Grassmann Research Articles

Grassmann manifold based framework for automated fall detection from a camera

In this paper, we propose a Grassmann manifold based novel, real-time framework for automated fall detection in an indoor environment using a single camera. Fall is an activity that is uncontrolled, unintentional, involuntary and can occur while a person is doing any daily living activity. This is especially true for the elderly and the sick, for whom a fall can lead to further complications that may cause irreversible damage to their health. Therefore, it is important to develop a non-intrusive, automated fall detection method such that an alert can be raised in case a fall occurs. We propose a Grassmann manifold based framework for fall detection from a single camera that is also capable of recognizing other daily living activities (DLA) such as walking, sitting, etc. We perform experiments using publicly available datasets and our experimental results show that the fall detection and recognition accuracy of our proposed framework is comparable with the state of the art.

A weighted sparse coding model on product Grassmann manifold for video-based human gesture recognition.

It is a challenging problem to classify multi-dimensional data with complex intrinsic geometry inherent, such as human gesture recognition based on videos. In particular, manifold structure is a good way to characterize intrinsic geometry of multi-dimensional data. The recently proposed sparse coding on Grassmann manifold shows high discriminative power in many visual classification tasks. It represents videos on Grassmann manifold using Singular Value Decomposition (SVD) of the data matrix by vectorizing each image in videos, while vectorization destroys the spatial structure of videos. To keep the spatial structure of videos, they can be represented as the form of data tensor. In this paper, we firstly represent human gesture videos on product Grassmann manifold (PGM) by Higher Order Singular Value Decomposition (HOSVD) of data tensor. Each factor manifold characterizes features of human gesture video from different perspectives and can be understood as appearance, horizontal motion and vertical motion of human gesture video respectively. We then propose a weighted sparse coding model on PGM, where weights can be understood as modeling the importance of factor manifolds. Furthermore, we propose an optimization algorithm for learning coding coefficients by embedding each factor Grassmann manifold into symmetric matrices space. Finally, we give a classification algorithm, and experimental results on three public datasets show that our method is competitive to some relevant excellent methods.

Grassmannian Diffusion Maps--Based Dimension Reduction and Classification for High-Dimensional Data

This work introduces the Grassmannian diffusion maps (GDMaps), a novel nonlinear dimensionality reduction technique that defines the affinity between points through their representation as low-dimensional subspaces corresponding to points on the Grassmann manifold. The method is designed for applications, such as image recognition and data-based classification of constrained high-dimensional data where each data point itself is a high-dimensional object (i.e., a large matrix) that can be compactly represented in a lower-dimensional subspace. The GDMaps is composed of two stages. The first is a pointwise linear dimensionality reduction wherein each high-dimensional object is mapped onto the Grassmann manifold representing the low-dimensional subspace on which it resides. The second stage is a multipoint nonlinear kernel-based dimension reduction using diffusion maps to identify the subspace structure of the points on the Grassmann manifold. To this end, an appropriate Grassmannian kernel is used to construct the transition matrix of a random walk on a graph connecting points on the Grassmann manifold. Spectral analysis of the transition matrix yields low-dimensional Grassmannian diffusion coordinates embedding the data into a low-dimensional reproducing kernel Hilbert space. Further, a novel data classification/recognition technique is developed based on the construction of an overcomplete dictionary of reduced dimension whose atoms are given by the Grassmannian diffusion coordinates. Three examples are considered. First, a “toy” example shows that the GDMaps can identify an appropriate parametrization of structured points on the unit sphere. The second example demonstrates the ability of the GDMaps to revealing the intrinsic subspace structure of high-dimensional random field data. In the last example, a face recognition problem is solved considering face images subject to varying illumination conditions, changes in face expressions, and occurrence of occlusions. The technique presented high recognition rates (i.e., 95% in the best case) using a fraction of the data required by conventional methods.

Reduced Order Modeling of Deformable Tire-Soil Interaction With Proper Orthogonal Decomposition

AbstractIn this study, model order reduction of high-fidelity off-road mobility models is explored to address the computational intensity of nonlinear finite element deformable tire–soil interaction models. To this end, a model order reduction procedure for the tire–soil interaction model is developed with the proper orthogonal decomposition (POD), and it is integrated into the off-road mobility simulation framework, leveraging high-performance computing. The POD is, however, limited in that the modes are dependent on snapshot data collected during the running of a full order model, limiting the modes to being accurate only for the specific scenario from which they were collected. Due to this limitation, a method of mode adaptation through interpolation on a tangent space of the Grassmann manifold is investigated to allow modes to be predicted for cases in which a full order model has not been run. It is demonstrated by several numerical examples that the POD modes are effective at retaining predictive accuracy while reducing computational time. The results show that adapted POD modes are more capable of characterizing the behavior of the model than modes produced at a different value of the simulation parameter. The POD-based reduced order modeling approach is further extended to the full vehicle simulation on deformable terrain through the co-simulation coupling algorithm by leveraging the high-performance computing technique.

GraSP: Local Grassmannian Spatio-Temporal Patterns for Unsupervised Pose Sequence Recognition

Many applications of action recognition, especially broad domains like surveillance or anomaly-detection, favor unsupervised methods considering that exhaustive labeling of actions is not possible. However, very limited work has happened in this domain. Moreover, the existing self-supervised approaches suffer from their dependency upon labeled data for finetuning. To this end, this paper puts forward a manifold based unsupervised pose-sequence recognition approach that leverages only the natural biases present in the data. It works by clustering the projections of temporal derivatives of the fragmented data on the Grassmann manifold. Temporal derivatives are formed by the inter-frame gradients with local and global metrics. To commensurate with this, a dynamic view-invariant pose representation is proposed. Additionally, a variable aggregation step is introduced for better feature vector quantization. Extensive empirical evaluation and ablations on several challenging datasets under three categories confirm the superiority of the proposed approach in contrast to current methods.

Nested Grassmannians for Dimensionality Reduction with Applications

In the recent past, nested structure of Riemannian manifolds has been studied in the context of dimensionality reduction as an alternative to the popular principal geodesic analysis (PGA) technique, for example, the principal nested spheres. In this paper, we propose a novel framework for constructing a nested sequence of homogeneous Riemannian manifolds. Common examples of homogeneous Riemannian manifolds include the spheres, the Stiefel manifolds, and the Grassmann manifolds. In particular, we focus on applying the proposed framework to the Grassmann manifolds, giving rise to the nested Grassmannians (NG). An important application in which Grassmann manifolds are encountered is planar shape analysis. Specifically, each planar (2D) shape can be represented as a point in the complex projective space which is a complex Grassmann manifold. Some salient features of our framework are: (i) it explicitly exploits the geometry of the homogeneous Riemannain manifolds and (ii) the nested lower-dimensional submanifolds need not be geodesic. With the proposed NG structure, we develop algorithms for the supervised and unsupervised dimensionality reduction problems respectively. The proposed algorithms are compared with PGA via simulation studies and real data experiments and are shown to achieve a higher ratio of expressed variance compared to PGA.<br>The code is available at <a href='https://github.com/cvgmi/NestedGrassmann'>https://github.com/cvgmi/NestedGrassmann</a>

A bottom-up ‘Game of Lie’ cellular automaton evolution in SO(3) root space both nucleating, crystallizing and space-filling the complete atomic realm

Re-examining Marius Sophus Lie’s Ph.D. thesis Over en Classe geometriske Transformationer from 1871 I found that its bearings on the “nature of Cartesian geometry” are that it “translates any geometric theorem into an algebraic one and thus…of the geometry of space a representation of the algebra of three variable quantities” with both the algebraic and “geometrical transformation…consisting of a transition from a point to a straight line as element…through a particularly remarkable transformation” in which “the Plücker line geometry can be transformed into a sphere geometry”, by its “straight lines of length equal to zero” partial derivative elements “turning…into the sphere’s rectilinear generatrices” both as “partial differential equations of the first order” and physical “geodetic curves” under the “general equation system f(x, y, z, dx, dy, dz) = 0” and its “spatial reciprocity” that “relative to the given line complex” in x, y, z “corresponds a certain curve-net” in the simultaneous “line element (dx, dy, dz)”. This is a revolutionary geometric algebra discovery of the constitution of our Universe as a stru ctural phase transition between Straight and Round, transcending Hermann Grassmann’s Ausdehnungslehre (1844/1862) and William Kingdom Clifford’s Space-theory of Matter (1876) by the infinitesimal transformations opening up spherical geometry in its real form identity state later assigned as SO(3). The continuous groups, whose vast field occupied the rest of his all too short life were not mentioned in the thesis but implied both in the term Classe (class) geometriske transformationer and the coverage of their principles in an “unlimited manifold of possible systems”; e.g. “turning rounded curves into rounded curves” and “straight lines into straight lines”, where “one can choose any space-curve which depends upon three parameters as the element of the geometry of the space”, and it “is possible to create a representation of an algebra that embraces an arbitrary number of variables.” Important examples were given, except of course of the still dormant elementary particles. Now, as the prime of matter’s appearance they step forth as the most natural to examine by the true Lie algebras in their original form. When over a long series of years I have scientifically carried out this, the result as here so far concluded is a concrete cellular automaton building kit of the Standard Model and all its features, and their mechanisms and dealings in a structural R3×SO(3) wave-packet organization, both inwards from the elementary particles and outwards via the periodic table of the atoms over the further hierarchical growth of this in molecular and crystal stages to an isotropic space-filling of the whole classical Euclidean Universe in harmonic exchange with its relativistic spherical moiety, and the dark mass and energy collectively worked out in the differential interstice between them.

Codebook Training for Trellis-Based Hierarchical Grassmannian Classification

We consider classification of points on a complex-valued Grassmann manifold of m-dimensional subspaces within the n-dimensional complex Euclidean space. We introduce a trellis-based hierarchical classification network, which is based on an orthogonal product decomposition of the orthogonal basis representing the m-dimensional subspace. Exploiting the similarity of the proposed trellis classifier with a neural network, we propose stochastic gradient-based training techniques. We apply the proposed methods to two important applications in wireless communication, namely Grassmannian channel state information quantization in multiple-input multiple-output communications and non-coherent Grassmannian multi-resolution transmission.

Shareability-Exclusivity Representation on Product Grassmann Manifolds for Multi-camera video clustering

With the rapid popularity of multi-camera networks, one human action is usually captured by multiple cameras located at different angles simultaneously. Multi-camera videos contain the distinct perspectives of one action, therefore multiple views can overcome the impacts of illumination and occlusion. In this paper, we propose a novel multi-camera video clustering model, named Shareability-Exclusivity Representation on Product Grassmann Manifolds (PGM-SER), to address two key issues in traditional multi-view clustering methods (MVC): (1) Most MVC methods directly construct a shared similarity matrix by fusing multi-view data or their corresponding similarity matrices, which ignores the exclusive information in each view; (2) Most MVC methods are designed for multi-view vectorial data, which cannot handle the nonlinear manifold structure hidden in multi-camera videos. The proposed PGM-SER firstly adopts Product Grassmann Manifolds to represent multi-camera videos, then simultaneously learn their shared and exclusive information in global structures to achieve multi-camera video clustering. We provide an effective optimization algorithm to solve PGM-SER and present the corresponding convergence analysis. Finally, PGM-SER is tested on three multi-camera human action video datasets and obtain satisfied experimental results.

An attention-based framework for multi-view clustering on Grassmann manifold

The key problem of multi-view clustering is to handle the inconsistency among multiple views. This article proposes an attention-based framework for multi-view clustering on Grassmann manifold (AMCGM). To be specific, the proposed AMCGM framework aims to learn a representative element on Grassmann manifold with the following four highlights: 1) AMCGM framework performs an attention-based weighted-learning scheme to capture the difference of views; 2) The clustering results can be directly generated by the structured graph learned via AMCGM, avoiding the randomness caused by traditional label-generation procedures, such as K-means clustering; 3) AMCGM has high extensibility since it can generate many multi-view clustering models on Grassmann manifold; 4) On Grassmann manifold, the relationship between the projection metric (PM)-based multi-view clustering model and squared projection metric (SPM)-based model is studied. Based on AMCGM framework, we propose some generated models and provide some useful conclusions. Moreover, to solve the optimization problems involved in the proposed AMCGM framework and generated models, we propose an efficiently iterative algorithm and provide rigorous convergence analysis. Extensive experimental results demonstrate the superb performance of our framework.

Video Domain Adaptation based on Optimal Transport in Grassmann Manifolds

Domain adaptation is a fundamental research field, which focuses on transforming knowledge between different domains. With the massive growth of video data, the video domain adaptation problem becomes increasingly significant for practical tasks. Motivated by the excellent performance of Grassmann manifolds representation in video recognition tasks, we propose an optimal transport based video domain adaptation model on Grassmann manifolds. The proposed model reduces the discrepancy between different domains for the frame and video level features. First, the frame level discrepancy is reduced by extracting domain consistency features. At the video level, a fixed number of frame features are formed and represented as points on Grassmann manifolds. These points are fused with predicted labels to form fusion features. Finally, the video level discrepancy is reduced by minimizing the distribution discrepancy of the fusion features between two domains. Cross-domain video recognition experiments demonstrate the validity of the proposed model. The experimental results demonstrate the excellent performance of the proposed algorithm compared with the state-of-art video domain adaptation models.

Motor imagery EEG decoding using manifold embedded transfer learning

BackgroundBrain computer interface (BCI) utilizes brain signals to help users interact with external devices directly. EEG is one of the most commonly used techniques for brain signal acquisition in BCI. However, it is notoriously difficult to build a generic EEG recognition model due to significant non-stationarity and subject-to-subject variations, and the requirement for long time training. Transfer learning (TL) is particularly useful because it can alleviate the calibration requirement in EEG-based BCI applications by transferring the calibration information from existing subjects to new subject. To take advantage of geometric properties in Riemann manifold and joint distribution adaptation, a manifold embedded transfer learning (METL) framework was proposed for motor imagery (MI) EEG decoding. New methodFirst, the covariance matrices of the EEG trials are first aligned on the SPD manifold. Then the features are extracted from both the symmetric positive definite (SPD) manifold and Grassmann manifold. Finally, the classification model is learned by combining the structural risk minimization (SRM) of source domain and joint distribution alignment of source and target domains. ResultExperimental results on two MI EEG datasets verify the effectiveness of the proposed METL. In particular, when there are a small amount of labeled samples in the target domain, METL demonstrated a more accurate and stable classification performance than conventional methods. Comparison with existing methodsCompared with several state-of-the-art methods, METL has achieved better classification accuracy, 71.81% and 69.06% in single-to-single (STS), 83.14% and 76.00% in multi-to-single (MTS) transfer tasks, respectively. ConclusionsMETL can cope with single source domain or multi-source domains and compared with single-source transfer learning, multi-source transfer learning can improve the performance effectively due to the data expansion. It is effective enough to achieve superior performance for classification of EEG signals.

$$L^2$$-Poisson integral representations of eigensections of invariant differential operators on a homogeneous line bundle over the complex Grassmann manifold $$SU(r,r+b)/S( U(r)\times U(r+b))$$

$$L^2$$-Poisson integral representations of eigensections of invariant differential operators on a homogeneous line bundle over the complex Grassmann manifold $$SU(r,r+b)/S( U(r)\times U(r+b))$$

From non-defectivity to identifiability

A point p in a projective space is h -identifiable via a variety X if there is a unique way to write p as a linear combination of h points of X . Identifiability is important both in algebraic geometry and in applications. In this paper we propose an entirely new approach to study identifiability, connecting it to the notion of secant defect for any smooth projective variety. In this way we are able to improve the known bounds on identifiability and produce new identifiability statements. In particular, we give optimal bounds for some Segre and Segre–Veronese varieties and provide the first identifiability statements for Grassmann varieties.

A Multistage Framework With Mean Subspace Computation and Recursive Feedback for Online Unsupervised Domain Adaptation.

In this paper, we address the Online Unsupervised Domain Adaptation (OUDA) problem and propose a novel multi-stage framework to solve real-world situations when the target data are unlabeled and arriving online sequentially in batches. Most of the traditional manifold-based methods on the OUDA problem focus on transforming each arriving target data to the source domain without sufficiently considering the temporal coherency and accumulative statistics among the arriving target data. In order to project the data from the source and the target domains to a common subspace and manipulate the projected data in real-time, our proposed framework institutes a novel method, called an Incremental Computation of Mean-Subspace (ICMS) technique, which computes an approximation of mean-target subspace on a Grassmann manifold and is proven to be a close approximate to the Karcher mean. Furthermore, the transformation matrix computed from the mean-target subspace is applied to the next target data in the recursive-feedback stage, aligning the target data closer to the source domain. The computation of transformation matrix and the prediction of next-target subspace leverage the performance of the recursive-feedback stage by considering the cumulative temporal dependency among the flow of the target subspace on the Grassmann manifold. The labels of the transformed target data are predicted by the pre-trained source classifier, then the classifier is updated by the transformed data and predicted labels. Extensive experiments on six datasets were conducted to investigate in depth the effect and contribution of each stage in our proposed framework and its performance over previous approaches in terms of classification accuracy and computational speed. In addition, the experiments on traditional manifold-based learning models and neural-network-based learning models demonstrated the applicability of our proposed framework for various types of learning models.

AoD-Adaptive Channel Feedback for FDD Massive MIMO Systems With Multiple-Antenna Users

In this paper, we propose an efficient feedback scheme for an angle of departure (AoD) based channel estimation in frequency division duplex (FDD) massive multiple-input multiple-output (MIMO) systems with multiple antennas at the users. The channel feedback scheme is based on zero-forcing block diagonalization (BD) and it is proposed for two distinct design cases; in case I, the number of streams intended for a user equals the number of antennas at that user; in case II, the number of streams is less than the number of receive antennas. Case I is applicable in scenarios where high data rate requirements are needed as it transmits data symbols over all of the available degrees of freedom of the system. Diversely, case II is applicable when reliability is a priority in the system as it uses the additional <i>receive</i> antennas at the user to achieve spatial diversity to enhance the link performance. The proposed scheme is analyzed for the two cases by quantifying the downlink rate gap from the case of perfect channel state information (CSI). Moreover, we design structured feedback codebooks based on optimal subspace packing in the Grassmannian manifold and show that these codes achieve close performance to the perfect CSI case. Additionally, a vector quantization scheme is proposed to quantize the user&#x2019;s channel matrix when optimal power allocation across multiple streams is adopted in the low signal-to-noise ratio (SNR) region. The feedback codebooks are based on optimal line packing in the Grassmannian manifold, where every vector of the user&#x2019;s channel matrix is quantized and sent to the BaseStation. The results demonstrate a fundamental trade-off between vector quantization, with power optimization across the data streams, and subspace quantization. Specifically, vector quantization codebooks outperform subspace-based codebooks in the low SNR region, while the situation is reversed in the high SNR region.

Analog MIMO Communication for One-Shot Distributed Principal Component Analysis

A fundamental algorithm for data analytics at the edge of wireless networks is distributed principal component analysis (DPCA), which finds the most important information embedded in a distributed high-dimensional dataset by distributed computation of a reduced-dimension data subspace, called principal components (PCs). In this paper, to support one-shot DPCA in wireless systems, we propose a framework of analog MIMO transmission featuring the uncoded analog transmission of local PCs for estimating the global PCs. To cope with channel distortion and noise, two maximum-likelihood (global) PC estimators are presented corresponding to the cases with and without receive channel state information (CSI). The first design, termed coherent PC estimator, is derived by solving a Procrustes problem and reveals the form of regularized channel inversion where the regulation attempts to alleviate the effects of both receiver noise and data noise. The second one, termed blind PC estimator, is designed based on the subspace channel-rotation-invariance property and computes a centroid of received local PCs on a Grassmann manifold. Using the manifold-perturbation theory, tight bounds on the mean square subspace distance (MSSD) of both estimators are derived for performance evaluation. The results reveal simple scaling laws of MSSD concerning device population, data and channel signal-to-noise ratios (SNRs), and array sizes. More importantly, both estimators are found to have identical scaling laws, suggesting the dispensability of CSI to accelerate DPCA. Simulation results validate the derived results and demonstrate the promising latency performance of the proposed analog MIMO

Tensor Discriminant Analysis on Grassmann Manifold with Application to Video Based Human Action Recognition

Tensor Discriminant Analysis on Grassmann Manifold with Application to Video Based Human Action Recognition

Progressive Domain Adaptive Network for Crater Detection

Existing crater detection methods are typically carried out in the same domain where the training and testing data are drawn from an identical distribution. However, this means that when we apply a trained crater detector to a new scene probably caused by changes in sensors, lighting or other factors will lead to a significant performance drop. In this work, we aim to improve the accuracy of the domain (scene) adaptive crater detection which will reduce labor costs caused by annotation. Firstly, we proposed a progressive domain adaptive network (PDAN) which can progressively learn the knowledge from the source domain to the target domain by the projected intermediate domain features generated by the subspace along the geodesic on the Grassmann manifold. Secondly, in order to further extract the domain agnostic feature and utilize the characteristics of craters, we proposed a low-level feature enhancement module (LFEM) and a circular boundary enhancement module (CBEM). Thirdly, a weighted instance-level alignment network (WIAN) is presented to adaptively align the instance level feature further reduce the differences between two domains, and enhance the performance. To evaluate the effectiveness of the proposed method, based on our previous Mars day crater dataset (MDCD), we present a domain adaptive crater detection (DACD) dataset which contains 1000 images in two domains and more than 20000 craters. The extensive experiments based on the DACD and open datasets demonstrated that the proposed network could effectively eliminate the impact of domain shift and achieve state-of-the-art results.

Geometric Optimization for Structure-Preserving Model Reduction of Hamiltonian Systems

Classical model reduction methods disregard the special symplectic structure associated with Hamiltonian systems. A key challenge in projection-based approaches is to construct a sym-plectic basis that captures the essential system information. This necessitates the computation of a so-called proper symplectic decomposition (PSD) of a given sample data set. The PSD problem allows for a canonical formulation as an optimization problem on the symplectic Stiefel manifold. However, as with their Euclidean counterparts, symplectic projectors only depend on the underlying symplectic subspaces and not on the selected symplectic bases. This motivates to tackle the PSD problem as a Riemannian optimization problem on the symplectic Grassmann manifold, i.e., the matrix manifold of symplectic projectors. Initial investigations on this manifold feature in a recent preprint of the authors. In this work, we investigate the feasibility and performance of this approach on two academic numerical examples. More precisely, we calculate an optimized PSD for snapshot matrices that stem from solving the one-dimensional linear wave equation and the one-dimensional nonlinear Schrödinger equation.