We introduce the Riemannian Proximal Sampler, a method for sampling from densities defined on Riemannian manifolds. The performance of this sampler critically depends on two key oracles: the Manifold Brownian Increments (MBI) oracle and the Riemannian Heat-kernel (RHK) oracle. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with ε-accuracy requires O(log(1/ε)) iterations in Kullback-Leibler divergence assuming access to exact oracles and O(log2(1/ε))iterations in the total variation metric assuming access to sufficiently accurate inexact oracles.

When this structure is known, at least locally, it can be exploited through reduction mappings that reparametrise part of the parameter space to lie on the solution manifold. These reductions naturally arise from inner optimisation problems and effectively remove redundant directions, yielding a lowerdimensional objective. In this work, we introduce a general framework to understand how such reductions influence the optimisation landscape. We show that well-designed reduction mappings improve curvature properties of the objective, leading to better-conditioned problems and theoretically faster convergence for gradient-based methods. Our analysis unifies a range of scenarios where structural information at optimality is leveraged to accelerate convergence, offering a principled explanation for the empirical gains observed in such optimisation algorithms.

Hyperbolic deep learning has become a growing research direction in computer vision due to the unique properties afforded by the alternate embedding space. The negative curvature and exponentially growing distance metric provide a natural framework for capturing hierarchical relationships between datapoints and allowing for finer separability between their embeddings. However, current hyperbolic learning approaches are still prone to overfitting, computationally expensive, and prone to instability, especially when attempting to learn the manifold curvature to adapt to tasks and different datasets. To address these issues, our paper presents a derivation for Riemannian AdamW that helps increase hyperbolic generalization ability. For improved stability, we introduce a novel fine-tunable hyperbolic scaling approach to constrain hyperbolic embeddings and reduce approximation errors. Using this along with our curvature-aware learning schema for Riemannian Optimizers enables the combination of curvature and non-trivialized hyperbolic parameter learning. Our approach demonstrates consistent performance improvements across Computer Vision, EEG classification, and hierarchical metric learning tasks while greatly reducing runtime.

The rapid advancement of generative AI enables highly realistic synthetic videos, posing significant challenges for content authentication and raising urgent concerns about misuse.

Many real-world graphs exhibit diverse and complex topological structures that are not well captured by geometric manifolds with uniform global curvature, such as hyperbolic or spherical spaces. Recently, there has been growing interest in embedding graphs into pseudo-Riemannian manifolds, which generalize both hyperbolic and spherical geometries. However, existing approaches face three significant limitations, including the ineffective pseudo-Riemannain framework, the shallow architectures, and the absence of clear guideline for selecting suitable pseudo-Riemannian manifolds. To address these issues, we introduce a novel diffeomorphic framework for graph embedding that aligns with the nature of pseudo-Riemannian manifolds. Subsequently, we propose the pseudo-Riemannian Graph Transformer for learning representations of complex graph structures. Our diffeomorphic framework in pseudo-Riemannian geometry enables the principled definitions of core transformer components, including linear attention, residual connection, and layer normalization. Finally, we develop a lightweight space searching algorithm to automatically identify the most suitable pseudo-Riemannian manifold for an input graph. Extensive experiments on diverse real-world graphs demonstrate that our model consistently outperforms other baselines in both node classification and link prediction tasks.

We analyse the convergence of one-hidden-layer ReLU networks trained by gradient flow on n data points. Our main contribution leverages the high dimensionality of the ambient space, which implies low correlation of the input samples, to demonstrate that a network with width of order log(n)neurons suffices for global convergence with high probability. Our analysis uses a Polyak-Łojasiewicz viewpoint along the gradient-flow trajectory, which provides an exponential rate of convergence of 1n. When the data are exactly orthogonal, we give further refined characterizations of the convergence speed, proving its asymptotic behavior lies between the orders 1n and 1 n, and exhibiting a phase-transition phenomenon in the convergence rate, during which it evolves from the lower bound to the upper, and in a relative time of order 1log(n).

Model merging offers a promising avenue for knowledge integration and parallel development without retraining. Yet, existing methods either ignore the geometry of the loss landscape or rely on intractable full-space Hessian approximations. We propose EpiMer, a framework that casts model merging as solving the Fréchet mean on a Riemannian manifold and restricts the computation to a low-rank subspace spanned by the task vectors. With the expected Hessian as the metric, we reveal a connection between local curvature and epistemic uncertainty of the parameters. Our theoretical analysis decomposes the merging error bound into the subspace Fréchet variance and the residual energy, and provides a closed-form characterization of when curvature-aware merging provably outperforms flat-geometry methods. In addition, our framework unifies both curvature-aware methods and recent spectral methods as special cases of the subspace Fréchet mean with different geometric metrics. Merging fine-tuned CLIP-ViT models on eight image classification tasks, Epistemic Merging strictly outperforms the baselines on all three CLIP-ViT backbones at matched rank, improving the across-task average accuracy and worst-task accuracy on every backbone.

We develop a geometric theory of projection heads in self-supervised learning by modeling the head as a trainable Riemannian metric on the backbone representation manifold. We show that linear heads perform implicit subspace whitening, while nonlinear heads adapt local metrics to satisfy the specific topological constraints of the loss, with head depth empirically dictating this capacity. Analyzing dimensional collapse, we prove that smooth nonlinear heads natively induce negative eigenvalues in the Hessian at collapsed equilibria, making them unstable. We empirically validate this by continuously tracking the optimization geometry during training, which reveals that smooth activations like Swish can generate explicit negative curvature to escape collapse, whereas linear and ReLU heads under continuous-time gradient flow cannot, relying instead on discrete-time optimization dynamics and BatchNorm. Finally, we geometrically characterize how metric degeneracy governs the information-invariance trade-off, explaining why the head must be discarded. Evaluated across contrastive and decorrelation-based objectives on foundation models, our results demonstrate that the projection head acts as a universal geometric buffer, decoupling the semantic backbone from the rigid, destructive constraints of the pretraining objective.

Learning of continuous exponential family distributions with unbounded support remains an important area of research for both theory and applications in high-dimensional statistics. In recent years, score matching has become a widely used method for learning exponential families with continuous variables due to its computational ease when compared against maximum likelihood estimation. However, theoretical understanding of the statistical properties of score matching is still lacking. In this work, we provide a non-asymptotic sample complexity analysis for learning the structure of exponential families of polynomials with score matching. The derived sample bounds show a polynomial dependence on the model dimension. These bounds are the first of its kind, as all prior work has shown only asymptotic bounds on the sample complexity.

We study bilevel optimization with a fixed polyhedral lower feasible set. Such problems are challenging for two reasons: active-set changes can make the upper objective nonsmooth, and existing hypergradient methods typically require lower-Hessian inversions or equivalent linear solves, which are computationally expensive. To address these issues, we adopt a logarithmic barrier smoothing of the lower problem to obtain a differentiable approximation of the constrained bilevel objective, and develop a proxy-gradient algorithm for the resulting barrier-smoothed surrogate. The algorithm uses only gradients of the upper and lower objectives; its only second-order object is the explicit logarithmic barrier Hessian determined by the fixed polyhedral constraints. Barrier smoothing restores differentiability, but Euclidean smoothness constants are not uniformly bounded near the boundary. We therefore develop a local Dikin-geometry analysis in which the barrier-metric provides an oracle-free curvature scale near the moving lower centers. This leads to barrier-aware schedules that keep the iterates inside locally well-behaved regions. For the barrier-smoothed objective, we prove stationarity rates of $\widetilde{O}(K^{-2/3})$ in the deterministic setting and $\widetilde{O}(K^{-2/5})$ under upper-level-only bounded stochastic noise after $K$ outer iterations, together with quantitative bias control as the barrier parameter decreases.