Relative Gottlieb Groups of Embeddings between Complex Grassmannians

Let Gr k , n be the complex Grassmann manifold of k -linear subspaces in ℂ n . We compute rational relative Gottlieb groups of the embedding i : Gr k , n ⟶ Gr k , n + r and show that the G -sequence is exact if r ≥ k n − k .

Knowledge Distillation of Grassmann Manifold Network for Remote Sensing Scene Classification

Due to device limitations, small networks are necessary for some real-world scenarios, such as satellites and micro-robots. Therefore, the development of a network with both good performance and small size is an important area of research. Deep networks can learn well from large amounts of data, while manifold networks have outstanding feature representation at small sizes. In this paper, we propose an approach that exploits the advantages of deep networks and shallow Grassmannian manifold networks. Inspired by knowledge distillation, we use the information learned from convolutional neural networks to guide the training of the manifold networks. Our approach leads to a reduction in model size, which addresses the problem of deploying deep learning on resource-limited embedded devices. Finally, a series of experiments were conducted on four remote sensing scene classification datasets. The method in this paper improved the classification accuracy by 2.31% and 1.73% on the UC Merced Land Use and SIRIWHU datasets, respectively, and the experimental results demonstrate the effectiveness of our approach.

Numerical Prediction of Two-Phase Flow through a Tube Bundle Based on Reduced-Order Model and a Void Fraction Correlation

Predicting the void fraction of a two-phase flow outside of tubes is essential to evaluate the thermohydraulic behaviour in steam generators. Indeed, it determines two-phase mixture properties and affects two-phase mixture velocity, which enable evaluating the pressure drop of the system. The two-fluid model for the numerical simulation of two-phase flows requires interaction laws between phases which are not known and/or reliable for a flow within a tube bundle. Therefore, the mixture model, for which it is easier to implement suitable correlations for tube bundles, is used. Indeed, by expressing the relative velocity as a function of slip, the void fraction model of Feenstra et al.and Hibiki et al. developed for upward cross-flow through horizontal tube bundles is introduced and compared. With the method suggested in this paper, the physical phenomena that occur in tube bundles are taken into consideration. Moreover, the tube bundle is modelled using a porous media approach where the Darcy–Forchheimer term is usually defined by correlations found in the literature. However, for some tube bundle geometries, these correlations are not available. The second goal of the paper is to quickly compute, in quasi-real-time, this term by a non-intrusive parametric reduced model based on Proper Orthogonal Decomposition. This method, named Bi-CITSGM (Bi-Calibrated Interpolation on the Tangent Subspace of the Grassmann Manifold), consists in interpolating the spatial and temporal bases by ITSGM (Interpolation on the Tangent Subspace of the Grassmann Manifold) in order to define the solution for a new parameter. The two developed methods are validated based on the experimental results obtained by Dowlati et al. for a two-phase cross-flow through a horizontal tube bundle.

Ensemble Learning Using Matrices of Classifier Interactions and Decision Profiles on Riemannian and Grassmann Manifolds

This paper introduces a new topic and research of geometric classifier ensemble learning using two types of objects: classifier prediction pairwise matrix (CPPM) and decision profiles (DPs). Learning from CPPM requires using Riemannian manifolds (R-manifolds) of symmetric positive definite (SPD) matrices. DPs can be used to build a Grassmann manifold (G-manifold). Experimental results show that classifier ensembles and their cascades built using R-manifolds are less dependent on some properties of individual classifiers (e.g. depth of decision trees in random forests (RFs) or extra trees (ETs)) in comparison to G-manifolds and Euclidean geometry. More independent individual classifiers allow obtaining R-manifolds with better properties for classification. Generally, the accuracy of classification in nonlinear geometry is higher than in Euclidean one. For multi-class problems, G-manifolds perform similarly to stacking-based classifiers built on R-manifolds of SPD matrices in terms of classification accuracy.

Rigidity theorems for holomorphic curves in a complex Grassmann manifold G(3,6)

In this paper, we prove some local rigidity theorems of holomorphic curves in a complex Grassmann manifold [Formula: see text] by moving frames. By applying our rigidity theorems, we also give a characterization of all homogeneous holomorphic two-spheres in [Formula: see text] classified by the second author.