote the radical of I. There is a unique decomposition I = P1 ∩⋯ ∩ Pk with each Pi a prime ideal corresponding to a minimal associated prime of I over R. Let V (𝒢) = V (I) denote the reduced algebraic set corresponding to the common zeroes of the elements of 𝒢. Techniques from numerical algebraic geometry can be used to determine the numerical irreducible decomposition of V (𝒢) over ℂ. This corresponds to producing a witness set for V (Pi) for each i = 1,…,k together with the degree and dimension of V (Pi) (a point in a witness set for V (Pi) can be considered as a numerical approximation for a general point on V (Pi)). The purpose of this paper is to show how to extend these results taking into account the field of definition for the polynomial system. In particular, let F be a number field (i.e. a finite field extension of ℚ) and let 𝒢 = {g1,…,gn} be a set ...

Constrained subspace estimation via convex optimization

Abstract: Given a collection of M experimentally measured subspaces, and a model-based subspace, this paper addresses the problem of finding a subspace that approximates the collection, under the constraint that it intersects the model-based subspace in a predetermined number of dimensions. This constrained subspace estimation (CSE) problem arises in applications such as beamforming, where the model-based subspace encodes prior information about the direction-of-arrival of some sources impinging on the array. In this paper, we formulate the constrained subspace estimation (CSE) problem, and present an approximation based on a semidefinite relaxation (SDR) of this non-convex problem. The performance of the proposed CSE algorithm is demonstrated via numerical simulation, and its application to beamforming is also discussed.

Numerical Polar calculus and cohomology of line bundles

Abstract: Let $L_1,\dots,L_s$ be line bundles on a smooth variety $X\subset \mathbb{P}^r$ and let $D_1,\dots,D_s$ be divisors on $X$ such that $D_i$ represents $L_i$. We give a probabilistic algorithm for computing the degree of intersections of polar classes which are in turn used for computing the Euler characteristic of linear combinations of $L_1,\dots,L_s$. The input consists of generators for the homogeneous ideals $I_X, I_{D_i} \subset \mathbb{C}[x_0,\ldots,x_r]$ defining $X$ and $D_i$.