Grassmann Research Articles

A note on the height of the third Stiefel--Whitney class of the canonical vector bundles over some Grassmann manifolds

We compute the height of the third Stiefel--Whitney characteristic class of the canonical bundles over some infinite classes of Grassmann manifolds of five dimensional vector subspaces of real vector spaces.

A matrix form of Grassmannian Ball–Barthe inequality

As one application of the Cauchy–Binet formula, the Ball–Barthe inequality plays a key role in solving reverse isoperimetric inequalities. Recently, a new Grassmannian Ball–Barthe inequality was established. With it, reverse isoperimetric inequalities on Grassmann manifolds were successfully proved. In this paper, we will establish its matrix form with a simpler proof than that of Grassmannian form. The matrix form also recovers the classical Ball–Barthe inequality in a special case.

Geodesics of projections in von Neumann algebras

Let A {\mathcal {A}} be a von Neumann algebra and P A {\mathcal {P}}_{\mathcal {A}} the manifold of projections in A {\mathcal {A}} . There is a natural linear connection in P A {\mathcal {P}}_{\mathcal {A}} , which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of C n \mathbb {C}^n . In this paper we show that two projections p , q p,q can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of A {\mathcal {A}} ), if and only if p ∧ q ⊥ ∼ p ⊥ ∧ q , \begin{equation*} p\wedge q^\perp \sim p^\perp \wedge q, \end{equation*} where ∼ \sim stands for the Murray-von Neumann equivalence of projections. It is shown that the minimal geodesic is unique if and only if p ∧ q ⊥ = p ⊥ ∧ q = 0 p\wedge q^\perp = p^\perp \wedge q=0 . If A {\mathcal {A}} is a finite factor, any pair of projections in the same connected component of P A {\mathcal {P}}_{\mathcal {A}} (i.e., with the same trace) can be joined by a minimal geodesic. We explore certain relations with Jones’ index theory for subfactors. For instance, it is shown that if N ⊂ M {\mathcal {N}}\subset {\mathcal {M}} are II 1 _1 factors with finite index [ M : N ] = t − 1 [{\mathcal {M}}:{\mathcal {N}}]={\mathbf {t}}^{-1} , then the geodesic distance d ( e N , e M ) d(e_{\mathcal {N}},e_{\mathcal {M}}) between the induced projections e N e_{\mathcal {N}} and e M e_{\mathcal {M}} is d ( e N , e M ) = arccos ⁡ ( t 1 / 2 ) d(e_{\mathcal {N}},e_{\mathcal {M}})=\arccos ({\mathbf {t}}^{1/2}) .

Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces \((M=G/H,g)\) whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (G/H, g), such that G is one of the compact classical Lie groups \({{\,\mathrm{SO}\,}}(n)\), \(\mathrm{U}(n)\), and H is a diagonally embedded product \(H_1\times \cdots \times H_s\), where \(H_j\) is of the same type as G. This class includes spheres, Stiefel manifolds, Grassmann manifolds and real flag manifolds. The present work is a contribution to the study of g.o. spaces (G/H, g) with H semisimple.

Symmetry of Dressed Photon

Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters n≥r≥s on dimensions. We discuss the symmetry on these spaces using classical invariant theory, orbit decomposition of prehomogeneous vector spaces, and compact reductive homogeneous space such as Grassmann manifold and flag variety. Finally, we go back to the original dressed photon with n=4,r=2,s=1.

Principal Bundle Structure of Matrix Manifolds

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold Gr(Rk) of linear subspaces of dimension r<k in Rk, which avoids the use of equivalence classes. The set Gr(Rk) is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on R(k−r)×r. Then, we define an atlas for the set Mr(Rk×r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk×k. Finally, we define an atlas for the set Mr(Rn×m) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rn)×Gr(Rm) and typical fibre GLr. The atlas of Mr(Rn×m) is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set Mr(Rn×m) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space Rn×m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space Rn×m, seen as the union of manifolds Mr(Rn×m), as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

How dynamics constrains probabilities in general probabilistic theories

We introduce a general framework for analysing general probabilistic theories, which emphasises the distinction between the dynamical and probabilistic structures of a system. The dynamical structure is the set of pure states together with the action of the reversible dynamics, whilst the probabilistic structure determines the measurements and the outcome probabilities. For transitive dynamical structures whose dynamical group and stabiliser subgroup form a Gelfand pair we show that all probabilistic structures are rigid (cannot be infinitesimally deformed) and are in one-to-one correspondence with the spherical representations of the dynamical group. We apply our methods to classify all probabilistic structures when the dynamical structure is that of complex Grassmann manifolds acted on by the unitary group. This is a generalisation of quantum theory where the pure states, instead of being represented by one-dimensional subspaces of a complex vector space, are represented by subspaces of a fixed dimension larger than one. We also show that systems with compact two-point homogeneous dynamical structures (i.e. every pair of pure states with a given distance can be reversibly transformed to any other pair of pure states with the same distance), which include systems corresponding to Euclidean Jordan Algebras, all have rigid probabilistic structures.

On a minimum enclosing ball of a collection of linear subspaces

This paper concerns the minimax center of a collection of linear subspaces. For k-dimensional subspaces of an n-dimensional vector space, this can be cast as finding the center of a minimum enclosing ball on a Grassmann manifold. For subspaces of differing dimension, the setting becomes a disjoint union of Grassmannians rather than a single manifold, and the problem is no longer well-defined. However, natural geometric maps exist between these manifolds with a well-defined notion of distance for the images of the subspaces under the mappings. Solving the problem in this context leads to a candidate minimax center on each of the constituent manifolds, but does not provide intuition about which candidate is the best representation of the data. Additionally, the solutions of different rank are generally not nested so a deflationary approach will not suffice, and the problem must be solved independently on each manifold. We propose an optimization problem parametrized by the rank of the minimax center. The solution is computed with a subgradient algorithm applied to the dual problem. By scaling the objective and penalizing the information lost by the rank-k minimax center, we jointly recover an optimal dimension, k⁎, and a subspace at the center of the minimum enclosing ball, U⁎, that best represents the data.

Grassmann Inequalities and Extremal Varieties in $${\mathbb {P}}\left( {{ \bigwedge ^p}{\mathbb {R}^n}} \right) $$

In continuation of the work in Leventides and Petroulakis (Adv Appl Clifford Algebras 27:1503–1515, 2016), Leventides et al. (J Optim Theory Appl 169(1):1–16, 2016), which defines extremal varieties in \(\mathbb {P}\left( {{ \bigwedge ^2}{\mathbb {R}^n}} \right) \), we define a more general concept of extremal varieties of the real Grassmannian \({G_p}\left( {{\mathbb {R}^n}} \right) \) in \(\mathbb {P}\left( {{ \bigwedge ^p}{\mathbb {R}^n}} \right) \). This concept is based on the minimization of the sums of squares of the quadratic Plücker relations defining the Grassmannian variety as well as the reverse maximisation problem. Such extremal problems define a set of Grassmannian inequalities on the set of Grassmann matrices, which are essential for the definition of the Grassmann variety and its dual extremal variety. We define and prove these inequalities for a general Grassmannian and we apply the existing results, in the cases \({{ \wedge ^2}{\mathbb {R}^{2n}}}\) and \({{ \wedge ^n}{\mathbb {R}^{2n}}}\). The resulting extremal varieties underline the fact which was demonstrated in Leventides et al. (2016, Linear Algebra Appl 461:139–162, 2014), that such varieties are represented by multi-vectors that acquire the property of a unique singular value with total multiplicity. Crucial to these inequalities are the numbers \(M_{n,p}\), which are calculated within the cases mentioned above.

Analytic bundle structure on the idempotent manifold

Let X be a (real or complex) Banach space (not necessarily a Hilbert space), and $$\mathcal {I}(X)$$ be the set of all non-trivial idempotents; i.e., bounded linear operators on X whose squares equal themselves. We show that, when equipped with the Banach submanifold structure induced from $$\mathcal {L}(X)$$ , the subset $$\mathcal {I}(X)$$ is a locally trivial analytic affine-Banach bundle over the Grassmann manifold $$\mathscr {G}(X)$$ , via the map $$\kappa $$ that sends $$Q\in \mathcal {I}(X)$$ to Q(X), such that the affine-Banach space structure on each fiber is the one induced from $$\mathcal {L}(X)$$ . Using this, we show that if H is a real or complex Hilbert space, then the assignment $$\begin{aligned} (E,T)\mapsto T^*\circ P_{E^\bot } + P_{E}, \quad \text {where}\quad E\in \mathscr {G}(H)\quad \text {and}\quad T\in \mathcal {L}(E,E^\bot ), \end{aligned}$$ induces a real bi-analytic bijection from the total space of the tangent bundle, $$\mathbf {T}(\mathscr {G}(H))$$ , of $$\mathscr {G}(H)$$ onto $$\mathcal {I}(H)$$ (here, $$E^\bot $$ is the orthogonal complement of E, $$P_E\in \mathcal {L}(H)$$ is the orthogonal projection onto E, and $$T^*$$ is the adjoint of T). Notice that this real bi-analytic bijection is an affine map on each tangent plane. Furthermore, if for every $$E\in \mathscr {G}(H)$$ , we identify $$\mathcal {L}(E,E^\bot )$$ with a subspace of $$\mathcal {L}(H)$$ via the embedding $$S\mapsto S\circ P_E$$ , then the inclusion map from $$\mathbf {T}(\mathscr {G}(H))$$ to the trivial Banach bundle $$\mathscr {G}(H)\times \mathcal {L}(H)$$ is a real analytic immersion. Through this, we give a concrete idempotent in $$M_{n^2}\big (C(\mathscr {G}(\mathbb {K}^n))\big )$$ that represents the K-theory class of the tangent bundle $$\mathbf {T}(\mathscr {G}(\mathbb {K}^n))$$ , when $$\mathbb {K}$$ is either the real field or the complex field.

Manifold Optimization-Assisted Gaussian Variational Approximation

Gaussian variational approximation is a popular methodology to approximate posterior distributions in Bayesian inference, especially in high-dimensional and large data settings. To control the computational cost, while being able to capture the correlations among the variables, the low rank plus diagonal structure was introduced in the previous literature for the Gaussian covariance matrix. For a specific Bayesian learning task, the uniqueness of the solution is usually ensured by imposing stringent constraints on the parameterized covariance matrix, which could break down during the optimization process. In this article, we consider two special covariance structures by applying the Stiefel manifold and Grassmann manifold constraints, to address the optimization difficulty in such factorization architectures. To speed up the updating process with minimum hyperparameter-tuning efforts, we design two new schemes of Riemannian stochastic gradient descent methods and compare them with other existing methods of optimizing on manifolds. In addition to fixing the identification issue, results from both simulation and empirical experiments prove the ability of the proposed methods of obtaining competitive accuracy and comparable converge speed in both high-dimensional and large-scale learning tasks. Supplementary materials for this article are available online.

Grassmannians and Cluster Structures

Cluster structures have been established on numerous algebraic varieties. These lectures focus on the Grassmannian variety and explain the cluster structures on it. The tools include dimer models on surfaces, associated algebras, and the study of associated module categories.

Generalization of the Neville–Aitken interpolation algorithm on Grassmann manifolds: Applications to reduced order model

AbstractAn extension of the well‐known Neville–Aitken's algorithm for interpolation on the Grassmann manifold in the framework of parametric model order reduction is presented. Interpolation points on are the subspaces spanned by bases obtained by Proper Orthogonal Decomposition of available solutions associated with the chosen parameter sampling. The Neville–Aitken's algorithm is performed recursively via the geodesic barycenter of two points. Three CFD applications are presented: (i) the Von Karman vortex shedding street, (ii) the lid‐driven cavity with inflow and (iii) the flow induced by a rotating solid. Numerical results are relevant with respect to accuracy while the asymptotic complexity is comparable to the state of the art.

Capacity of Remote Classification Over Wireless Channels

Remote classification involves offloading complex object-recognition tasks from mobile devices to servers at the network edge. It brings to the mobile device the capability of discerning hundreds of object classes by using the computational and storage capabilities of the infrastructure. Remote classification is challenged by the finite and variable data rate of the wireless channel, which affects the capability to transfer high-dimensional features and thus limits the classification resolution. We introduce a set of metrics under the name of classification capacity that are defined as the maximum number of classes that can be discerned over a given communication channel while meeting a target probability for classification error. We treat both the cases of a channel where the instantaneous rate is known and unknown. The objective is to choose a subset of classes from a class library that offers satisfactory performance over a given channel. We treat two different cases of subset selection. First, a device can select the subset by pruning the class library until arriving at a subset that meets the targeted error probability while maximizing the classification capacity. Adopting a subspace data model, we prove the equivalence of classification capacity maximization to the problem of packing on the Grassmann manifold. The results show that the classification capacity grows exponentially with the instantaneous communication rate, and super-exponentially with the dimensions of each data cluster. This also holds for ergodic and outage capacities with fading if the instantaneous rate is replaced with an average rate and a fixed rate, respectively. In the second case, a device has a unique preference of class subset for every communication rate, which is modeled as an instance of uniformly sampling the library. Without class selection, the classification capacity and its ergodic and outage counterparts are proved to scale linearly with their corresponding communication rates instead of the exponential growth in the last case.

Time Series Data Classification on Grassmann Manifold

Time Series Classification has become one of the most challenging problems in many signal processing and machine learning applications, e.g., audio/video signal processing and EEG signal processing. We propose a novel method of extracting features from data for classification and representing data on a Grassmann manifold by parameterizing it using the autoregressive moving model (ARMA). Then we perform classification on these extracted features by training support vector machines (SVM), with appropriate kernels on the Grassmann manifold. We performed tests on several publicly available datasets. We found that an SVM with a proper kernel on the Grassmann manifold consistently performs better than an SVM using a typical Gaussian kernel that acts on the data in Euclidean space. Furthermore, we found that the Grassmann SVM technique overperforms the literature in some high-dimensional datasets, without the need for any other preprocessing techniques. This work demonstrates the power of data-based manifold techniques in improving the performance of existing algorithms.

Multi-view clustering with interactive mechanism

Existing multi-view clustering methods either seek to directly learn a consistent spectral embedding, or to learn a consistent graph. This work presents a novel model, called Multi-view Clustering with Interactive Mechanism (MCIM). Using the interactive mechanism, the uniform graph and spectral embedding can be learned alternatively and promote to each other. Furthermore, we perform spectral embedding learning on Grassmann manifold via an implicitly weighted-learning scheme and reveal the clustering result via graph learning. To solve the proposed model, we propose an efficient optimization method and provide the corresponding convergence analysis. The experimental results on real image datasets demonstrate the superiorities of MCIM compared to several SOTA methods.

Holomorphic maps into Grassmann manifolds (harmonic maps into Grassmann manifolds III)

A well-known Calabi’s rigidity theorem on holomorphic isometric immersions into the complex projective space is generalized to the case that the target is the complex Grassmann manifolds. Our strategy is to use the differential geometry of vector bundles and a generalization of do Carmo and Wallach theory developed in Nagatomo (Harmonic maps into Grassmann manifolds. arXiv:mathDG/1408.1504 ). We introduce the associated maps with holomorphic maps to obtain a general rigidity theorem (Theorem 5.6). As applications, several rigidity results on Einstein–Hermitian holomorphic maps are exhibited and we also give an interpretation of the existence of a Kahler structure with an $$S^1$$ -action on the moduli spaces of holomorphic isometric embeddings of a compact Kahler manifold into complex quadrics. Theorem 5.6 also implies classification theorems for equivariant holomorphic maps.

Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces (M=G∕H,g) whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (Sp(n)∕Sp(n1)×⋯×Sp(ns),g), with 0<n1+⋯+ns≤n. Such spaces include spheres, quaternionic Stiefel manifolds, Grassmann manifolds and quaternionic flag manifolds. The present work is a contribution to the study of g.o. spaces (G∕H,g) with H semisimple.

Text classification based on the word subspace representation

In this paper, we propose a novel framework for text classification based on subspace-based methods. Recent studies showed the advantages of modeling texts as linear subspaces in a high-dimensional word vector space, to which we refer as word subspace. Therefore, we propose solving topic classification and sentiment analysis by using the word subspace along with different subspace-based methods. We explore the word embeddings geometry to decide which subspace-based method is more suitable for each task. We empirically demonstrate that a word subspace generated from sets of texts is a unique representation of a semantic topic that can be spanned by basis vectors derived from different texts. Therefore, texts can be classified by comparing their word subspace with the topic class subspaces. We achieve this framework by using the mutual subspace method that effectively handles multiple subspaces for classification. For sentiment analysis, as word embeddings do not necessarily consider sentiment information (i.e., opposite sentiment words have similar word vectors), we introduce the orthogonal mutual subspace method, to push opposite sentiment words apart. Furthermore, as there may be overlap between the sentiment class subspaces due to overlapping topics, we propose modeling a sentiment class by a set of multiple word subspaces, generated from each text belonging to the class. We further model the sentiment classes on a Grassmann manifold by using the Grassmann subspace method and its discriminative extension, the Grassmann orthogonal subspace method. We show the validity of each framework through experiments on four widely used datasets.

Examples of Mironov Cycles in Grassmannians

Providing some examples of Lagrangian cycles that arise as a generalization of Mironov’s construction to the case of Grassmann manifolds $ \operatorname{Gr}_{{𝔺}}(k,n+1) $ , we show that these manifolds enjoy all data necessary for this generalization, the natural real structure, and an incomplete toric action. We also provide new concrete examples.