Related papers: Iso-Riemannian Optimization on Learned Data Manifo…
We study Cartan-Schouten metrics, explore invariant dual connections, and propose them as models for Information Geometry. Based on the underlying Riemannian barycenter and the biinvariant mean of Lie groups, we subsequently propose a new…
Convex optimization is a vibrant and successful area due to the existence of a variety of efficient algorithms that leverage the rich structure provided by convexity. Convexity of a smooth set or a function in a Euclidean space is defined…
The Symmetric Positive Definite (SPD) matrices have received wide attention for data representation in many scientific areas. Although there are many different attempts to develop effective deep architectures for data processing on the…
Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of…
It is shown that applying manifold learning techniques to Poincar\'e sections of high-dimensional, chaotic dynamical systems can uncover their low-dimensional topological organization. Manifold learning provides a low-dimensional embedding…
Modern generative modeling methods have demonstrated strong performance in learning complex data distributions from clean samples. In many scientific and imaging applications, however, clean samples are unavailable, and only noisy or…
Despite the recent success of Bayesian optimization (BO) in a variety of applications where sample efficiency is imperative, its performance may be seriously compromised in settings characterized by high-dimensional parameter spaces. A…
We are concerned with decentralized optimization over a compact submanifold, where the loss functions of local datasets are defined by their respective local datasets. A key challenge in decentralized optimization is mitigating the…
Decentralized optimization has become a fundamental tool for large-scale learning systems; however, most existing methods rely on the classical Lipschitz smoothness assumption, which is often violated in problems with rapidly varying…
Latent space geometry provides a rigorous and empirically valuable framework for interacting with the latent variables of deep generative models. This approach reinterprets Euclidean latent spaces as Riemannian through a pull-back metric,…
Block majorization-minimization (BMM) is a simple iterative algorithm for nonconvex optimization that sequentially minimizes a majorizing surrogate of the objective function in each block coordinate while the other block coordinates are…
Bayesian optimization (BO) recently became popular in robotics to optimize control parameters and parametric policies in direct reinforcement learning due to its data efficiency and gradient-free approach. However, its performance may be…
For optimization problems on Riemannian manifolds, many types of globally convergent algorithms have been proposed, and they are often equipped with the Riemannian version of the Armijo line search for global convergence. Such existing…
In machine learning, data is usually represented in a (flat) Euclidean space where distances between points are along straight lines. Researchers have recently considered more exotic (non-Euclidean) Riemannian manifolds such as hyperbolic…
We study nonconvex optimization landscapes for learning overcomplete representations, including learning (i) sparsely used overcomplete dictionaries and (ii) convolutional dictionaries, where these unsupervised learning problems find many…
Decentralized optimization on Riemannian manifolds is foundational for many modern machine learning and signal processing applications in which data are non-Euclidean and generated and processed in a distributed manner. Although intrinsic…
Let $f \colon \mathcal{M} \to \mathbb{R}$ be a Lipschitz and geodesically convex function defined on a $d$-dimensional Riemannian manifold $\mathcal{M}$. Does there exist a first-order deterministic algorithm which (a) uses at most…
Learning-to-optimize is an emerging framework that seeks to speed up the solution of certain optimization problems by leveraging training data. Learned optimization solvers have been shown to outperform classical optimization algorithms in…
In this paper, we propose a new global analysis framework for a class of low-rank matrix recovery problems on the Riemannian manifold. We analyze the global behavior for the Riemannian optimization with random initialization. We use the…
Randomly initialized first-order optimization algorithms are the method of choice for solving many high-dimensional nonconvex problems in machine learning, yet general theoretical guarantees cannot rule out convergence to critical points of…