This paper introduces a novel method for visualizing high-dimensional Grassmannians through 2D embeddings on the Poincaré disk. The proposed approach involves the construction of affinity matrices on each manifold, followed by the minimization of KL-divergence between the geodesics affinity. This process enables the identification of an optimal projection that effectively preserves the geometry of the original high-dimensional Grassmannian. Our main theoretical contribution lies in bounding the embedding loss with respect to two factors: the logarithm of the number of subspaces and a term dependent on the distribution of subspaces within the Grassmannian. Notably, this term is smaller when subspaces exhibit well-defined clusters, and larger in the absence of any discernible structure. We complement our theoretical analysis with comprehensive experiments on both synthetic and real datasets. The experimental results showcase the superiority of our embedding in accurately visualizing Grassmannians compared to existing representations.
GLSMs for exotic Grassmannians
A bstract In this paper we explore nonabelian gauged linear sigma models (GLSMs) for symplectic and orthogonal Grassmannians and flag manifolds, checking e.g. global symmetries, Witten indices, and Calabi-Yau conditions, following up a proposal in the math community. For symplectic Grassmannians, we check that Coulomb branch vacua of the GLSM are consistent with ordinary and equivariant quantum cohomology of the space.
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- Award ID(s):
- 1720321
- NSF-PAR ID:
- 10219754
- Date Published:
- Journal Name:
- Journal of High Energy Physics
- Volume:
- 2020
- Issue:
- 10
- ISSN:
- 1029-8479
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract -
Gaussian processes (GPs) are very widely used for modeling of unknown functions or surfaces in applications ranging from regression to classification to spatial processes. Although there is an increasingly vast literature on applications, methods, theory and algorithms related to GPs, the overwhelming majority of this literature focuses on the case in which the input domain corresponds to a Euclidean space. However, particularly in recent years with the increasing collection of complex data, it is commonly the case that the input domain does not have such a simple form. For example, it is common for the inputs to be restricted to a non-Euclidean manifold, a case which forms the motivation for this article. In particular, we propose a general extrinsic framework for GP modeling on manifolds, which relies on embedding of the manifold into a Euclidean space and then constructing extrinsic kernels for GPs on their images. These extrinsic Gaussian processes (eGPs) are used as prior distributions for unknown functions in Bayesian inferences. Our approach is simple and general, and we show that the eGPs inherit fine theoretical properties from GP models in Euclidean spaces. We consider applications of our models to regression and classification problems with predictors lying in a large class of manifolds, including spheres, planar shape spaces, a space of positive definite matrices, and Grassmannians. Our models can be readily used by practitioners in biological sciences for various regression and classification problems, such as disease diagnosis or detection. Our work is also likely to have impact in spatial statistics when spatial locations are on the sphere or other geometric spaces.more » « less
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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 Grassmann-valued 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 Linde-Buzo-Grey (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_2-median 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_2-median on the Mind's Eye dataset.more » « less
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Commuting integral and differential operators connect the topics of signal processing, random matrix theory, and integrable systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the Korteweg–de Vries (KdV) equation. We prove a general theorem that the integral operator associated to every wave function in the infinite-dimensional adelic Grassmannian G r a d of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intrinsic property is shown to follow from the symmetries of Grassmannians of Kadomtsev–Petviashvili (KP) wave functions, where the direct commutativity property holds for operators associated to wave functions fixed by Wilson’s sign involution but is violated in general. Based on this result, we prove a second main theorem that the integral operators in the computation of the singular values of the truncated generalized Laplace transforms associated to all bispectral wave functions of rank 1 reflect a differential operator. A 9 0 ○ rotation argument is used to prove a third main theorem that the integral operators in the computation of the singular values of the truncated generalized Fourier transforms associated to all such KP wave functions commute with a differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, including as special cases the integral operators associated to all rational solutions of the KdV and KP hierarchies considered by [Airault, McKean, and Moser, Commun. Pure Appl. Math. 30, 95–148 (1977)] and [Krichever, Funkcional. Anal. i Priložen. 12, 76–78 (1978)], respectively, in the late 1970s. Many examples are presented.more » « less
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null (Ed.)In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of all such invariants using finite-difference operators, the role of the q-hypergeometric series arising in the context of quasimap compactifications of spaces of rational curves in such varieties, the theory of twisted GW-invariants including level structures, as well as the Jackson-type integrals playing the role of equivariant K-theoretic mirrors.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/JHEP10(2020)200
