arXiv subjects

Michael Kirby

Publications and source records attributed to Michael Kirby.

At least 37 records · Page 2Linked to original sources

arXiv2019

Total variation vs L1 regularization: a comparison of compressive sensing optimization methods for chemical detection

One of the fundamental assumptions of compressive sensing (CS) is that a signal can be reconstructed from a small number of samples by solving an optimization problem with the appropriate regularization term. Two standard regularization terms are the L1 norm and the total variation (TV) norm. We present a comparison of CS reconstruction results based on these two approaches in the context of chemical detection, and we demonstrate that optimization based on the L1 norm outperforms optimization based on the TV norm. Our comparison is driven by CS sampling, reconstruction, and chemical detection in two real-world datasets: the Physical Sciences Inc. Fabry-Pérot interferometer sensor multispectral dataset and the Johns Hopkins Applied Physics Lab FTIR-based longwave infrared sensor hyperspectral dataset. Both datasets contain the release of a chemical simulant such as glacial acetic acid, triethyl phosphate, and sulfur hexafluoride. For chemical detection we use the adaptive coherence estimator (ACE) and bulk coherence, and we propose algorithmic ACE thresholds to define the presence or absence of a chemical of interest in both un-compressed data cubes and reconstructed data cubes. The un-compressed data cubes provide an approximate ground truth. We demonstrate that optimization based on either the L1 norm or TV norm results in successful chemical detection at a compression rate of 90%, but we show that L1 optimization is preferable. We present quantitative comparisons of chemical detection on reconstructions from the two methods, with an emphasis on the number of pixels with an ACE value above the threshold.

eess.IV↗

arXiv2019

A data-driven approach to sampling matrix selection for compressive sensing

Sampling is a fundamental aspect of any implementation of compressive sensing. Typically, the choice of sampling method is guided by the reconstruction basis. However, this approach can be problematic with respect to certain hardware constraints and is not responsive to domain-specific context. We propose a method for defining an order for a sampling basis that is optimal with respect to capturing variance in data, thus allowing for meaningful sensing at any desired level of compression. We focus on the Walsh-Hadamard sampling basis for its relevance to hardware constraints, but our approach applies to any sampling basis of interest. We illustrate the effectiveness of our method on the Physical Sciences Inc. Fabry-Pérot interferometer sensor multispectral dataset, the Johns Hopkins Applied Physics Lab FTIR-based longwave infrared sensor hyperspectral dataset, and a Colorado State University Swiss Ranger depth image dataset. The spectral datasets consist of simulant experiments, including releases of chemicals such as GAA and SF6. We combine our sampling and reconstruction with the adaptive coherence estimator (ACE) and bulk coherence for chemical detection and we incorporate an algorithmic threshold for ACE values to determine the presence or absence of a chemical. We compare results across sampling methods in this context. We have successful chemical detection at a compression rate of 90%. For all three datasets, we compare our sampling approach to standard orderings of sampling basis such as random, sequency, and an analog of sequency that we term `frequency.' In one instance, the peak signal to noise ratio was improved by over 30% across a test set of depth images.

eess.SP↗

arXiv2019

A fractal dimension for measures via persistent homology

We use persistent homology in order to define a family of fractal dimensions, denoted $\mathrm{dim}_{\mathrm{PH}}^i(μ)$ for each homological dimension $i\ge 0$, assigned to a probability measure $μ$ on a metric space. The case of $0$-dimensional homology ($i=0$) relates to work by Michael J Steele (1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if $μ$ is supported on a compact subset of Euclidean space $\mathbb{R}^m$ for $m\ge2$, then Steele's work implies that $\mathrm{dim}_{\mathrm{PH}}^0(μ)=m$ if the absolutely continuous part of $μ$ has positive mass, and otherwise $\mathrm{dim}_{\mathrm{PH}}^0(μ)<m$. Experiments suggest that similar results may be true for higher-dimensional homology $0<i<m$, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points $n$ goes to infinity, of the total sum of the $i$-dimensional persistent homology interval lengths for $n$ random points selected from $μ$ in an i.i.d. fashion. To some measures $μ,$ we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of $0$-dimensional homology when $μ$ is the uniform distribution over the unit interval, and conjecture that it exists when $μ$ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.

math.DS↗

arXiv2018

Monitoring the shape of weather, soundscapes, and dynamical systems: a new statistic for dimension-driven data analysis on large data sets

Dimensionality-reduction methods are a fundamental tool in the analysis of large data sets. These algorithms work on the assumption that the "intrinsic dimension" of the data is generally much smaller than the ambient dimension in which it is collected. Alongside their usual purpose of mapping data into a smaller dimension with minimal information loss, dimensionality-reduction techniques implicitly or explicitly provide information about the dimension of the data set. In this paper, we propose a new statistic that we call the $κ$-profile for analysis of large data sets. The $κ$-profile arises from a dimensionality-reduction optimization problem: namely that of finding a projection into $k$-dimensions that optimally preserves the secants between points in the data set. From this optimal projection we extract $κ,$ the norm of the shortest projected secant from among the set of all normalized secants. This $κ$ can be computed for any $k$; thus the tuple of $κ$ values (indexed by dimension) becomes a $κ$-profile. Algorithms such as the Secant-Avoidance Projection algorithm and the Hierarchical Secant-Avoidance Projection algorithm, provide a computationally feasible means of estimating the $κ$-profile for large data sets, and thus a method of understanding and monitoring their behavior. As we demonstrate in this paper, the $κ$-profile serves as a useful statistic in several representative settings: weather data, soundscape data, and dynamical systems data.

cs.LG↗

arXiv2018

Too many secants: a hierarchical approach to secant-based dimensionality reduction on large data sets

A fundamental question in many data analysis settings is the problem of discerning the "natural" dimension of a data set. That is, when a data set is drawn from a manifold (possibly with noise), a meaningful aspect of the data is the dimension of that manifold. Various approaches exist for estimating this dimension, such as the method of Secant-Avoidance Projection (SAP). Intuitively, the SAP algorithm seeks to determine a projection which best preserves the lengths of all secants between points in a data set; by applying the algorithm to find the best projections to vector spaces of various dimensions, one may infer the dimension of the manifold of origination. That is, one may learn the dimension at which it is possible to construct a diffeomorphic copy of the data in a lower-dimensional Euclidean space. Using Whitney's embedding theorem, we can relate this information to the natural dimension of the data. A drawback of the SAP algorithm is that a data set with $T$ points has $O(T^2)$ secants, making the computation and storage of all secants infeasible for very large data sets. In this paper, we propose a novel algorithm that generalizes the SAP algorithm with an emphasis on addressing this issue. That is, we propose a hierarchical secant-based dimensionality-reduction method, which can be employed for data sets where explicitly calculating all secants is not feasible.

cs.CV↗

arXiv2018

A GPU-Oriented Algorithm Design for Secant-Based Dimensionality Reduction

Dimensionality-reduction techniques are a fundamental tool for extracting useful information from high-dimensional data sets. Because secant sets encode manifold geometry, they are a useful tool for designing meaningful data-reduction algorithms. In one such approach, the goal is to construct a projection that maximally avoids secant directions and hence ensures that distinct data points are not mapped too close together in the reduced space. This type of algorithm is based on a mathematical framework inspired by the constructive proof of Whitney's embedding theorem from differential topology. Computing all (unit) secants for a set of points is by nature computationally expensive, thus opening the door for exploitation of GPU architecture for achieving fast versions of these algorithms. We present a polynomial-time data-reduction algorithm that produces a meaningful low-dimensional representation of a data set by iteratively constructing improved projections within the framework described above. Key to our algorithm design and implementation is the use of GPUs which, among other things, minimizes the computational time required for the calculation of all secant lines. One goal of this report is to share ideas with GPU experts and to discuss a class of mathematical algorithms that may be of interest to the broader GPU community.

cs.CV↗

arXiv2018

Endmember Extraction on the Grassmannian

Endmember extraction plays a prominent role in a variety of data analysis problems as endmembers often correspond to data representing the purest or best representative of some feature. Identifying endmembers then can be useful for further identification and classification tasks. In settings with high-dimensional data, such as hyperspectral imagery, it can be useful to consider endmembers that are subspaces as they are capable of capturing a wider range of variations of a signature. The endmember extraction problem in this setting thus translates to finding the vertices of the convex hull of a set of points on a Grassmannian. In the presence of noise, it can be less clear whether a point should be considered a vertex. In this paper, we propose an algorithm to extract endmembers on a Grassmannian, identify subspaces of interest that lie near the boundary of a convex hull, and demonstrate the use of the algorithm on a synthetic example and on the 220 spectral band AVIRIS Indian Pines hyperspectral image.

cs.CV↗

arXiv2018

Integral Invariants from Covariance Analysis of Embedded Riemannian Manifolds

Principal Component Analysis can be performed over small domains of an embedded Riemannian manifold in order to relate the covariance analysis of the underlying point set with the local extrinsic and intrinsic curvature. We show that the volume of domains on a submanifold of general codimension, determined by the intersection with higher-dimensional cylinders and balls in the ambient space, have asymptotic expansions in terms of the mean and scalar curvatures. Moreover, we propose a generalization of the classical third fundamental form to general submanifolds and prove that the eigenvalue decomposition of the covariance matrices of the domains have asymptotic expansions with scale that contain the curvature information encoded by the traces of this tensor. In the case of hypersurfaces, this covariance analysis recovers the principal curvatures and principal directions, which can be used as descriptors at scale to build up estimators of the second fundamental form, and thus the Riemann tensor, of general submanifolds.

math.DG↗

arXiv2018

Manifold Curvature Descriptors from Hypersurface Integral Invariants

Integral invariants obtained from Principal Component Analysis on a small kernel domain of a submanifold encode important geometric information classically defined in differential-geometric terms. We generalize to hypersurfaces in any dimension major results known for surfaces in space, which in turn yield a method to estimate the extrinsic and intrinsic curvature of an embedded Riemannian submanifold of general codimension. In particular, integral invariants are defined by the volume, barycenter, and the EVD of the covariance matrix of the domain. We obtain the asymptotic expansion of such invariants for a spherical volume component delimited by a hypersurface and for the hypersurface patch created by ball intersetions, showing that the eigenvalues and eigenvectors can be used as multi-scale estimators of the principal curvatures and principal directions. This approach may be interpreted as performing statistical analysis on the underlying point-set of a submanifold in order to obtain geometric descriptors at scale with potential applications to Manifold Learning and Geometry Processing of point clouds.

math.DG↗

arXiv2017

Geometry of Curves in $\mathbb R^n$, Singular Value Decomposition, and Hankel Determinants

Let $γ: I \rightarrow \mathbb R^n$ be a parametric curve of class $C^{n+1}$, regular of order $n$. The Frenet-Serret apparatus of $γ$ at $γ(t)$ consists of a frame $e_1(t), \dots , e_n(t)$ and generalized curvature values $κ_1(t), \dots, κ_{n-1}(t)$. Associated with each point of $γ$ there are also local singular vectors $u_1(t), \dots, u_n(t)$ and local singular values $σ_1(t), \dots, σ_{n}(t)$. This local information is obtained by considering a limit, as $ε$ goes to zero, of covariance matrices defined along $γ$ within an $ε$-ball centered at $γ(t)$. We prove that for each $t\in I$, the Frenet-Serret frame and the local singular vectors agree at $γ(t)$ and that the values of the curvature functions at $t$ can be expressed as a fixed multiple of a ratio of local singular values at $t$. More precisely, we show that if $γ(t)\subset \mathbb R^n$ for any $n\in\mathbb N$ then, for each $i$ between $2$ and $n$, $κ_{i-1}(t)=\sqrt{a_{i-1}}\frac{σ_{i}(t)}{σ_1(t) σ_{i-1}(t)}$ with $a_{i-1} = \left(\frac{i}{i+(-1)^i}\right)^2 {\frac{4i^2-1}{3}}$. For this we prove a general formula for the recursion relation of a certain class of sequences of Hankel determinants using the theory of monic orthogonal polynomials and moment sequences.

math.DG↗

arXiv2016

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

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.

cs.CG↗

arXiv2016

Persistence Images: A Stable Vector Representation of Persistent Homology

Many datasets 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 dataset. 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.

cs.CG↗

arXiv2016

Persistent Homology on Grassmann Manifolds for Analysis of Hyperspectral Movies

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.

cs.CV↗

arXiv2015

High Energy Physics Forum for Computational Excellence: Working Group Reports (I. Applications Software II. Software Libraries and Tools III. Systems)

Computing plays an essential role in all aspects of high energy physics. As computational technology evolves rapidly in new directions, and data throughput and volume continue to follow a steep trend-line, it is important for the HEP community to develop an effective response to a series of expected challenges. In order to help shape the desired response, the HEP Forum for Computational Excellence (HEP-FCE) initiated a roadmap planning activity with two key overlapping drivers -- 1) software effectiveness, and 2) infrastructure and expertise advancement. The HEP-FCE formed three working groups, 1) Applications Software, 2) Software Libraries and Tools, and 3) Systems (including systems software), to provide an overview of the current status of HEP computing and to present findings and opportunities for the desired HEP computational roadmap. The final versions of the reports are combined in this document, and are presented along with introductory material.

physics.comp-ph↗

arXiv2015

HEP-FCE Working Group on Libraries and Tools

This is a report from the Libraries and Tools Working Group of the High Energy Physics Forum for Computational Excellence. It presents the vision of the working group for how the HEP software community may organize and be supported in order to more efficiently share and develop common software libraries and tools across the world's diverse set of HEP experiments. It gives prioritized recommendations for achieving this goal and provides a survey of a select number of areas in the current HEP software library and tools landscape. The survey identifies aspects which support this goal and areas with opportunities for improvements. The survey covers event processing software frameworks, software development, data management, workflow and workload management, geometry information management and conditions databases.

hep-ex↗

arXiv2015

Outcome prediction in mathematical models of immune response to infection

Clinicians need to predict patient outcomes with high accuracy as early as possible after disease inception. In this manuscript, we show that patient-to-patient variability sets a fundamental limit on outcome prediction accuracy for a general class of mathematical models for the immune response to infection. However, accuracy can be increased at the expense of delayed prognosis. We investigate several systems of ordinary differential equations (ODEs) that model the host immune response to a pathogen load. Advantages of systems of ODEs for investigating the immune response to infection include the ability to collect data on large numbers of `virtual patients', each with a given set of model parameters, and obtain many time points during the course of the infection. We implement patient-to-patient variability $v$ in the ODE models by randomly selecting the model parameters from Gaussian distributions with variance $v$ that are centered on physiological values. We use logistic regression with one-versus-all classification to predict the discrete steady-state outcomes of the system. We find that the prediction algorithm achieves near $100\%$ accuracy for $v=0$, and the accuracy decreases with increasing $v$ for all ODE models studied. The fact that multiple steady-state outcomes can be obtained for a given initial condition, i.e. the basins of attraction overlap in the space of initial conditions, limits the prediction accuracy for $v>0$. Increasing the elapsed time of the variables used to train and test the classifier, increases the prediction accuracy, while adding explicit external noise to the ODE models decreases the prediction accuracy. Our results quantify the competition between early prognosis and high prediction accuracy that is frequently encountered by clinicians.

q-bio.QM↗

arXiv2015

Classification of Hyperspectral Imagery on Embedded Grassmannians

We propose an approach for capturing the signal variability in hyperspectral imagery using the framework of the Grassmann manifold. Labeled points from each class are sampled and used to form abstract points on the Grassmannian. The resulting points on the Grassmannian have representations as orthonormal matrices and as such do not reside in Euclidean space in the usual sense. There are a variety of metrics which allow us to determine a distance matrices that can be used to realize the Grassmannian as an embedding in Euclidean space. We illustrate that we can achieve an approximately isometric embedding of the Grassmann manifold using the chordal metric while this is not the case with geodesic distances. However, non-isometric embeddings generated by using a pseudometric on the Grassmannian lead to the best classification results. We observe that as the dimension of the Grassmannian grows, the accuracy of the classification grows to 100% on two illustrative examples. We also observe a decrease in classification rates if the dimension of the points on the Grassmannian is too large for the dimension of the Euclidean space. We use sparse support vector machines to perform additional model reduction. The resulting classifier selects a subset of dimensions of the embedding without loss in classification performance.

cs.CV↗

arXiv2012

Locally Linear Embedding Clustering Algorithm for Natural Imagery

The ability to characterize the color content of natural imagery is an important application of image processing. The pixel by pixel coloring of images may be viewed naturally as points in color space, and the inherent structure and distribution of these points affords a quantization, through clustering, of the color information in the image. In this paper, we present a novel topologically driven clustering algorithm that permits segmentation of the color features in a digital image. The algorithm blends Locally Linear Embedding (LLE) and vector quantization by mapping color information to a lower dimensional space, identifying distinct color regions, and classifying pixels together based on both a proximity measure and color content. It is observed that these techniques permit a significant reduction in color resolution while maintaining the visually important features of images.

math.GT↗