Grassmann Research Articles

A diagonalization-free optimization algorithm for solving Kohn-Sham equations of closed-shell molecules.

A local optimization algorithm for solving the Kohn-Sham equations is presented. It is based on a direct minimization of the energy functional under the equality constraints representing the Grassmann Manifold. The algorithm does not require an eigendecomposition, which may be advantageous in large-scale computations. It is optimized to reduce the number of Kohn-Sham matrix evaluations to one per iteration to be competitive with standard self-consistent field (SCF) approach accelerated by direct inversion of the iterative subspace (DIIS). Numerical experiments include a comparison of the algorithm with DIIS. A high reliability of the algorithm is observed in configurations where SCF iterations fail to converge or find a wrong solution corresponding to a stationary point different from the global minimum. The local optimization algorithm itself does not guarantee that the found minimum is global. However, a randomization of the initial approximation shows a convergence to the right minimum in the vast majority of cases.

A new transferable bearing fault diagnosis approach with adaptive manifold embedded distribution alignment

The conditional and marginal distributions of bearing vibration signal data collected under different working conditions generally may not obey the same distribution. Moreover, feature distortions are difficult to eliminate when performing distribution alignment in the original space. Based on these challenges, this study proposes a novel transferable fault diagnosis approach with adaptive manifold embedded distribution alignment (AMEDA). AMEDA can learn transformed feature representations in the Grassmann manifold space by constructing a geodesic flow kernel to avoid feature distortions. In addition, this paper initially aims to construct an adaptive factor defined by -distance to adjust the relative importance of the conditional and marginal distributions dynamically. After manifold feature learning and dynamic distribution alignment, a cross-domain classifier f with structural risk minimization can be learned to predict unlabeled target domain data. Experimental results from multiple indicators based on two datasets demonstrate the superiority of AMEDA over the other existing methods.

An existence theorem for generalized quasi-variational inequalities involving the Grassmannian manifold with an application

We present an existence theorem for a class of generalized quasi-variational problem involving Grassmannian manifolds. This class is directly inspired by a general equilibrium problem with time, uncertainty and incomplete financial market with real assets. The problem of the existence of this equilibrium cannot be analyzed using standard techniques employed in similar models. Then, we show how the concept of equilibrium is strictly related to the concept of Grassmannian manifolds. Finally, we present a variational inequality problem, whose solutions are equilibria of the proposed model.

How many simplices are needed to triangulate a Grassmannian?

In this paper we use recently developed methods to compute a lower bound for the number of simplices that are needed to triangulate the Grassmann manifold $\G_k(\mathbb{R}^n)$. We first estimate the number of vertices that are needed for such a triangulation by giving a general lower bound and some more precise bounds for $k=2,3,4$. By applying the Lower Bound Theorem (LBT) of Barnette for triangulated manifolds, we then obtain estimates for the number of simplices in all dimensions. For higher-dimensional simplices these estimates can be considerably improved by using the recent progress on the Generalized Lower Bound Theorem for triangulated manifolds, which states that the $h''$-numbers of triangulated manifolds are unimodal, together with the computation of the Poincare polynomial. For example, we are able to prove that the number of top-dimensional simplices in a triangulation of $\G_k(\mathbb{R}^n)$ grows exponentially with $n$. Our method can be used to estimate the minimal size of triangulations for other spaces, like Lie groups, flag manifolds, Stiefel manifolds etc.

Parametric reduced order modeling-based discrete velocity method for simulation of steady rarefied flows

In this work, a parametric reduced order modeling-based discrete velocity method (PROM-DVM) is developed for simulation of steady rarefied flows. This method aims to reduce the number of discrete velocity points so as to improve the computational efficiency of DVM. For solving similar problems with different initial parameters, the developed method can generate a sought-after reduced discrete velocity space for the target parameter value from some pre-computed cases and solve the Boltzmann equation directly in the reduced discrete velocity space. At first, the singular-value decomposition (SVD) method is used to find the reduced-order bases for the cases of pre-computed parameter values and the reduced-order basis for the target parameter value is then constructed from the pre-computed bases by the interpolation method based on the Grassmann manifold and its tangent space. The reduced discrete velocity space and its connection to the original discrete velocity space are further established by the discrete empirical interpolation method (DEIM) based on the interpolated reduced-order basis. Since most points in the original discrete velocity space which are of negligible importance are removed in the computation of the present method, a marked improvement in computational efficiency with respect to the DVM in the original discrete velocity space is achieved. Numerical results show that the PROM-DVM can reach 13 times speed-up in CPU time for lid-driven cavity flow.

Video-based person re-identification by intra-frame and inter-frame graph neural network

In the past few years, video-based person re-identification (Re-ID) have attracted growing research attention. The crucial problem for this task is how to learn robust video feature representation, which can weaken the influence of factors such as occlusion, illumination, and background etc. A great deal of previous works utilize spatio-temporal information to represent pedestrian video, but the correlations between parts of human body are ignored. In order to take advantage of the relationship among different parts, we propose a novel Intra-frame and Inter-frame Graph Neural Network (I2GNN) to solve the video-based person Re-ID task. Specifically, (1) the features from each part are treated as graph nodes from each frame; (2) the intra-frame edges are established by the correlation between different parts; (3) the inter-frame edges are constructed between the same parts across adjacent frames. I2GNN learns video representations by employing the adjacent matrix of the graph and input features to conduct graph convolution, and then adopts projection metric learning on Grassman manifold to measure the similarities between learned pedestrian features. Moreover, this paper proposes a novel occlusion-invariant term to make the part features close to their center, which can relive several uncontrolled complicated factors, such as occlusion and pose invariance. Besides, we have carried out extensive experiments on four widely used datasets: MARS, DukeMTMC-VideoReID, PRID2011, and iLIDS-VID. The experimental results demonstrate that our proposed I2GNN model is more competitive than other state-of-the-art methods.

The feedback invariant measures of distance to uncontrollability and unobservability

ABSTRACT The selection of systems of inputs and outputs forms part of the early system design that is important since it preconditions the potential for control design. Existing methodologies for input, output structure selection rely on criteria expressing distance to uncontrollability, unobservability. Although controllability is invariant under state feedback, its corresponding degrees expressing distance to uncontrollability is not. The paper introduces new criteria for distance to uncontrollability (dually for unobservability) which is invariant under feedback transformations. The approach uses the restricted matrix pencils developed for the characterisation of invariant spaces of the geometric theory and then deploys exterior algebra to define the invariant input and output decoupling polynomials. This reduces the overall problem of distance to uncontrollability (unobservability) to two optimisation problems: the distance from the Grassmann variety and distance of a set of polynomials from non-coprimeness. Results on the distance of Sylvester Resultants from singularity provide the new measures.

Berezin-Karpelevich formula for chi-spherical functions on complex Grassmannians

In [5], Berezin and Karpelevich gave, without a proof, an explicit formula for spherical functions on complex Grassmannian manifolds. A first attempt to give a proof of Berezin-Karpelevich formula was taken, in [16], by Takahashi. His proof contained a gap, which was fixed later, in [10], by Hoogenboom. The aim of this paper is to generalize Berezin-Karpelevich formula to the case of $\chi$-spherical functions on complex Grassmannian manifolds $ SU(p+q)/S\left(U(p)\times U(q)\right)$.

Non intrusive method for parametric model order reduction using a bi-calibrated interpolation on the Grassmann manifold

Approximating solutions of non-linear parametrized physical problems by interpolation presents a major challenge in terms of accuracy. In fact, pointwise interpolation of such solutions is rarely efficient and generally leads to incorrect predictions. To overcome this issue, instead of interpolating solutions directly by a straightforward approach, reduced order models (ROMs) can be efficiently used. To this end, the ITSGM (Interpolation On a Tangent Space of the Grassmann Manifold) is an efficient technique used to interpolate parameterized POD (Proper Orthogonal Decomposition) bases. The temporal dynamics is afterwards determined by the Galerkin projection of the predicted basis onto the high fidelity model. However, such interpolated ROMs based on ITSGM/Galerkin are intrusive, given the fact that their construction requires access to the equations of the underlying high fidelity model. In the present paper a non-intrusive approach (Galerkin free) for the construction of reduced order models is proposed. This method, referred to as Bi-CITSGM (Bi-Calibrated ITSGM) consists of two major steps. First, the untrained spatial and temporal bases are predicted by the ITSGM method and the POD eigenvalues by spline cubic. Then, two orthogonal matrices, determined as analytical solutions of two optimization problems, are introduced in order to calibrate the interpolated bases with their corresponding eigenvalues. Results on the flow problem past a circular cylinder where the parameter of interpolation is the Reynolds number, show that for new untrained Reynolds number values, the developed approach produces sufficiently accurate solutions in a real-computational time.

Tropical geometry and Newton–Okounkov cones for Grassmannian of planes from compactifications

AbstractWe construct a family of compactifications of the affine cone of the Grassmannian variety of $2$ -planes. We show that both the tropical variety of the Plücker ideal and familiar valuations associated to the construction of Newton–Okounkov bodies for the Grassmannian variety can be recovered from these compactifications. In this way, we unite various perspectives for constructing toric degenerations of flag varieties.

Network clustering via kernel-ARMA modeling and the Grassmannian: The brain-network case

This paper demonstrates that all clustering tasks in a dynamic (brain) network, i.e., state clustering, community detection, and subnetwork state-sequence clustering, can be addressed by a novel unifying network-clustering framework. The connecting threads of the components of the proposed framework are: a novel kernel-based autoregressive-moving-average (ARMA) model which propels feature extraction from the network time-series, and the Riemannian geometry of the Grassmann manifold (Grassmannian) into which the extracted features are mapped. Clustering of the Grassmannian features is performed via the novel extension of a recently introduced algorithm which capitalizes on the Grassmannian distances and angular information of the point-cloud of features. Numerical tests on synthetic and real functional-magnetic-resonance-imaging (fMRI) data showcase the favorable performance of the proposed scheme against state-of-the-art network-clustering and manifold-learning methods, and corroborate the claim of this paper that the proposed framework can serve as a useful data-analytic toolbox for network(-neuroscience) research.

On the topological complexity of Grassmann manifolds

Abstract We prove that the topological complexity of a quaternionic flag manifold is half of its real dimension. For the real oriented Grassmann manifolds G͠ n,k , 3 ≤ k ≤ [n/2], the zero-divisor cup-length of the rational cohomology of G͠ n,k is computed in terms of n and k which gives a lower bound for the topological complexity of G͠ n,k , TC(G͠ n,k ). When k = 3, it is observed in certain cases that better lower bounds for TC(G͠ n,3) are obtained using ℤ2-cohomology.

Integrable matrix models in discrete space-time

We introduce a class of integrable dynamical systems of interacting classical matrix-valued fields propagating on a discrete space-time lattice, realized as many-body circuits built from elementary symplectic two-body maps. The models provide an efficient integrable Trotterization of non-relativistic \sigmaσ-models with complex Grassmannian manifolds as target spaces, including, as special cases, the higher-rank analogues of the Landau–Lifshitz field theory on complex projective spaces. As an application, we study transport of Noether charges in canonical local equilibrium states. We find a clear signature of superdiffusive behavior in the Kardar–Parisi–Zhang universality class, irrespectively of the chosen underlying global unitary symmetry group and the quotient structure of the compact phase space, providing a strong indication of superuniversal physics.

Junctions of mass-deformed nonlinear sigma models on SO(2N)/U(N) and Sp(N)/U(N). Part II

We construct three-pronged junctions of mass-deformed nonlinear sigma models on SO(2N)/U(N) and Sp(N )/U(N ) for generic N. We study the nonlinear sigma models on the Grassmann manifold or on the complex projective space. We discuss the relation between the nonlinear sigma model constructed in the harmonic superspace for- malism and the nonlinear sigma model constructed in the projective superspace formalism by comparing each model with the mathcal{N} = 2 nonlinear sigma model constructed in the mathcal{N} = 1 superspace formalism.

Gradient Projection Method on Matrix Manifolds

The minimization of a function with a Lipschitz continuous gradient on a proximally smooth subset of a finite-dimensional Euclidean space is considered. Under the restricted secant inequality, the gradient projection method as applied to the problem converges linearly. In certain cases, the linear convergence of the gradient projection method is proved for the real Stiefel or Grassmann manifolds.

Three-Pronged Junctions on SO(2N)/U(N) and Sp(N)/U(N)

We report on the results of our work [1], which discusses $$\mathcal{N} = 2$$ nonlinear sigma models on the quadrics of the Grassmann manifold and three-pronged junctions of the mass-deformed nonlinear sigma models on $$SO(8){\text{/}}U(4)$$ and $$Sp(3){\text{/}}U(3)$$ . This article is prepared for the Proceedings of International Workshop “Supersymmetries and Quantum Symmetries—SQS’2019”, which was held in Yerevan from 26 to 31 August, 2019. The talk was based on [1].

Construction of spread codes based on Abelian non-cyclic orbit codes

Let Fq be the finite field with q elements (where q is a prime power). Since any invertible matrix maps subspaces to subspaces of the same dimension we have a group action of the general linear group, GLn(Fq), on Gq(k,n) (Grassmann variety). The orbits of a subgroup of GLn(Fq) acting on the Grassmann variety are called (subspace) orbit codes. When the subgroup acting on Gq(k,n) is cyclic the associated codes are called cyclic orbit codes. We make a construction abelian non-cyclic orbit codes by making full use of the companion matrix of a primitive polynomial over finite fields, it is a partial spread code. Based on this code, an optimal partial spread code is obtained. Our results answer the first of two open problems presented by Climent et al.

Gradient-based Learning Methods Extended to Smooth Manifolds Applied to Automated Clustering

Grassmann manifold based sparse spectral clustering is a classification technique that consists in learning a latent representation of data, formed by a subspace basis, which is sparse. In order to learn a latent representation, spectral clustering is formulated in terms of a loss minimization problem over a smooth manifold known as Grassmannian. Such minimization problem cannot be tackled by one of traditional gradient-based learning algorithms, which are only suitable to perform optimization in absence of constraints among parameters. It is, therefore, necessary to develop specific optimization/learning algorithms that are able to look for a local minimum of a loss function under smooth constraints in an efficient way. Such need calls for manifold optimization methods. In this paper, we extend classical gradient-based learning algorithms on at parameter spaces (from classical gradient descent to adaptive momentum) to curved spaces (smooth manifolds) by means of tools from manifold calculus. We compare clustering performances of these methods and known methods from the scientific literature. The obtained results confirm that the proposed learning algorithms prove lighter in computational complexity than existing ones without detriment in clustering efficacy.

Learning a consensus affinity matrix for multi-view clustering via subspaces merging on Grassmann manifold

Integrative multi-view subspace clustering aims to partition observed samples into underlying clusters through fusing representative subspace information from different views into a latent space. The clustering performance relies on the accuracy of sample affinity measurement. However, existing approaches leverage the subspace representation of each view and overlook the learning of appropriate sample affinities. This paper proposes to learn a consensus affinity directly by merging subspace representations of different views on a Grassmann manifold while maintaining their geometric structures across these views. The proposed method not only preserves the structure of the most informative individual view, but also discovers a latent common structure across all views. The associated constrained optimization problem is solved using the alternating direction method of multipliers. Extensive experiments on synthetic and real-world datasets show that the proposed method outperforms several state-of-the-art multi-view subspace clustering methods. The affinity matrix obtained by our method can extract highly representative and latent common information to enhance the clustering performance.

Pizzetti formula on the Grassmannian of 2-planes

This paper is devoted to the role played by the Higgs algebra $$H_3$$ in the generalisation of classical harmonic analysis from the sphere $$S^{m-1}$$ to the (oriented) Grassmann manifold $${{\text {Gr}}}_o(m,2)$$ of 2-planes. This algebra is identified as the dual partner (in the sense of Howe duality) of the orthogonal group $${\text {SO}}(m)$$ acting on functions on the Grassmannian. This is then used to obtain a Pizzetti formula for integration over this manifold. The resulting formulas are finally compared to formulas obtained earlier for the Pizzetti integration over Stiefel manifolds, using an argument involving symmetry reduction.