Related papers: The Grassmannian of affine subspaces
Convex geometry has recently attracted great attention as a framework to formulate general probabilistic theories. In this framework, convex sets and affine maps represent the state spaces of physical systems and the possible dynamics,…
The shape and orientation of data clouds reflect variability in observations that can confound pattern recognition systems. Subspace methods, utilizing Grassmann manifolds, have been a great aid in dealing with such variability. However,…
Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed…
The real Grassmannian is both a projective variety (via Pl\"ucker coordinates) and an affine variety (via orthogonal projections). We connect these two representations, and we develop the commutative algebra of the latter variety. We…
Grassmann angles improve upon similar concepts of angle between subspaces that measure volume contraction in orthogonal projections, working for real or complex subspaces, and being more efficient when dimensions are different. Their…
By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all…
Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization. Here a new tool to restrict their…
It is well known that the space of oriented lines of Euclidean space has a natural symplectic structure. Moreover, given an immersed, oriented hypersurface S the set of oriented lines that cross S orthogonally is a Lagrangian submanifold.…
Learning representations on Grassmann manifolds is popular in quite a few visual recognition tasks. In order to enable deep learning on Grassmann manifolds, this paper proposes a deep network architecture by generalizing the Euclidean…
In this paper, we study the Grassmannian of n-dimensional subspaces of a 2n-dimensional vector space and its infinite-dimensional analogues. Such a Grassmannian can be endowed with two binary relations (adjacent and distant), with pencils…
Motivated by the concept of Euclidean Distance Degree, which measures the complexity of finding the nearest point to an algebraic set in Euclidean space, we introduce the notion of Grassmann Distance Complexity (GDC). This concept…
We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These…
Let G be a simply-connected simple compact Lie group over the complex numbers. The affine Grassmannian is a projective ind-variety, homotopy-equivalent to the loop space of G and closely analogous to a maximal flag variety of the classical…
Subspace learning and matrix factorization problems have great many applications in science and engineering, and efficient algorithms are critical as dataset sizes continue to grow. Many relevant problem formulations are non-convex, and in…
Geometrical properties of holonomic and non holonomic varieties defined by the Pfaff equations connected with a first order systems of differential equations are studied. The Riemann extensions of affine connected spaces for investigation…
We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an…
We prove a characterization of profinite algebras, i.e., topological algebras that are isomorphic to a projective limit of finite discrete algebras. In general profiniteness concerns both the topological and algebraic characteristics of a…
We introduce a Bayesian model for inferring mixtures of subspaces of different dimensions. The key challenge in such a mixture model is specification of prior distributions over subspaces of different dimensions. We address this challenge…
Due to advancements in digital cameras, it is easy to gather multiple images (or videos) from an object under different conditions. Therefore, image-set classification has attracted more attention, and different solutions were proposed to…
In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G_2. Some of these results are new. We give self-contained proofs here. One often encounters these…