Showing papers by "Chris Peterson published in 2016"

Nerve Complexes of Circular Arcs

Abstract: We show that the nerve and clique complexes of n arcs in the circle are homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension. Moreover this homotopy type can be computed in time $$O(n\log n)$$O(nlogn). For the particular case of the nerve complex of evenly-spaced arcs of the same length, we determine explicit homology bases and we relate the complex to a cyclic polytope with n vertices. We give three applications of our knowledge of the homotopy types of nerve complexes of circular arcs. First, we show that the Lovasz bound on the chromatic number of a circular complete graph is either sharp or off by one. Second, we use the connection to cyclic polytopes to give a novel topological proof of a known upper bound on the distance between successive roots of a homogeneous trigonometric polynomial. Third, we show that the Vietoris---Rips or ambient Cech simplicial complex of n points in the circle is homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension, and furthermore this homotopy type can be computed in time $$O(n\log n)$$O(nlogn).

Algorithms and basic asymptotics for generalized numerical semigroups in {\mathbb {N}}^d

Abstract: Let \({\mathbb {N}}\) denote the monoid of natural numbers. A numerical semigroup is a cofinite submonoid \(S\subseteq {\mathbb {N}}\). For the purposes of this paper, a generalized numerical semigroup (GNS) is a cofinite submonoid \(S\subseteq {\mathbb {N}}^d\). The cardinality of \({\mathbb {N}}^d \setminus S\) is called the genus. We describe a family of algorithms, parameterized by (relaxed) monomial orders, that can be used to generate trees of semigroups with each GNS appearing exactly once. Let \(N_{g,d}\) denote the number of generalized numerical semigroups \(S\subseteq {\mathbb {N}}^d\) of genus \(g\). We compute \(N_{g,d}\) for small values of \(g,d\) and provide coarse asymptotic bounds on \(N_{g,d}\) for large values of \(g,d\). For a fixed \(g\), we show that \(F_g(d)=N_{g,d}\) is a polynomial function of degree \(g\). We close with several open problems/conjectures related to the asymptotic growth of \(N_{g,d}\) and with suggestions for further avenues of research.

An order fitting rule for optimal subspace averaging

Abstract: The problem of estimating a low-dimensional subspace from a collection of experimentally measured subspaces arises in many applications of statistical signal processing. In this paper we address this problem, and give a solution for the average subspace that minimizes an extrinsic mean-squared error, defined by the squared Frobenius norm between projection matrices. The solution automatically returns the dimension of the optimal average subspace, which is the novel result of the paper. The proposed order fitting rule is based on thresholding the eigenvalues of the average projection matrix, and thus it is free of penalty terms or other tuning parameters commonly used by other rank estimation techniques. Several numerical examples demonstrate the usefulness and applicability of the proposed criterion, showing how the dimension of the average subspace captures the variability of the measured subspaces.

Flag-based detection of weak gas signatures in long-wave infrared hyperspectral image sequences

Abstract: We present a flag manifold based method for detecting chemical plumes in long-wave infrared hyperspectral movies. The method encodes temporal and spatial information related to a hyperspectral pixel into a flag, or nested sequence of linear subspaces. The technique used to create the flags pushes information about the background clutter, ambient conditions, and potential chemical agents into the leading elements of the flags. Exploiting this temporal information allows for a detection algorithm that is sensitive to the presence of weak signals. This method is compared to existing techniques qualitatively on real data and quantitatively on synthetic data to show that the flag-based algorithm consistently performs better on data when the SINRdB is low, and beats the ACE and MF algorithms in probability of detection for low probabilities of false alarm even when the SINRdB is high.

Persistent Homology on Grassmann Manifolds for Analysis of Hyperspectral Movies

Abstract: The existence of characteristic structure, or shape, in complex data sets has been recognized as increasingly important for mathematical data analysis. This realization has motivated the development of new tools such as persistent homology for exploring topological invariants, or features, in large data sets. In this paper, we apply persistent homology to the characterization of gas plumes in time dependent sequences of hyperspectral cubes, i.e. the analysis of 4-way arrays. We investigate hyperspectral movies of Long-Wavelength Infrared data monitoring an experimental release of chemical simulant into the air. Our approach models regions of interest within the hyperspectral data cubes as points on the real Grassmann manifold Gk,i¾?n whose points parameterize the k-dimensional subspaces of $$\mathbb {R}^n$$Rn, contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows a sequence of time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmann manifold. This motivates the search for topological features, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the Grassmann manifold. The proposed mathematical model affords the processing of large data sets while retaining valuable discriminatory information. In this paper, we discuss how embedding our data in the Grassmann manifold, together with topological data analysis, captures dynamical events that occur as the chemical plume is released and evolves.

The average number of real lines on a random cubic

Abstract: We derive a formula expressing the average number En of real lines on a random hypersurface of degree 2n − 3 in RP in terms of the expected modulus of the determinant of a special random matrix. In the case n = 3 we prove that the average number of real lines on a random cubic surface in RP is 6 √ 2− 3. Our technique can also be used to express the number Cn of complex lines on a generic hypersurface of degree 2n − 3 in CP in terms of the determinant of a random Hermitian matrix. As a special case we obtain a new proof of the classical statement C3 = 27.

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 completed on reasonably large, high dimensional data sets.

Random fields and the enumerative geometry of lines on real and complex hypersurfaces

Abstract: We derive a formula expressing the average number $E_n$ of real lines on a random hypersurface of degree $2n-3$ in $\mathbb{R}\textrm{P}^n$ in terms of the expected modulus of the determinant of a special random matrix. In the case $n=3$ we prove that the average number of real lines on a random cubic surface in $\mathbb{R}\textrm{P}^3$ equals: $$E_3=6\sqrt{2}-3.$$ Our technique can also be used to express the number $C_n$ of complex lines on a generic hypersurface of degree $2n-3$ in $\mathbb{C}\textrm{P}^n$ in terms of the determinant of a random Hermitian matrix. As a special case we obtain a new proof of the classical statement $C_3=27.$ We determine, at the logarithmic scale, the asymptotic of the quantity $E_n$, by relating it to $C_n$ (whose asymptotic has been recently computed D. Zagier). Specifically we prove that: $$\lim_{n\to \infty}\frac{\log E_n}{\log C_n}=\frac{1}{2}.$$ Finally we show that this approach can be used to compute the number $R_n=(2n-3)!!$ of real lines, counted with their intrinsic signs, on a generic real hypersurface of degree $2n-3$ in $\mathbb{R}\textrm{P}^n$.

Persistent Homology on Grassmann Manifolds for Analysis of Hyperspectral Movies

Abstract: The existence of characteristic structure, or shape, in complex data sets has been recognized as increasingly important for mathematical data analysis. This realization has motivated the development of new tools such as persistent homology for exploring topological invariants, or features, in large data sets. In this paper we apply persistent homology to the characterization of gas plumes in time dependent sequences of hyperspectral cubes, i.e. the analysis of 4-way arrays. We investigate hyperspectral movies of Long-Wavelength Infrared data monitoring an experimental release of chemical simulant into the air. Our approach models regions of interest within the hyperspectral data cubes as points on the real Grassmann manifold $G(k, n)$ (whose points parameterize the $k$-dimensional subspaces of $\mathbb{R}^n$), contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows a sequence of time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmann manifold. This motivates the search for topological features, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the Grassmann manifold. The proposed mathematical model affords the processing of large data sets while retaining valuable discriminatory information. In this paper, we discuss how embedding our data in the Grassmann manifold, together with topological data analysis, captures dynamical events that occur as the chemical plume is released and evolves.

Reduced dimension estimators in matched subspace detection

Abstract: The standard matched subspace detector is modified by replacing a full-rank oblique pseudoinverse with a reduced-dimension version that trades bias for variance. Minimization of the bias squared plus variance produces an order determination rule relating signal to noise ratio and noise gain. The reduced dimension detector is then applied to several detection tasks in hyperspectral imagery yielding promising results.