arXiv:2603.06651v1 Announce Type: cross Abstract: Euclidean gradient descent algorithms barely capture the geometry of objective function-induced hypersurfaces and risk driving update trajectories off the hypersurfaces. Riemannian gradient descent algorithms address these issues but fail to represent complex hypersurfaces via a single classic manifold. We propose geodesic gradient descent (GGD), a generic and learning-rate-free Riemannian gradient descent algorithm. At each iteration, GGD uses an n-dimensional sphere to approximate a local neighborhood on the objective function-induced hypersurface, adapting to arbitrarily complex geometries. A tangent vector derived from the Euclidean gradient is projected onto the sphere to form a geodesic, ensuring the update trajectory stays on the hypersurface. Parameter updates are performed using the endpoint of the geodesic. The maximum step size of the gradient in GGD is equal to a quarter of the arc length on the n-dimensional sphere, thus eliminating the need for a learning rate. Experimental results show that compared with the classic Adam algorithm, GGD achieves test MSE reductions ranging from 35.79% to 48.76% for fully connected networks on the Burgers' dataset, and cross-entropy loss reductions ranging from 3.14% to 11.59% for convolutional neural networks on the MNIST dataset.

arXiv:2507.00790v4 Announce Type: replace-cross Abstract: Unified image restoration is a significantly challenging task in low-level vision. Existing methods either make tailored designs for specific tasks, limiting their generalizability across various types of degradation, or rely on training with paired datasets, thereby suffering from closed-set constraints. To address these issues, we propose a novel, dataset-free, and unified approach through recurrent posterior sampling utilizing a pretrained latent diffusion model. Our method incorporates the multimodal understanding model to provide sematic priors for the generative model under a task-blind condition. Furthermore, it utilizes a lightweight module to align the degraded input with the generated preference of the diffusion model, and employs recurrent refinement for posterior sampling. Extensive experiments demonstrate that our method outperforms state-of-the-art methods, validating its effectiveness and robustness. Our code and data are available at https://github.com/AMAP-ML/LD-RPS.

Nature, Published online: 10 March 2026; doi:10.1038/s41586-026-10351-8

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