As a significant direction in this line, hyperbolic representation learning has been shown to offer several advantages over conventional Euclidean geometry.

In this paper, we comparatively analyze the Bures-Wasserstein (BW) geometry with the popular Affine-Invariant (AI) geometry for Riemannian optimization on the symmetric positive definite (SPD) matrix manifold. Our study begins with an observation that the BW metric has a linear dependence on SPD matrices in contrast to the quadratic dependence of the AI metric. We build on this to show that the BW metric is a more suitable and robust choice for several Riemannian optimization problems over ill-conditioned SPD matrices. We show that the BW geometry has a non-negative curvature, which further improves convergence rates of algorithms over the non-positively curved AI geometry. Finally, we verify that several popular cost functions, which are known to be geodesic convex under the AI geometry, are also geodesic convex under the BW geometry. Extensive experiments on various applications support our findings.

Neural Information Processing Systems http://nips.cc/

Under the data manifold hypothesis, high-dimensional data are concentrated near a low-dimensional manifold. We study the problem of Riemannian optimization over such manifolds when they are given only implicitly through the data distribution, and the standard manifold operations required by classical algorithms are unavailable. This formulation captures a broad class of data-driven design problems that are central to modern generative AI. Our key idea is to introduce a link function that connects the data distribution to the geometric operations needed for optimization. We show that this function enables the recovery of essential manifold operations, such as retraction and Riemannian gradient computation. Moreover, we establish a direct connection between our construction and the score function in diffusion models of the data distribution. This connection allows us to leverage well-studied parameterizations, efficient training procedures, and even pretrained score networks from the diffusion model literature to perform optimization. Building on this foundation, we propose two efficient inference-time algorithms -- Denoising Landing Flow (DLF) and Denoising Riemannian Gradient Descent (DRGD) -- and provide theoretical guarantees for both feasibility (approximate manifold adherence) and optimality (small Riemannian gradient norm). Finally, we demonstrate the effectiveness of our approach on finite-horizon reference tracking tasks in data-driven control, highlighting its potential for practical generative and design applications.

We study online Riemannian optimization on Hadamard manifolds under the framework of horospherical convexity (h-convexity). Prior work mostly relies on the geodesic convexity (g-convexity), leading to regret bounds scaling poorly with the manifold curvature. To address this limitation, we analyze Riemannian online gradient descent for h-convex and strongly h-convex functions and establish $O(\sqrt{T})$ and $O(\log(T))$ regret guarantees, respectively. These bounds are curvature-independent and match the results in the Euclidean setting. We validate our approach with experiments on the manifold of symmetric positive definite (SPD) matrices equipped with the affine-invariant metric. In particular, we investigate online Tyler's $M$-estimation and online Fréchet mean computation, showing the application of h-convexity in practice.

We study decentralized online Riemannian optimization over manifolds with possibly positive curvature, going beyond the Hadamard manifold setting. Decentralized optimization techniques rely on a consensus step that is well understood in Euclidean spaces because of their linearity. However, in positively curved Riemannian spaces, a main technical challenge is that geodesic distances may not induce a globally convex structure. In this work, we first analyze a curvature-aware Riemannian consensus step that enables a linear convergence beyond Hadamard manifolds. Building on this step, we establish a $O(\sqrt{T})$ regret bound for the decentralized online Riemannian gradient descent algorithm. Then, we investigate the two-point bandit feedback setup, where we employ computationally efficient gradient estimators using smoothing techniques, and we demonstrate the same $O(\sqrt{T})$ regret bound through the subconvexity analysis of smoothed objectives.

Reduced Rank Regression (RRR) is a widely used method for multi-response regression. However, RRR assumes a linear relationship between features and responses. While linear models are useful and often provide a good approximation, many real-world problems involve more complex relationships that cannot be adequately captured by simple linear interactions. One way to model such relationships is via multilinear transformations. This paper introduces Higher Order Reduced Rank Regression (HORRR), an extension of RRR that leverages multi-linear transformations, and as such is capable of capturing nonlinear interactions in multi-response regression. HORRR employs tensor representations for the coefficients and a Tucker decomposition to impose multilinear rank constraints as regularization akin to the rank constraints in RRR. Encoding these constraints as a manifold allows us to use Riemannian optimization to solve this HORRR problems. We theoretically and empirically analyze the use of Riemannian optimization for solving HORRR problems.

Learned denoisers play a fundamental role in various signal generation (e.g., diffusion models) and reconstruction (e.g., compressed sensing) architectures, whose success derives from their ability to leverage low-dimensional structure in data. Existing denoising methods, however, either rely on local approximations that require a linear scan of the entire dataset or treat denoising as generic function approximation problems, often sacrificing efficiency and interpretability. We consider the problem of efficiently denoising a new noisy data point sampled from an unknown $d$-dimensional manifold $M \in \mathbb{R}^D$, using only noisy samples. This work proposes a framework for test-time efficient manifold denoising, by framing the concept of "learning-to-denoise" as "learning-to-optimize". We have two technical innovations: (i) online learning methods which learn to optimize over the manifold of clean signals using only noisy data, effectively "growing" an optimizer one sample at a time. (ii) mixed-order methods which guarantee that the learned optimizers achieve global optimality, ensuring both efficiency and near-optimal denoising performance. We corroborate these claims with theoretical analyses of both the complexity and denoising performance of mixed-order traversal. Our experiments on scientific manifolds demonstrate significantly improved complexity-performance tradeoffs compared to nearest neighbor search, which underpins existing provable denoising approaches based on exhaustive search.