Related papers: Numerical algorithms on the affine Grassmannian

We show that singular Riemannian foliations, or, more generally, manifold submetries, defined on a compact normal homogeneous space, have algebraic nature. Moreover, in this case there exists a one-to-one correspondence between algebras of…

Inferring time-varying graph structures from high-dimensional nodal observations is a fundamental problem arising in neuroscience, finance, climatology, and beyond. Two intrinsic challenges govern this problem: maintaining the…

Matrix-valued optimization tasks, including those involving symmetric positive definite (SPD) matrices, arise in a wide range of applications in machine learning, data science and statistics. Classically, such problems are solved via…

We classify all smooth flat Riemannian metrics on the two-dimensional plane. In the complete case, it is well-known that these metrics are isometric to the Euclidean metric. In the incomplete case, there is an abundance of…

A generalization of the notion of a (pseudo-) Riemannian space is proposed in a framework of noncommutative geometry. In particular, there are parametrized families of generalized Riemannian spaces which are deformations of classical…

We use filtrations of the Grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a Riemann surface, and so all harmonic maps from the 2-sphere, to the unitary group for a general class…

In image set classification, a considerable advance has been made by modeling the original image sets by second order statistics or linear subspace, which typically lie on the Riemannian manifold. Specifically, they are Symmetric Positive…

Optimization over the Stiefel manifold is a fundamental computational problem in many scientific and engineering applications. Despite considerable research effort, high-dimensional optimization problems over the Stiefel manifold remain…

We construct a Riemannian metric $g$ on $\mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $\Gamma\subset \mathbb R^4$ such that the unique area minimizing surface spanned by $\Gamma$ has infinite…

We introduce a Riemannian optimistic online learning algorithm for Hadamard manifolds based on inexact implicit updates. Unlike prior work, our method can handle in-manifold constraints, and matches the best known regret bounds in the…

This paper describes a quantum algorithm for efficiently decomposing finite Abelian groups. Such a decomposition is needed in order to apply the Abelian hidden subgroup algorithm. Such a decomposition (assuming the Generalized Riemann…

We introduce a generalized Rayleigh-quotient on the tensor product of Grassmannians enabling a unified approach to well-known optimization tasks from different areas of numerical linear algebra, such as best low-rank approximations of…

Shape optimization is commonly applied in engineering to optimize shapes with respect to an objective functional relying on PDE solutions. In this paper, we view shape optimization as optimization on Riemannian shape manifolds. We consider…

In Riemannian optimization, it is well known that the condition number of the Riemannian Hessian at an optimum strongly influences the asymptotic convergence behavior of optimization algorithms. On the manifold of symmetric positive…

Understanding how systems built out of modular components can be jointly optimized is an important problem in biology, engineering, and machine learning. The backpropagation algorithm is one such solution and has been instrumental in the…

High-dimensional data with intrinsic low-dimensional structure is ubiquitous in machine learning and data science. While various approaches allow one to learn a data manifold with a Riemannian structure from finite samples, performing…

We provide physics-inspired derivations of a number of algorithms for computing the permanent of a matrix. In particular we formulate the computation of the permanent as a Grassmann integral that may be viewed as an interacting many-fermion…

The effectiveness of dimensionality reduction with quadratic manifolds hinges on the choice of a reduced basis and the associated quadratic correction terms. Existing approaches typically rely on subspaces spanned by the leading principal…

In this paper, we develop a variant of the well-known Gauss-Newton (GN) method to solve a class of nonconvex optimization problems involving low-rank matrix variables. As opposed to the standard GN method, our algorithm allows one to handle…

Recently, studies on machine learning have focused on methods that use symmetry implicit in a specific manifold as an inductive bias. Grassmann manifolds provide the ability to handle fundamental shapes represented as shape spaces, enabling…