Expressivity of Bi-Lipschitz Normalizing Flows: A Score-Based Diffusion Perspective

Many normalizing flow architectures impose regularity constraints, yet their distributional approximation properties are not fully characterized. We study the expressivity of bi-Lipschitz normalizing flows through the lens of score-based diffusion models. For the probability flow ODE of a variance-preserving diffusion, Lipschitz regularity of the score induces a flow of bi-Lipschitz diffeomorphic transport maps. This ODE bridge allows us to analyze the distributional approximation power of bi-Lipschitz normalizing flows and, conversely, derive deterministic convergence guarantees for diffusion-based transport. Our key idea is to use the probability flow ODE to link regularity of the score to regularity of the induced transport maps. We verify score regularity for broad target densities, including compactly supported densities, Gaussian convolutions of compactly supported measures and finite Gaussian mixtures. We obtain a universal distributional approximation result: Gaussian pullbacks induced by bi-Lipschitz variance-preserving transport maps are $L^1$-dense among all probability densities. For Gaussian convolution targets, we further obtain convergence in Kullback-Leibler divergence without early stopping.

Bridging Input Feature Spaces Towards Graph Foundation Models

Unlike vision and language domains, graph learning lacks a shared input space, as input features differ across graph datasets not only in semantics, but also in value ranges and dimensionality. This misalignment prevents graph models from generalizing across datasets, limiting their use as foundation models. In this work, we propose ALL-IN, a simple and theoretically grounded method that enables transferability across datasets with different input features. Our approach projects node features into a shared random space and constructs representations via covariance-based statistics, thus eliminating dependence on the original feature space. We show that the computed node-covariance operators and the resulting node representations are invariant in distribution to permutations of the input features. We further demonstrate that the expected operator exhibits invariance to general orthogonal transformations of the input features. Empirically, ALL-IN achieves strong performance across diverse node- and graph-level tasks on unseen datasets with new input features, without requiring architecture changes or retraining. These results point to a promising direction for input-agnostic, transferable graph models.

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.

No-reference based automatic parameter optimization for iterative reconstruction using a novel search space aware crow search algorithm

Iterative reconstruction technique's ability to reduce radiation exposure by using fewer projections has attracted significant attention. However, these methods typically require a precise tuning of several hyperparameters, which can have a major impact on reconstruction quality. Manually setting these parameters is time-consuming and increases the workload for human operators. In this paper, we introduce a novel fully automatic parameter optimization framework that can be applied to a wide range of Cone-beam computed tomography (CBCT) iterative reconstruction algorithms to determine optimal parameters without requiring a reference reconstruction. The proposed method incorporates a modified crow search algorithm (CSA) featuring a superior set-dependent local search mechanism, a search-space-aware global search strategy, and an objective-driven balance between local and global search. Additionally, to ensure an effective initial population, we propose a chaotic diagonal linear uniform initialization scheme that accelerates algorithm convergence. The performance of the proposed framework was evaluated on three imaging machines and four real datasets, as well as three different iterative reconstruction methods with the highest number of tunable parameters, representing the most challenging senario. The results indicate that the proposed method could outperform manual settings and CSA, with an 4.19% improvement in average fitness and 4.89% and 3.82% improvements on CHILL@UK and RPI_AXIS, respectively, which are two benchmark no-reference learning-based quality metrics. In addition, the qualitative results clearly show the superiority of the proposed method by maintaining fine details sharply. The overall performance of the proposed framework across different comparison scenarios demonstrates its effectiveness and robustness across all cases.

Hierarchical Multiscale Structure-Function Coupling for Brain Connectome Integration

Integrating structural and functional connectomes remains challenging because their relationship is non-linear and organized over nested modular hierarchies. We propose a hierarchical multiscale structure-function coupling framework for connectome integration that jointly learns individualized modular organization and hierarchical coupling across structural connectivity (SC) and functional connectivity (FC). The framework includes: (i) Prototype-based Modular Pooling (PMPool), which learns modality-specific multiscale communities by selecting prototypical ROIs and optimizing a differentiable modularity-inspired objective; (ii) an Attention-based Hierarchical Coupling Module (AHCM) that models both within-hierarchy and cross-hierarchy SC-FC interactions to produce enriched hierarchical coupling representations; and (iii) a Coupling-guided Clustering loss (CgC-Loss) that regularizes SC and FC community assignments with coupling signals, allowing cross-modal interactions to shape community alignment across hierarchies. We evaluate the model's performance across four cohorts for predicting brain age, cognitive score, and disease classification. Our model consistently outperforms baselines and other state-of-the-art approaches across three tasks. Ablation and sensitivity analyses verify the contributions of key components. Finally, the visualizations of learned coupling reveal interpretable differences, suggesting that the framework captures biologically meaningful structure-function relationships.

ProSMA-UNet: Decoder Conditioning for Proximal-Sparse Skip Feature Selection

Medical image segmentation commonly relies on U-shaped encoder-decoder architectures such as U-Net, where skip connections preserve fine spatial detail by injecting high-resolution encoder features into the decoder. However, these skip pathways also propagate low-level textures, background clutter, and acquisition noise, allowing irrelevant information to bypass deeper semantic filtering -- an issue that is particularly detrimental in low-contrast clinical imaging. Although attention gates have been introduced to address this limitation, they typically produce dense sigmoid masks that softly reweight features rather than explicitly removing irrelevant activations. We propose ProSMA-UNet (Proximal-Sparse Multi-Scale Attention U-Net), which reformulates skip gating as a decoder-conditioned sparse feature selection problem. ProSMA constructs a multi-scale compatibility field using lightweight depthwise dilated convolutions to capture relevance across local and contextual scales, then enforces explicit sparsity via an $\ell_1$ proximal operator with learnable per-channel thresholds, yielding a closed-form soft-thresholding gate that can remove noisy responses. To further suppress semantically irrelevant channels, ProSMA incorporates decoder-conditioned channel gating driven by global decoder context. Extensive experiments on challenging 2D and 3D benchmarks demonstrate state-of-the-art performance, with particularly large gains ($\approx20$\%) on difficult 3D segmentation tasks. Project page: https://math-ml-x.github.io/ProSMA-UNet/

Towards Improved Sentence Representations using Token Graphs

Obtaining a single-vector representation from a Large Language Model's (LLM) token-level outputs is a critical step for nearly all sentence-level tasks. However, standard pooling methods like mean or max aggregation treat tokens as an independent set, discarding the rich relational structure captured by the model's self-attention layers and making them susceptible to signal dilution. To address this, we introduce GLOT, a lightweight, structure-aware pooling module that reframes pooling as relational learning followed by aggregation. Operating on the outputs of a frozen LLM, GLOT first constructs a latent token-similarity graph, then refines token representations with a graph neural network, and finally aggregates them using a readout layer. Experimentally, our approach is remarkably robust and efficient: on a diagnostic stress test where 90% of tokens are random distractors, GLOT maintains over 97% accuracy while baseline methods collapse. Furthermore, it is competitive with state-of-the-art techniques on benchmarks like GLUE and MTEB with 20x fewer trainable parameters and speeds up the training time by over 100x compared with parameter-efficient fine-tuning methods. Supported by a theoretical analysis of its expressive power, our work shows that learning over token graphs is a powerful paradigm for the efficient adaptation of frozen LLMs. Our code is published at https://github.com/ipsitmantri/GLOT.

Approximation Theory for Lipschitz Continuous Transformers

Stability and robustness are critical for deploying Transformers in safety-sensitive settings. A principled way to enforce such behavior is to constrain the model's Lipschitz constant. However, approximation-theoretic guarantees for architectures that explicitly preserve Lipschitz continuity have yet to be established. In this work, we bridge this gap by introducing a class of gradient-descent-type in-context Transformers that are Lipschitz-continuous by construction. We realize both MLP and attention blocks as explicit Euler steps of negative gradient flows, ensuring inherent stability without sacrificing expressivity. We prove a universal approximation theorem for this class within a Lipschitz-constrained function space. Crucially, our analysis adopts a measure-theoretic formalism, interpreting Transformers as operators on probability measures, to yield approximation guarantees independent of token count. These results provide a rigorous theoretical foundation for the design of robust, Lipschitz continuous Transformer architectures.

Trajectory Stitching for Solving Inverse Problems with Flow-Based Models

Flow-based generative models have emerged as powerful priors for solving inverse problems. One option is to directly optimize the initial latent code (noise), such that the flow output solves the inverse problem. However, this requires backpropagating through the entire generative trajectory, incurring high memory costs and numerical instability. We propose MS-Flow, which represents the trajectory as a sequence of intermediate latent states rather than a single initial code. By enforcing the flow dynamics locally and coupling segments through trajectory-matching penalties, MS-Flow alternates between updating intermediate latent states and enforcing consistency with observed data. This reduces memory consumption while improving reconstruction quality. We demonstrate the effectiveness of MS-Flow over existing methods on image recovery and inverse problems, including inpainting, super-resolution, and computed tomography.

Diffeomorphism-Equivariant Neural Networks

Incorporating group symmetries via equivariance into neural networks has emerged as a robust approach for overcoming the efficiency and data demands of modern deep learning. While most existing approaches, such as group convolutions and averaging-based methods, focus on compact, finite, or low-dimensional groups with linear actions, this work explores how equivariance can be extended to infinite-dimensional groups. We propose a strategy designed to induce diffeomorphism equivariance in pre-trained neural networks via energy-based canonicalisation. Formulating equivariance as an optimisation problem allows us to access the rich toolbox of already established differentiable image registration methods. Empirical results on segmentation and classification tasks confirm that our approach achieves approximate equivariance and generalises to unseen transformations without relying on extensive data augmentation or retraining.