Grassmann Research Articles

Robust Grassmann manifold convex hull collaborative representation learning and its kernel extension for image set analysis

Robust Grassmann manifold convex hull collaborative representation learning and its kernel extension for image set analysis

Symplectic Grassmannians and cyclic quivers

AbstractThe goal of this paper is to extend the quiver Grassmannian description of certain degenerations of Grassmann varieties to the symplectic case. We introduce a symplectic version of quiver Grassmannians studied in our previous papers and prove a number of results on these projective algebraic varieties. First, we construct a cellular decomposition of the symplectic quiver Grassmannians in question and develop combinatorics needed to compute Euler characteristics and Poincaré polynomials. Second, we show that the number of irreducible components of our varieties coincides with the Euler characteristic of the classical symplectic Grassmannians. Third, we describe the automorphism groups of the underlying symplectic quiver representations and show that the cells are the orbits of this group. Lastly, we provide an embedding into the affine flag varieties for the affine symplectic group.

Multi-kernel clustering with tensor fusion on Grassmann manifold for high-dimensional genomic data

The high dimensionality and noise challenges in genomic data make it difficult for traditional clustering methods. Existing multi-kernel clustering methods aim to improve the quality of the affinity matrix by learning a set of base kernels, thereby enhancing clustering performance. However, directly learning from the original base kernels presents challenges in handling errors and redundancies when dealing with high-dimensional data, and there is still a lack of feasible multi-kernel fusion strategies. To address these issues, we propose a Multi-Kernel Clustering method with Tensor fusion on Grassmann manifolds, called MKCTM. Specifically, we maximize the clustering consensus among base kernels by imposing tensor low-rank constraints to eliminate noise and redundancy. Unlike traditional kernel fusion approaches, our method fuses learned base kernels on the Grassmann manifold, resulting in a final consensus matrix for clustering. We integrate tensor learning and fusion processes into a unified optimization model and propose an effective iterative optimization algorithm for solving it. Experimental results on ten datasets, comparing against 12 popular baseline clustering methods, confirm the superiority of our approach. Our code is available at https://github.com/foureverfei/MKCTM.git.

Geodesic Convexity of the Symmetric Eigenvalue Problem and Convergence of Steepest Descent

AbstractWe study the convergence of the Riemannian steepest descent algorithm on the Grassmann manifold for minimizing the block version of the Rayleigh quotient of a symmetric matrix. Even though this problem is non-convex in the Euclidean sense and only very locally convex in the Riemannian sense, we discover a structure for this problem that is similar to geodesic strong convexity, namely, weak-strong convexity. This allows us to apply similar arguments from convex optimization when studying the convergence of the steepest descent algorithm but with initialization conditions that do not depend on the eigengap $$\delta $$ δ . When $$\delta >0$$ δ > 0 , we prove exponential convergence rates, while otherwise the convergence is algebraic. Additionally, we prove that this problem is geodesically convex in a neighbourhood of the global minimizer of radius $${\mathcal {O}}(\sqrt{\delta })$$ O ( δ ) .

Federated PCA on Grassmann Manifold for IoT Anomaly Detection

Federated PCA on Grassmann Manifold for IoT Anomaly Detection

Polynomial chaos expansions on principal geodesic Grassmannian submanifolds for surrogate modeling and uncertainty quantification

In this work we introduce a manifold learning-based surrogate modeling framework for uncertainty quantification in high-dimensional stochastic systems. Our first goal is to perform data mining on the available simulation data to identify a set of low-dimensional (latent) descriptors that efficiently parameterize the response of the high-dimensional computational model. To this end, we employ Principal Geodesic Analysis on the Grassmann manifold of the response to identify a set of disjoint principal geodesic submanifolds, of possibly different dimension, that captures the variation in the data. Since operations on the Grassmann require the data to be concentrated, we propose an adaptive algorithm based on Riemannian K-means and the minimization of the sample Fréchet variance on the Grassmann manifold to identify “local” principal geodesic submanifolds that represent different system behavior across the parameter space. Polynomial chaos expansion is then used to construct a mapping between the random input parameters and the projection of the response on these local principal geodesic submanifolds. The method is demonstrated on four test cases, a toy-example that involves points on a hypersphere, a Lotka-Volterra dynamical system, a continuous-flow stirred-tank chemical reactor system, and a two-dimensional Rayleigh-Bénard convection problem.

Riemannian Newton Methods for Energy Minimization Problems of Kohn–Sham Type

This paper is devoted to the numerical solution of constrained energy minimization problems arising in computational physics and chemistry such as the Gross–Pitaevskii and Kohn–Sham models. In particular, we introduce Riemannian Newton methods on the infinite-dimensional Stiefel and Grassmann manifolds. We study the geometry of these two manifolds, its impact on the Newton algorithms, and present expressions of the Riemannian Hessians in the infinite-dimensional setting, which are suitable for variational spatial discretizations. A series of numerical experiments illustrates the performance of the methods and demonstrates their supremacy compared to other well-established schemes such as the self-consistent field iteration and gradient descent schemes.

Model order reduction of thermal-dynamic coupled flexible multibody system with multiple varying parameters

Spacecrafts usually suffer from the thermal shock during operation, leading to large control error or even failure. At the same time, spacecrafts are often rigid-flexible coupled multibody systems with large degrees of freedom and multiple varying parameters. The real-time control and condition monitoring require effective order reduction methods. In this investigation, the displacement and temperature fields are discretized by the Absolute Node Coordinate Formulation (ANCF) to achieve unified description. The coupled thermal-dynamic reduced-order model (ROM) is established based on the Proper Orthogonal Decomposition (POD) method. For the purpose of further increase efficiency, a linearization method is introduced so that the calculation times of the elastic force can be significantly reduced. In order to predict the response of the thermal-dynamic coupled system with multiple varying parameters, the interpolation on Grassmann manifold is introduced to maintain the orthogonality of the basis. As verification and validation, three numerical examples are presented to show the feasibility and efficacy of the proposed method.

The algorithm for denoising point clouds of annular forgings based on Grassmann manifold and density clustering

Abstract In the industrial sector, annular forgings serve as critical load-bearing components in mechanical equipment. During the production process, the precise measurement of the dimensional parameters of annular forgings is of paramount importance to ensure their quality and safety. However, owing to the influence of the measurement environment, the manufacturing process of annular forgings can introduce varying degrees of noise, resulting in inaccurate dimensional measurements. Therefore, researching methods for three-dimensional point cloud data to eliminate noise in annular forging point clouds is of significant importance for improving the accuracy of forging measurements. This paper presents a denoising approach for three-dimensional point cloud data of annular forgings based on Grassmann manifold and density clustering (GDAD). First, within the Grassmann manifold, the core points for density clustering are determined using density parameters. Second, density clustering is performed within the Grassmann manifold, with the Cauchy distance replacing the Euclidean distance to reduce the impact of noise and outliers on the analysis results. Finally, a search tree model was constructed to filter out incorrect point cloud clusters. The fusion of clustering results and the search tree model achieved denoising of point cloud data. Simulation experiments on annular forgings demonstrate that GDAD effectively eliminates edge noise in annular forgings and performs well in denoising point-cloud models with varying levels of noise intensity.

Boosting Spectral Efficiency With Data-Carrying Reference Signals on the Grassmann Manifold

Boosting Spectral Efficiency With Data-Carrying Reference Signals on the Grassmann Manifold

Generic Stabilizers in Actions of Simple Algebraic Groups

In this paper we treat faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties. By explicit calculation, we show that in each case, with essentially one exception, there is a dense open subset any point of which has stabilizer conjugate to a fixed subgroup, called the generic stabilizer. We provide tables listing generic stabilizers in the cases where they are non-trivial; in addition we decide whether or not there is a dense orbit, or a regular orbit for the action on the module.

Signal latent subspace: A new representation for environmental sound classification

In this study, we propose Signal Latent Subspace (SLS), a flexible method that classifies environmental sound events using the subspace representations of latent features obtained from various neural network-based models. Our main goal is to leverage the high expressiveness of neural networks while retaining the advantages of subspace representation, such as its robustness to noise and ability to work under small sample size (SSS) conditions. We also propose an ensemble strategy native to the subspace representation, to achieve increased performance and reduce the generalization error. We do this through product Grassmann manifold (PGM), resulting in SLS-PGM. Each subspace constructed from latent features of a network can be seen as a point on a factor Grassmann manifold (GM) of a neural network; through PGM, it is possible to unify factor manifolds into a singular representation, and perform classification through a similarity metric on the manifold. We further improve SLS and SLS-PGM in two ways: (1) by using generalized difference subspace (GDS) projection to address the lack of between-class discrimination of subspace representation and (2) by leveraging finetuning regimes to better adapt neural network models to the ESC task. We evaluate our proposed methods, factoring various neural networks, on ESC-10, ESC-50 and UrbanSound environmental sound datasets, and provide extensive ablation experiments and notes for practical use.

Pinched Constantly Curved Holomorphic Two-Spheres in the Complex Grassmann Manifolds

Pinched Constantly Curved Holomorphic Two-Spheres in the Complex Grassmann Manifolds

Exploring geometry of genome space via Grassmann manifolds

It is important to understand the geometry of genome space in biology. After transforming genome sequences into frequency matrices of the chaos game representation (FCGR), we regard a genome sequence as a point in a suitable Grassmann manifold by analyzing the column space of the corresponding FCGR. To assess the sequence similarity, we employ the generalized Grassmannian distance, an intrinsic geometric distance that differs from the traditional Euclidean distance used in the classical k-mer frequency-based methods. With this method, we constructed phylogenetic trees for various genome datasets, including influenza A virus hemagglutinin gene, Orthocoronavirinae genome, and SARS-CoV-2 complete genome sequences. Our comparative analysis with multiple sequence alignment and alignment-free methods for large-scale sequences revealed that our method, which employs the subspace distance between the column spaces of different FCGRs (FCGR-SD), outperformed its competitors in terms of both speed and accuracy. In addition, we used low-dimensional visualization of the SARS-CoV-2 genome sequences and spike protein nucleotide sequences with our methods, resulting in some intriguing findings. We not only propose a novel and efficient algorithm for comparing genome sequences but also demonstrate that genome data have some intrinsic manifold structures, providing a new geometric perspective for molecular biology studies.

Deep Fusion of Intrinsic Vibration Information and Grassmann Manifold-based Similarity for Fault Identification of Reciprocating Compressor

This paper introduces a new method based on deep belief networks (DBN) to integrate intrinsic vibration information and assess the similarity of subspaces established on the Grassmann manifold for intelligent fault diagnosis of a reciprocating compressor (RC). Initially, raw vibration signals undergo empirical mode decomposition (EMD) to break them down into multiple intrinsic mode functions (IMFs). This operation can reveal inherent vibration patterns of fault and other components hidden in the original signals. Subsequently, features are refined from all the IMFs and concatenated into a high-dimensional representative vector, offering localized and comprehensive insights into RC operation. Through DBN, the fault-sensitive information are further refined from the features to enhance their performance in fault identification. Finally, similarities among subspaces on the Grassmann manifold are computed to match fault types. The efficacy of the method is validated using field data. Comparative analysis with traditional approaches for feature dimension reduction, feature extraction, and Euclidean distance-based fault identification underscores the effectiveness and superiority of the proposed method in RC fault diagnosis. Conflict of Interest Statement The authors declare no conflicts of interest.

On a generalized notion of metrics

In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n-metric which assigns a value to a tuple of n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 2$$\\end{document} points. Some elementary properties of pseudo n-metrics are provided and their construction via exterior products is investigated. We discuss some examples from the geometry of Euclidean vector spaces leading to pseudo n-metrics on the unit sphere, on the Stiefel manifold, and on the Grassmann manifold. Further, we construct a pseudo n-metric on hypergraphs and discuss the problem of generalizing the Hausdorff metric for closed sets to a pseudo n-metric.

On the Geometry of Grassmannian Manifold

On the Geometry of Grassmannian Manifold

Distributed Learning for Principal Eigenspaces without Moment Constraints

Distributed Principal Component Analysis (PCA) has been studied to deal with the case when data are stored across multiple machines and communication cost or privacy concerns prohibit the computation of PCA in a central location. However, the sub-Gaussian assumption in the related literature is restrictive in real application where outliers or heavy-tailed data are common in areas such as finance and macroeconomics. In this article, we propose a distributed algorithm for estimating the principal eigenspaces without any moment constraints on the underlying distribution. We study the problem under the elliptical family framework and adopt the sample multivariate Kendall’s tau matrix to extract eigenspace estimators from all submachines, which can be viewed as points in the Grassmann manifold. We then find the “center” of these points as the final distributed estimator of the principal eigenspace. We investigate the bias and variance for the distributed estimator and derive its convergence rate which depends on the effective rank, eigengap of the scatter matrix and the number of submachines. We show that the distributed estimator performs as if we have full access to the whole data. Simulation studies show that the distributed algorithm performs comparably with the existing one for light-tailed data, while showing great advantages for heavy-tailed data. We also extend the distributed algorithm to cases with limited communication constraints and with elliptical factor structure. Thorough simulation studies and a real application to a macroeconomic dataset verify the advantages of the proposed distributed algorithms. Supplementary materials for this article are available online.

Infinite Families of (q^2+1)-Tight Sets of Quadrics with an Automorphism Group PSp(n,q), n=4,6

We present new infinite families of $(q^2+1)$-tight sets of quadrics admitting the symplectic group ${\rm PSp}(n,q)$, $n=4,6$, as an automorphism group. Our constructions rely on the geometry of Veronese and Grassmann varieties.

Parametric Nonlinear Model Reduction Using Machine Learning on Grassmann Manifold with an Application on a Flow Simulation

Parametric Nonlinear Model Reduction Using Machine Learning on Grassmann Manifold with an Application on a Flow Simulation