GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains

Score-based diffusion models in infinite-dimensional function spaces provide a mathematically principled framework for modelling function-valued data, offering key advantages such as resolution invariance and the ability to handle irregular discretisations. However, practical implementations have struggled to fully realise these benefits. Existing backbones like Fourier neural operators are often biased towards regular grids and fail to generalise to complex domain topologies. We propose a novel architecture for function-space diffusion models that represents generalised graph convolutional kernels as finite element functions, enabling the model to naturally handle unstructured meshes and complex geometries. We demonstrate the efficacy of our network architecture through a series of unconditional and conditional sampling experiments across diverse geometries, including non-convex and multiply-connected domains. Our results show that the proposed method maintains resolution invariance and achieves high fidelity in capturing functional distributions on non-trivial geometries.

TMD-Bench: A Multi-Level Evaluation Paradigm for Music-Dance Co-Generation

Unified audio-visual generation is rapidly gaining industrial and creative relevance, enabling applications in virtual production and interactive media. However, when moving from general audio-video synthesis to music-dance co-generation, the task becomes substantially harder: musical rhythm, phrasing, and accents must drive choreographic motion at fine temporal resolution, and such rhythmic coupling is not captured by unimodal metrics or generic audiovisual consistency scores used in current evaluation practice. We introduce TMD-Bench, a benchmark for text-driven music-dance co-generation that assesses systems across unimodal generation quality, instruction adherence, and cross-modal rhythmic alignment. The benchmark integrates computable physical metrics with perceptual multimodal judgments, and is supported by a curated rhythm-aligned music-dance dataset and a fine-grained Music Captioner for structured music semantics. TMD-Bench further reveals that (i) modern commercial audio-visual models, such as Veo 3 and Sora 2, produce high-quality music and video, while rhythmic coupling remains less consistently optimized and leaves room for improvement, and (ii) our unified baseline RhyJAM trained on rhythm-aligned data achieves competitive beat-level synchronization while maintaining competitive unimodal fidelity. This presents prospects for building next-generation music-dance models that explicitly optimize rhythmic and kinetic coherence.

How many measurements are enough? Bayesian recovery in inverse problems with general distributions

We study the sample complexity of Bayesian recovery for solving inverse problems with general prior, forward operator and noise distributions. We consider posterior sampling according to an approximate prior $\mathcal{P}$, and establish sufficient conditions for stable and accurate recovery with high probability. Our main result is a non-asymptotic bound that shows that the sample complexity depends on (i) the intrinsic complexity of $\mathcal{P}$, quantified by its so-called approximate covering number, and (ii) concentration bounds for the forward operator and noise distributions. As a key application, we specialize to generative priors, where $\mathcal{P}$ is the pushforward of a latent distribution via a Deep Neural Network (DNN). We show that the sample complexity scales log-linearly with the latent dimension $k$, thus establishing the efficacy of DNN-based priors. Generalizing existing results on deterministic (i.e., non-Bayesian) recovery for the important problem of random sampling with an orthogonal matrix $U$, we show how the sample complexity is determined by the coherence of $U$ with respect to the support of $\mathcal{P}$. Hence, we establish that coherence plays a fundamental role in Bayesian recovery as well. Overall, our framework unifies and extends prior work, providing rigorous guarantees for the sample complexity of solving Bayesian inverse problems with arbitrary distributions.