This paper aims to provide a novel design of a multiscale framelet convolution for spectral graph neural networks (GNNs). While current spectral methods excel in various graph learning tasks, they often lack the flexibility to adapt to noisy, incomplete, or perturbed graph signals, making them fragile in such conditions. Our newly proposed framelet convolution addresses these limitations by decomposing graph data into low-pass and high-pass spectra through a finely-tuned multiscale approach. Our approach directly designs filtering functions within the spectral domain, allowing for precise control over the spectral components. The proposed design excels in filtering out unwanted spectral information and significantly reduces the adverse effects of noisy graph signals. Our approach not only enhances the robustness of GNNs but also preserves crucial graph features and structures. Through extensive experiments on diverse, real-world graph datasets, we demonstrate that our framelet convolution achieves superior performance in node classification tasks. It exhibits remarkable resilience to noisy data and adversarial attacks, highlighting its potential as a robust solution for real-world graph applications. This advancement opens new avenues for more adaptive and reliable spectral GNN architectures.
The University of Sydney（悉尼大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学数学研究所）；Shanghai Jiao Tong University（上海交通大学）；The University of New South Wales（新南威尔士大学）；University of Cambridge（剑桥大学）；Zhejiang Normal University（浙江师范大学）；University of California, Los Angeles（加州大学洛杉矶分校）
This paper presents a new approach for assembling graph neural networks based on framelet transforms. The latter provides a multi-scale representation for graph-structured data. We decompose an input graph into low-pass and high-pass frequencies coefficients for network training, which then defines a framelet-based graph convolution. The framelet decomposition naturally induces a graph pooling strategy by aggregating the graph feature into low-pass and high-pass spectra, which considers both the feature values and geometry of the graph data and conserves the total information. The graph neural…
The University of Sydney（悉尼大学）；Shanghai Jiao Tong University（上海交通大学）；Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）；The University of New South Wales（新南威尔士大学）
As graph data collected from the real world is merely noise-free, a practical representation of graphs should be robust to noise. Existing research usually focuses on feature smoothing but leaves the geometric structure untouched. Furthermore, most work takes L2-norm that pursues a global smoothness, which limits the expressivity of graph neural networks. This paper tailors regularizers for graph data in terms of both feature and structure noises, where the objective function is efficiently solved with the alternating direction method of multipliers (ADMM). The proposed scheme allows to take m…
In this paper, we develop a general theoretical framework for constructing Haar-type tight framelets on any compact set with a hierarchical partition. In particular, we construct a novel area-regular hierarchical partition on the 2-sphere and establish its corresponding spherical Haar tight framelets with directionality. We conclude by evaluating and illustrating the effectiveness of our area-regular spherical Haar tight framelets in several denoising experiments. Furthermore, we propose a convolutional neural network (CNN) model for spherical signal denoising which employs the fast framelet d…
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）；Seoul National University Bundang Hospital（首尔国立大学盆唐医院）
Model based iterative reconstruction (MBIR) algorithms for low-dose X-ray CT are computationally expensive. To address this problem, we recently proposed a deep convolutional neural network (CNN) for low-dose X-ray CT and won the second place in 2016 AAPM Low-Dose CT Grand Challenge. However, some of the texture were not fully recovered. To address this problem, here we propose a novel framelet-based denoising algorithm using wavelet residual network which synergistically combines the expressive power of deep learning and the performance guarantee from the framelet-based denoising algorithms. …
Korea Advanced Institute of Science and Technology（韩国科学技术院）；Bio Imaging and Signal Processing Lab.（生物成像与信号处理实验室）；Department of Bio and Brain Engineering（生物与脑工程系）
Recently, deep learning approaches with various network architectures have achieved significant performance improvement over existing iterative reconstruction methods in various imaging problems. However, it is still unclear why these deep learning architectures work for specific inverse problems. To address these issues, here we show that the long-searched-for missing link is the convolution framelets for representing a signal by convolving local and non-local bases. The convolution framelets was originally developed to generalize the theory of low-rank Hankel matrix approaches for inverse pr…
The University of Sydney（悉尼大学）；Shanghai Jiao Tong University（上海交通大学）；University of Cambridge（剑桥大学）；The University of New South Wales（新南威尔士大学）
Many real-world relational systems, such as social networks and biological systems, contain dynamic interactions. When learning dynamic graph representation, it is essential to employ sequential temporal information and geometric structure. Mainstream work achieves topological embedding via message passing networks (e.g., GCN, GAT). The temporal evolution, on the other hand, is conventionally expressed via memory units (e.g., LSTM or GRU) that possess convenient information filtration in a gate mechanism. Though, such a design prevents large-scale input sequence due to the over-complicated enc…
Recurrent neural networks are a powerful tool for modeling sequential data, but the dependence of each timestep's computation on the previous timestep's output limits parallelism and makes RNNs unwieldy for very long sequences. We introduce quasi-recurrent neural networks (QRNNs), an approach to neural sequence modeling that alternates convolutional layers, which apply in parallel across timesteps, and a minimalist recurrent pooling function that applies in parallel across channels. Despite lacking trainable recurrent layers, stacked QRNNs have better predictive accuracy than stacked LSTMs of …
Encoder-decoder networks using convolutional neural network (CNN) architecture have been extensively used in deep learning literatures thanks to its excellent performance for various inverse problems. However, it is still difficult to obtain coherent geometric view why such an architecture gives the desired performance. Inspired by recent theoretical understanding on generalizability, expressivity and optimization landscape of neural networks, as well as the theory of convolutional framelets, here we provide a unified theoretical framework that leads to a better understanding of geometry of en…
This paper presents a novel way to apply mathematical finance and machine learning (ML) to forecast stock options prices. Following results from the paper Quasi-Reversibility Method and Neural Network Machine Learning to Solution of Black-Scholes Equations (appeared on the AMS Contemporary Mathematics journal), we create and evaluate new empirical mathematical models for the Black-Scholes equation to analyze data for 92,846 companies. We solve the Black-Scholes (BS) equation forwards in time as an ill-posed inverse problem, using the Quasi-Reversibility Method (QRM), to predict option price fo…
The paper presents a comparative study of state-of-the-art approaches for question classification task: Logistic Regression, Convolutional Neural Networks (CNN), Long Short-Term Memory Network (LSTM) and Quasi-Recurrent Neural Networks (QRNN). All models use pre-trained GLoVe word embeddings and trained on human-labeled data. The best accuracy is achieved using CNN model with five convolutional layers and various kernel sizes stacked in parallel, followed by one fully connected layer. The model reached 90.7% accuracy on TREC 10 test set. All the model architectures in this paper were developed…
Causal Convolutional Neural Networks as Finite Impulse Response Filters
Future Resilient Systems, Singapore ETH Centre（新加坡未来韧性系统，新加坡ETH中心）；Department of Industrial Systems Engineering and Management（工业系统工程与管理系）；ETHZ；Department of Civil, Environmental and Geomatic Engineering（土木、环境与测绘工程系）
The abundance of gaps in satellite image time series often complicates the application of deep learning models such as convolutional neural networks for spatiotemporal modeling. Based on previous work in computer vision on image inpainting, this paper shows how three-dimensional spatiotemporal partial convolutions can be used as layers in neural networks to fill gaps in satellite image time series. To evaluate the approach, we apply a U-Net-like model on incomplete image time series of quasi-global carbon monoxide observations from the Sentinel-5P satellite. Prediction errors were comparable t…
Graph Convolutional Network (GCN) has achieved extraordinary success in learning effective task-specific representations of nodes in graphs. However, regarding Heterogeneous Information Network (HIN), existing HIN-oriented GCN methods still suffer from two deficiencies: (1) they cannot flexibly explore all possible meta-paths and extract the most useful ones for a target object, which hinders both effectiveness and interpretability; (2) they often need to generate intermediate meta-path based dense graphs, which leads to high computational complexity. To address the above issues, we propose an…
Deep neural network is difficult to train and this predicament becomes worse as the depth increases. The essence of this problem exists in the magnitude of backpropagated errors that will result in gradient vanishing or exploding phenomenon. We show that a variant of regularizer which utilizes orthonormality among different filter banks can alleviate this problem. Moreover, we design a backward error modulation mechanism based on the quasi-isometry assumption between two consecutive parametric layers. Equipped with these two ingredients, we propose several novel optimization solutions that can…
IRCNN$^{+}$: An Enhanced Iterative Residual Convolutional Neural Network for Non-stationary Signal Decomposition
Guangdong University of Finance and Economics（广东财经大学）；Istituto di Astrofisica e Planetologia Spaziali（天体物理学与行星科学研究所）；Istituto Nazionale di Geofisica e Vulcanologia（国家地球物理与火山学研究所）；Georgia Institute of Technology（佐治亚理工学院）；Sun Yat-sen University（中山大学）
