Related papers: Tame Riemannian Stochastic Approximation

Covariance matrices have attracted attention for machine learning applications due to their capacity to capture interesting structure in the data. The main challenge is that one needs to take into account the particular geometry of the…

Optimization on Riemannian manifolds widely arises in eigenvalue computation, density functional theory, Bose-Einstein condensates, low rank nearest correlation, image registration, and signal processing, etc. We propose an adaptive…

We present a new class of stochastic, geometrically-driven optimization algorithms on the orthogonal group $O(d)$ and naturally reductive homogeneous manifolds obtained from the action of the rotation group $SO(d)$. We theoretically and…

We study the Riemannian optimization methods on the embedded manifold of low rank matrices for the problem of matrix completion, which is about recovering a low rank matrix from its partial entries. Assume $m$ entries of an $n\times n$ rank…

A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations $|Ax|^2=y$. The algorithms are developed by exploiting the inherent low rank structure of the problem based on the…

In the paper, we study a class of useful minimax problems on Riemanian manifolds and propose a class of effective Riemanian gradient-based methods to solve these minimax problems. Specifically, we propose an effective Riemannian gradient…

We propose practical deep Gaussian process models on Riemannian manifolds, similar in spirit to residual neural networks. With manifold-to-manifold hidden layers and an arbitrary last layer, they can model manifold- and scalar-valued…

For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives.…

The stochastic gradient descent (SGD) algorithm is the algorithm we use to train neural networks. However, it remains poorly understood how the SGD navigates the highly nonlinear and degenerate loss landscape of a neural network. In this…

We prove that Riemannian contraction in a supervised learning setting implies generalization. Specifically, we show that if an optimizer is contracting in some Riemannian metric with rate $\lambda > 0$, it is uniformly algorithmically…

From optimal transport to robust dimensionality reduction, a plethora of machine learning applications can be cast into the min-max optimization problems over Riemannian manifolds. Though many min-max algorithms have been analyzed in the…

Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required non-trivial smoothness…

Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…

We propose a stochastic variance-reduced cubic regularized Newton algorithm to optimize the finite-sum problem over a Riemannian submanifold of the Euclidean space. The proposed algorithm requires a full gradient and Hessian update at the…

Stochastic Gradient Descent (SGD) with adaptive steps is widely used to train deep neural networks and generative models. Most theoretical results assume that it is possible to obtain unbiased gradient estimators, which is not the case in…

Many supervised learning tasks have intrinsic symmetries, such as translational and rotational symmetry in image classifications. These symmetries can be exploited to enhance performance. We formulate the symmetry constraints into a concise…

Solving the inverse kinematics problem is a fundamental challenge in motion planning, control, and calibration for articulated robots. Kinematic models for these robots are typically parametrized by joint angles, generating a complicated…

Large-scale optimization problems arising from the discretization of problems involving PDEs sometimes admit solutions that can be well approximated by low-rank matrices. In this paper, we will exploit this low-rank approximation property…

In this paper, we consider a class of structured nonsmooth optimization problems over an embedded submanifold of a Euclidean space, where the first part of the objective is the sum of a difference-of-convex (DC) function and a smooth…

Direct search methods represent a robust and reliable class of algorithms for solving black-box optimization problems. In this paper, we explore the application of those strategies to Riemannian optimization, wherein minimization is to be…