相关论文: Nystr\"om Approximation on Manifolds

In this paper, we study alternating projections on nontangential manifolds based on the tangent spaces. The main motivation is that the projection of a point onto a manifold can be computational expensive. We propose to use the tangent…

This work puts forth low-complexity Riemannian subspace descent algorithms for the minimization of functions over the symmetric positive definite (SPD) manifold. Different from the existing Riemannian gradient descent variants, the proposed…

This work is on constrained large-scale non-convex optimization where the constraint set implies a manifold structure. Solving such problems is important in a multitude of fundamental machine learning tasks. Recent advances on Riemannian…

Constrained optimization plays a crucial role in the fields of quantum physics and quantum information science and becomes especially challenging for high-dimensional complex structure problems. One specific issue is that of quantum process…

The numerical solution of partial differential equations on high-dimensional domains gives rise to computationally challenging linear systems. When using standard discretization techniques, the size of the linear system grows exponentially…

We present a framework for efficiently approximating differential-geometric primitives on arbitrary manifolds via construction of an atlas graph representation, which leverages the canonical characterization of a manifold as a finite…

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…

Several tensor networks are built of isometric tensors, i.e. tensors satisfying $W^\dagger W = \mathrm{I}$. Prominent examples include matrix product states (MPS) in canonical form, the multiscale entanglement renormalization ansatz (MERA),…

Optimization with orthogonality constraints frequently arises in various fields such as machine learning. Riemannian optimization offers a powerful framework for solving these problems by equipping the constraint set with a Riemannian…

When generalizing schemes for real-valued data approximation or decomposition to data living in Riemannian manifolds, tangent space-based schemes are very attractive for the simple reason that these spaces are linear. An open challenge is…

Low-rank approximation of a matrix function, $f(A)$, is an important task in computational mathematics. Most methods require direct access to $f(A)$, which is often considerably more expensive than accessing $A$. Persson and Kressner (SIMAX…

In recent years, various subspace algorithms have been developed to handle large-scale optimization problems. Although existing subspace Newton methods require fewer iterations to converge in practice, the matrix operations and full…

The techniques and analysis presented in this paper provide new methods to solve optimization problems posed on Riemannian manifolds. A new point of view is offered for the solution of constrained optimization problems. Some classical…

We examine a wide class of stochastic approximation algorithms for solving (stochastic) nonlinear problems on Riemannian manifolds. Such algorithms arise naturally in the study of Riemannian optimization, game theory and optimal transport,…

Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks.…

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…

Manifold learning seeks a low dimensional representation that faithfully captures the essence of data. Current methods can successfully learn such representations, but do not provide a meaningful set of operations that are associated with…

Optimization on Hadamard manifolds -- the natural Riemannian setting for globally geodesically convex problems -- relies on exponential maps to retract tangent vectors and parallel transport to connect tangent spaces across the manifold.…

The natural gradient method is widely used in statistical optimization, but its standard formulation assumes a Euclidean parameter space. This paper proposes an inversion-free stochastic natural gradient method for probability distributions…

We propose a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function…