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A flag is a nested sequence of vector spaces. The type of the flag encodes the sequence of dimensions of the vector spaces making up the flag. A flag manifold is a manifold whose points parameterize all flags of a fixed type in a fixed vector space. This paper provides the mathematical framework necessary for implementing selforganizing mappings on flag manifolds. Flags arise implicitly in many data analysis contexts including wavelet, Fourier, and singular value decompositions. The proposed geometric framework in this paper enables the computation of distances between flags, the computation of geodesics between flags, and the ability to move one flag a prescribed distance in the direction of another flag. Using these operations as building blocks, we implement the SOM algorithm on a flag manifold. The basic algorithm is applied to the problem of parameterizing a set of flags of a fixed type.more » « less

Finding prototypes (e.g., mean and median) for a dataset is central to a number of common machine learning algorithms. Subspaces have been shown to provide useful, robust representations for datasets of images, videos and more. Since subspaces correspond to points on a Grassmann manifold, one is led to consider the idea of a subspace prototype for a Grassmannvalued dataset. While a number of different subspace prototypes have been described, the calculation of some of these prototypes has proven to be computationally expensive while other prototypes are affected by outliers and produce highly imperfect clustering on noisy data. This work proposes a new subspace prototype, the flag median, and introduces the FlagIRLS algorithm for its calculation. We provide evidence that the flag median is robust to outliers and can be used effectively in algorithms like LindeBuzoGrey (LBG) to produce improved clusterings on Grassmannians. Numerical experiments include a synthetic dataset, the MNIST handwritten digits dataset, the Mind's Eye video dataset and the UCF YouTube action dataset. The flag median is compared the other leading algorithms for computing prototypes on the Grassmannian, namely, the l_2median and to the flag mean. We find that using FlagIRLS to compute the flag median converges in 4 iterations on a synthetic dataset. We also see that Grassmannian LBG with a codebook size of 20 and using the flag median produces at least a 10% improvement in cluster purity over Grassmannian LBG using the flag mean or l_2median on the Mind's Eye dataset.more » « less

Abstract In this paper, we review scientific opportunities and challenges related to detection and reconstruction of lowenergy (less than 100 MeV) signatures in liquid argon timeprojection chamber (LArTPC) neutrino detectors. LArTPC neutrino detectors designed for performing precise longbaseline oscillation measurements with GeVscale accelerator neutrino beams also have unique sensitivity to a range of physics and astrophysics signatures via detection of event features at and below the few tens of MeV range. In addition, lowenergy signatures are an integral part of GeVscale accelerator neutrino interaction finalstates, and their reconstruction can enhance the oscillation physics sensitivities of LArTPC experiments. New physics signals from accelerator and natural sources also generate diverse signatures in the lowenergy range, and reconstruction of these signatures can increase the breadth of Beyond the Standard Model scenarios accessible in LArTPCbased searches. A variety of experimental and theoryrelated challenges remain to realizing this full range of potential benefits. Neutrino interaction crosssections and other nuclear physics processes in argon relevant to subhundredMeV LArTPC signatures are poorly understood, and improved theory and experimental measurements are needed; pion decayatrest sources and charged particle and neutron test beams are ideal facilities for improving this understanding. There are specific calibration needs in the lowenergy range, as well as specific needs for control and understanding of radiological and cosmogenic backgrounds. Lowenergy signatures, whether steadystate or part of a supernova burst or larger GeVscale event topology, have specific triggering, DAQ and reconstruction requirements that must be addressed outside the scope of conventional GeVscale data collection and analysis pathways. Novel concepts for future LArTPC technology that enhance lowenergy capabilities should also be explored to help address these challenges.more » « lessFree, publiclyaccessible full text available January 1, 2024

A flag is a nested sequence of vector spaces. The type of the flag is determined by the sequence of dimensions of the vector spaces making up the flag. A flag manifold is a manifold whose points parameterize all flags of a particular type in a fixed vector space. This paper provides the mathematical framework necessary for implementing selforganizing mappings on flag manifolds. Flags arise implicitly in many data analysis techniques for instance in wavelet, Fourier, and singular value decompositions. The proposed geometric framework in this paper enables the computation of distances between flags, the computation of geodesics between flags, and the ability to move one flag a prescribed distance in the direction of another flag. Using these operations as building blocks, we implement the SOM algorithm on a flag manifold. The basic algorithm is applied to the problem of parameterizing a set of flags of a fixed type.more » « less

Free, publiclyaccessible full text available June 1, 2024

Principal component analysis of cylindrical neighborhoods is proposed to study the local geometry of embedded Riemannian manifolds. At every generic point and scale, a highdimensional cylinder orthogonal to the tangent space at the point cuts out a pathconnected patch whose pointset distribution in ambient space encodes the intrinsic and extrinsic curvature. The covariance matrix of the points from that neighborhood has eigenvectors whose scale limit tends to the FrenetSerret frame for curves, and to what we call the RicciWeingarten principal directions for submanifolds. More importantly, the limit of differences and products of eigenvalues can be used to recover curvature information at the point. The formula for hypersurfaces in terms of principal curvatures is particularly simple and plays a crucial role in the study of highercodimension cases.more » « less

Abstract—Given a collection of M experimentally measured subspaces, and a modelbased subspace, this paper addresses the problem of finding a subspace that approximates the collection, under the constraint that it intersects the modelbased subspace in a predetermined number of dimensions. This constrained subspace estimation (CSE) problem arises in applications such as beamforming, where the modelbased subspace encodes prior information about the directionofarrival 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 nonconvex problem. The performance of the proposed CSE algorithm is demonstrated via numerical simulation, and its application to beamforming is also discussed.more » « less

Free, publiclyaccessible full text available January 1, 2024