Efficient Training of Neural Fractional-Order Differential Equation via Adjoint Backpropagation

Fractional-order differential equations (FDEs) enhance traditional differential equations by extending the order of differential operators from integers to real numbers, offering greater flexibility in modeling complex dynamical systems with nonlocal characteristics. Recent progress at the intersection of FDEs and deep learning has catalyzed a new wave of innovative models, demonstrating the potential to address challenges such as graph representation learning. However, training neural FDEs has primarily relied on direct differentiation through forward-pass operations in FDE numerical solvers, leading to increased memory usage and computational complexity, particularly in large-scale applications. To address these challenges, we propose a scalable adjoint backpropagation method for training neural FDEs by solving an augmented FDE backward in time, which substantially reduces memory requirements. This approach provides a practical neural FDE toolbox and holds considerable promise for diverse applications. We demonstrate the effectiveness of our method in several tasks, achieving performance comparable to baseline models while significantly reducing computational overhead.

Neural Variable-Order Fractional Differential Equation Networks

Neural differential equation models have garnered significant attention in recent years for their effectiveness in machine learning applications.Among these, fractional differential equations (FDEs) have emerged as a promising tool due to their ability to capture memory-dependent dynamics, which are often challenging to model with traditional integer-order approaches.While existing models have primarily focused on constant-order fractional derivatives, variable-order fractional operators offer a more flexible and expressive framework for modeling complex memory patterns. In this work, we introduce the Neural Variable-Order Fractional Differential Equation network (NvoFDE), a novel neural network framework that integrates variable-order fractional derivatives with learnable neural networks.Our framework allows for the modeling of adaptive derivative orders dependent on hidden features, capturing more complex feature-updating dynamics and providing enhanced flexibility. We conduct extensive experiments across multiple graph datasets to validate the effectiveness of our approach.Our results demonstrate that NvoFDE outperforms traditional constant-order fractional and integer models across a range of tasks, showcasing its superior adaptability and performance.

Distributed-Order Fractional Graph Operating Network

We introduce the Distributed-order fRActional Graph Operating Network (DRAGON), a novel continuous Graph Neural Network (GNN) framework that incorporates distributed-order fractional calculus. Unlike traditional continuous GNNs that utilize integer-order or single fractional-order differential equations, DRAGON uses a learnable probability distribution over a range of real numbers for the derivative orders. By allowing a flexible and learnable superposition of multiple derivative orders, our framework captures complex graph feature updating dynamics beyond the reach of conventional models. We provide a comprehensive interpretation of our framework's capability to capture intricate dynamics through the lens of a non-Markovian graph random walk with node feature updating driven by an anomalous diffusion process over the graph. Furthermore, to highlight the versatility of the DRAGON framework, we conduct empirical evaluations across a range of graph learning tasks. The results consistently demonstrate superior performance when compared to traditional continuous GNN models. The implementation code is available at \url{https://github.com/zknus/NeurIPS-2024-DRAGON}.

PRFusion: Toward Effective and Robust Multi-Modal Place Recognition with Image and Point Cloud Fusion

Place recognition plays a crucial role in the fields of robotics and computer vision, finding applications in areas such as autonomous driving, mapping, and localization. Place recognition identifies a place using query sensor data and a known database. One of the main challenges is to develop a model that can deliver accurate results while being robust to environmental variations. We propose two multi-modal place recognition models, namely PRFusion and PRFusion++. PRFusion utilizes global fusion with manifold metric attention, enabling effective interaction between features without requiring camera-LiDAR extrinsic calibrations. In contrast, PRFusion++ assumes the availability of extrinsic calibrations and leverages pixel-point correspondences to enhance feature learning on local windows. Additionally, both models incorporate neural diffusion layers, which enable reliable operation even in challenging environments. We verify the state-of-the-art performance of both models on three large-scale benchmarks. Notably, they outperform existing models by a substantial margin of +3.0 AR@1 on the demanding Boreas dataset. Furthermore, we conduct ablation studies to validate the effectiveness of our proposed methods. The codes are available at: https://github.com/sijieaaa/PRFusion

Generalized Graph Signal Reconstruction via the Uncertainty Principle

We introduce a novel uncertainty principle for generalized graph signals that extends classical time-frequency and graph uncertainty principles into a unified framework. By defining joint vertex-time and spectral-frequency spreads, we quantify signal localization across these domains, revealing a trade-off between them. This framework allows us to identify a class of signals with maximal energy concentration in both domains, forming the fundamental atoms for a new joint vertex-time dictionary. This dictionary enhances signal reconstruction under practical constraints, such as incomplete or intermittent data, commonly encountered in sensor and social networks. Numerical experiments on real-world datasets demonstrate the effectiveness of the proposed approach, showing improved reconstruction accuracy and noise robustness compared to existing methods.

A Graph Signal Processing Perspective of Network Multiple Hypothesis Testing with False Discovery Rate Control

We consider a multiple hypothesis testing problem in a sensor network over the joint spatial-time domain. The sensor network is modeled as a graph, with each vertex representing a sensor and a signal over time associated with each vertex. We assume a hypothesis test and an associated p-value for every sample point in the joint spatial-time domain. Our goal is to determine which points have true alternative hypotheses. By parameterizing the unknown alternative distribution of $p$-values and the prior probabilities of hypotheses being null with a bandlimited generalized graph signal, we can obtain consistent estimates for them. Consequently, we also obtain an estimate of the local false discovery rates (lfdr). We prove that by using a step-up procedure on the estimated lfdr, we can achieve asymptotic false discovery rate control at a pre-determined level. Numerical experiments validate the effectiveness of our approach compared to existing methods.

FedSheafHN: Personalized Federated Learning on Graph-structured Data

Personalized subgraph Federated Learning (FL) is a task that customizes Graph Neural Networks (GNNs) to individual client needs, accommodating diverse data distributions. However, applying hypernetworks in FL, while aiming to facilitate model personalization, often encounters challenges due to inadequate representation of client-specific characteristics. To overcome these limitations, we propose a model called FedSheafHN, using enhanced collaboration graph embedding and efficient personalized model parameter generation. Specifically, our model embeds each client's local subgraph into a server-constructed collaboration graph. We utilize sheaf diffusion in the collaboration graph to learn client representations. Our model improves the integration and interpretation of complex client characteristics. Furthermore, our model ensures the generation of personalized models through advanced hypernetworks optimized for parallel operations across clients. Empirical evaluations demonstrate that FedSheafHN outperforms existing methods in most scenarios, in terms of client model performance on various graph-structured datasets. It also has fast model convergence and effective new clients generalization.

Hybrid Event-Frame Neural Spike Detector for Neuromorphic Implantable BMI

This work introduces two novel neural spike detection schemes intended for use in next-generation neuromorphic brain-machine interfaces (iBMIs). The first, an Event-based Spike Detector (Ev-SPD) which examines the temporal neighborhood of a neural event for spike detection, is designed for in-vivo processing and offers high sensitivity and decent accuracy (94-97%). The second, Neural Network-based Spike Detector (NN-SPD) which operates on hybrid temporal event frames, provides an off-implant solution using shallow neural networks with impressive detection accuracy (96-99%) and minimal false detections. These methods are evaluated using a synthetic dataset with varying noise levels and validated through comparison with ground truth data. The results highlight their potential in next-gen neuromorphic iBMI systems and emphasize the need to explore this direction further to understand their resource-efficient and high-performance capabilities for practical iBMI settings.

Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND

We introduce the FRactional-Order graph Neural Dynamical network (FROND), a new continuous graph neural network (GNN) framework. Unlike traditional continuous GNNs that rely on integer-order differential equations, FROND employs the Caputo fractional derivative to leverage the non-local properties of fractional calculus. This approach enables the capture of long-term dependencies in feature updates, moving beyond the Markovian update mechanisms in conventional integer-order models and offering enhanced capabilities in graph representation learning. We offer an interpretation of the node feature updating process in FROND from a non-Markovian random walk perspective when the feature updating is particularly governed by a diffusion process. We demonstrate analytically that oversmoothing can be mitigated in this setting. Experimentally, we validate the FROND framework by comparing the fractional adaptations of various established integer-order continuous GNNs, demonstrating their consistently improved performance and underscoring the framework's potential as an effective extension to enhance traditional continuous GNNs. The code is available at \url{https://github.com/zknus/ICLR2024-FROND}.

PointDifformer: Robust Point Cloud Registration With Neural Diffusion and Transformer

Point cloud registration is a fundamental technique in 3-D computer vision with applications in graphics, autonomous driving, and robotics. However, registration tasks under challenging conditions, under which noise or perturbations are prevalent, can be difficult. We propose a robust point cloud registration approach that leverages graph neural partial differential equations (PDEs) and heat kernel signatures. Our method first uses graph neural PDE modules to extract high dimensional features from point clouds by aggregating information from the 3-D point neighborhood, thereby enhancing the robustness of the feature representations. Then, we incorporate heat kernel signatures into an attention mechanism to efficiently obtain corresponding keypoints. Finally, a singular value decomposition (SVD) module with learnable weights is used to predict the transformation between two point clouds. Empirical experiments on a 3-D point cloud dataset demonstrate that our approach not only achieves state-of-the-art performance for point cloud registration but also exhibits better robustness to additive noise or 3-D shape perturbations.