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Recent advancements in Large Language Models (LLMs) have enhanced efficiency across various domains, including protein engineering, where they offer promising opportunities for dry lab and wet lab experiment workflow automation. Previous work, namely TourSynbio-Agent, integrates a protein-specialized multimodal LLM (i.e. TourSynbio-7B) with domain-specific deep learning (DL) models to streamline both computational and experimental protein engineering tasks. While initial validation demonstrated TourSynbio-7B's fundamental protein property understanding, the practical effectiveness of the complete TourSynbio-Agent framework in real-world applications remained unexplored. This study presents a comprehensive validation of TourSynbio-Agent through five diverse case studies spanning both computational (dry lab) and experimental (wet lab) protein engineering. In three computational case studies, we evaluate the TourSynbio-Agent's capabilities in mutation prediction, protein folding, and protein design. Additionally, two wet-lab validations demonstrate TourSynbio-Agent's practical utility: engineering P450 proteins with up to 70% improved selectivity for steroid 19-hydroxylation, and developing reductases with 3.7x enhanced catalytic efficiency for alcohol conversion. Our findings from the five case studies establish that TourSynbio-Agent can effectively automate complex protein engineering workflows through an intuitive conversational interface, potentially accelerating scientific discovery in protein engineering.

The exponential growth in protein-related databases and scientific literature, combined with increasing demands for efficient biological information retrieval, has created an urgent need for unified and accessible search methods in protein engineering research. We present TourSynbio-Search, a novel bioinformatics search agent framework powered by the TourSynbio-7B protein multimodal large language model (LLM), designed to address the growing challenges of information retrieval across rapidly expanding protein databases and corresponding online research literature. The agent's dual-module architecture consists of PaperSearch and ProteinSearch components, enabling comprehensive exploration of both scientific literature and protein data across multiple biological databases. At its core, TourSynbio-Search employs an intelligent agent system that interprets natural language queries, optimizes search parameters, and executes search operations across major platforms including UniProt, PDB, ArXiv, and BioRxiv. The agent's ability to process intuitive natural language queries reduces technical barriers, allowing researchers to efficiently access and analyze complex biological data without requiring extensive bioinformatics expertise. Through detailed case studies in literature retrieval and protein structure visualization, we demonstrate TourSynbio-Search's effectiveness in streamlining biological information retrieval and enhancing research productivity. This framework represents an advancement in bridging the accessibility gap between complex biological databases and researchers, potentially accelerating progress in protein engineering applications. Our codes are available at: https://github.com/tsynbio/Toursynbio-Search

Protein engineering is important for biomedical applications, but conventional approaches are often inefficient and resource-intensive. While deep learning (DL) models have shown promise, their training or implementation into protein engineering remains challenging for biologists without specialized computational expertise. To address this gap, we propose AutoProteinEngine (AutoPE), an agent framework that leverages large language models (LLMs) for multimodal automated machine learning (AutoML) for protein engineering. AutoPE innovatively allows biologists without DL backgrounds to interact with DL models using natural language, lowering the entry barrier for protein engineering tasks. Our AutoPE uniquely integrates LLMs with AutoML to handle model selection for both protein sequence and graph modalities, automatic hyperparameter optimization, and automated data retrieval from protein databases. We evaluated AutoPE through two real-world protein engineering tasks, demonstrating substantial performance improvements compared to traditional zero-shot and manual fine-tuning approaches. By bridging the gap between DL and biologists' domain expertise, AutoPE empowers researchers to leverage DL without extensive programming knowledge. Our code is available at https://github.com/tsynbio/AutoPE.

Enzymes are biological catalysts that can accelerate chemical reactions compared to uncatalyzed reactions in aqueous environments. Their catalytic efficiency is quantified by the turnover number (kcat), a parameter in enzyme kinetics. Enhancing enzyme activity is important for optimizing slow chemical reactions, with far-reaching implications for both research and industrial applications. However, traditional wet-lab methods for measuring and optimizing enzyme activity are often resource-intensive and time-consuming. To address these limitations, we introduce kcatDiffuser, a novel regressor-guided diffusion model designed to predict and improve enzyme turnover numbers. Our approach innovatively reformulates enzyme mutation prediction as a protein inverse folding task, thereby establishing a direct link between structural prediction and functional optimization. kcatDiffuser is a graph diffusion model guided by a regressor, enabling the prediction of amino acid mutations at multiple random positions simultaneously. Evaluations on BERENDA dataset shows that kcatDiffuser can achieve a {\Delta} log kcat of 0.209, outperforming state-of-the-art methods like ProteinMPNN, PiFold, GraDe-IF in improving enzyme turnover numbers. Additionally, kcatDiffuser maintains high structural fidelity with a recovery rate of 0.716, pLDDT score of 92.515, RMSD of 3.764, and TM-score of 0.934, demonstrating its ability to generate enzyme variants with enhanced activity while preserving essential structural properties. Overall, kcatDiffuser represents a more efficient and targeted approach to enhancing enzyme activity. The code is available at https://github.com/xz32yu/KcatDiffuser.

Protein inverse folding aims to identify viable amino acid sequences that can fold into given protein structures, enabling the design of novel proteins with desired functions for applications in drug discovery, enzyme engineering, and biomaterial development. Diffusion probabilistic models have emerged as a promising approach in inverse folding, offering both feasible and diverse solutions compared to traditional energy-based methods and more recent protein language models. However, existing diffusion models for protein inverse folding operate in discrete data spaces, necessitating prior distributions for transition matrices and limiting smooth transitions and gradients inherent to continuous spaces, leading to suboptimal performance. Drawing inspiration from the success of diffusion models in continuous domains, we introduce the Latent Graph Diffusion Model for Protein Inverse Folding (LaGDif). LaGDif bridges discrete and continuous realms through an encoder-decoder architecture, transforming protein graph data distributions into random noise within a continuous latent space. Our model then reconstructs protein sequences by considering spatial configurations, biochemical attributes, and environmental factors of each node. Additionally, we propose a novel inverse folding self-ensemble method that stabilizes prediction results and further enhances performance by aggregating multiple denoised output protein sequence. Empirical results on the CATH dataset demonstrate that LaGDif outperforms existing state-of-the-art techniques, achieving up to 45.55% improvement in sequence recovery rate for single-chain proteins and maintaining an average RMSD of 1.96 {\AA} between generated and native structures. The code is public available at https://github.com/TaoyuW/LaGDif.

Recent breakthroughs in large language models (LLMs) offer unprecedented natural language understanding and generation capabilities. However, existing surveys on LLMs in biomedicine often focus on specific applications or model architectures, lacking a comprehensive analysis that integrates the latest advancements across various biomedical domains. This review, based on an analysis of 484 publications sourced from databases including PubMed, Web of Science, and arXiv, provides an in-depth examination of the current landscape, applications, challenges, and prospects of LLMs in biomedicine, distinguishing itself by focusing on the practical implications of these models in real-world biomedical contexts. Firstly, we explore the capabilities of LLMs in zero-shot learning across a broad spectrum of biomedical tasks, including diagnostic assistance, drug discovery, and personalized medicine, among others, with insights drawn from 137 key studies. Then, we discuss adaptation strategies of LLMs, including fine-tuning methods for both uni-modal and multi-modal LLMs to enhance their performance in specialized biomedical contexts where zero-shot fails to achieve, such as medical question answering and efficient processing of biomedical literature. Finally, we discuss the challenges that LLMs face in the biomedicine domain including data privacy concerns, limited model interpretability, issues with dataset quality, and ethics due to the sensitive nature of biomedical data, the need for highly reliable model outputs, and the ethical implications of deploying AI in healthcare. To address these challenges, we also identify future research directions of LLM in biomedicine including federated learning methods to preserve data privacy and integrating explainable AI methodologies to enhance the transparency of LLMs.

The structural similarities between protein sequences and natural languages have led to parallel advancements in deep learning across both domains. While large language models (LLMs) have achieved much progress in the domain of natural language processing, their potential in protein engineering remains largely unexplored. Previous approaches have equipped LLMs with protein understanding capabilities by incorporating external protein encoders, but this fails to fully leverage the inherent similarities between protein sequences and natural languages, resulting in sub-optimal performance and increased model complexity. To address this gap, we present TourSynbio-7B, the first multi-modal large model specifically designed for protein engineering tasks without external protein encoders. TourSynbio-7B demonstrates that LLMs can inherently learn to understand proteins as language. The model is post-trained and instruction fine-tuned on InternLM2-7B using ProteinLMDataset, a dataset comprising 17.46 billion tokens of text and protein sequence for self-supervised pretraining and 893K instructions for supervised fine-tuning. TourSynbio-7B outperforms GPT-4 on the ProteinLMBench, a benchmark of 944 manually verified multiple-choice questions, with 62.18% accuracy. Leveraging TourSynbio-7B's enhanced protein sequence understanding capability, we introduce TourSynbio-Agent, an innovative framework capable of performing various protein engineering tasks, including mutation analysis, inverse folding, protein folding, and visualization. TourSynbio-Agent integrates previously disconnected deep learning models in the protein engineering domain, offering a unified conversational user interface for improved usability. Finally, we demonstrate the efficacy of TourSynbio-7B and TourSynbio-Agent through two wet lab case studies on vanilla key enzyme modification and steroid compound catalysis.

Graph neural networks (GNNs) have become pivotal tools for processing graph-structured data, leveraging the message passing scheme as their core mechanism. However, traditional GNNs often grapple with issues such as instability, over-smoothing, and over-squashing, which can degrade performance and create a trade-off dilemma. In this paper, we introduce a discriminatively trained, multi-layer Deep Scattering Message Passing (DSMP) neural network designed to overcome these challenges. By harnessing spectral transformation, the DSMP model aggregates neighboring nodes with global information, thereby enhancing the precision and accuracy of graph signal processing. We provide theoretical proofs demonstrating the DSMP's effectiveness in mitigating these issues under specific conditions. Additionally, we support our claims with empirical evidence and thorough frequency analysis, showcasing the DSMP's superior ability to address instability, over-smoothing, and over-squashing., Comment: 35 pages, 6 figures

Antibodies, crucial for immune defense, primarily rely on complementarity-determining regions (CDRs) to bind and neutralize antigens, such as viruses. The design of these CDRs determines the antibody's affinity and specificity towards its target. Generative models, particularly denoising diffusion probabilistic models (DDPMs), have shown potential to advance the structure-based design of CDR regions. However, only a limited dataset of bound antibody-antigen structures is available, and generalization to out-of-distribution interfaces remains a challenge. Physics based force-fields, which approximate atomic interactions, offer a coarse but universal source of information to better mold designs to target interfaces. Integrating this foundational information into diffusion models is, therefore, highly desirable. Here, we propose a novel approach to enhance the sampling process of diffusion models by integrating force field energy-based feedback. Our model, DiffForce, employs forces to guide the diffusion sampling process, effectively blending the two distributions. Through extensive experiments, we demonstrate that our method guides the model to sample CDRs with lower energy, enhancing both the structure and sequence of the generated antibodies.

The parallels between protein sequences and natural language in their sequential structures have inspired the application of large language models (LLMs) to protein understanding. Despite the success of LLMs in NLP, their effectiveness in comprehending protein sequences remains an open question, largely due to the absence of datasets linking protein sequences to descriptive text. Researchers have then attempted to adapt LLMs for protein understanding by integrating a protein sequence encoder with a pre-trained LLM. However, this adaptation raises a fundamental question: "Can LLMs, originally designed for NLP, effectively comprehend protein sequences as a form of language?" Current datasets fall short in addressing this question due to the lack of a direct correlation between protein sequences and corresponding text descriptions, limiting the ability to train and evaluate LLMs for protein understanding effectively. To bridge this gap, we introduce ProteinLMDataset, a dataset specifically designed for further self-supervised pretraining and supervised fine-tuning (SFT) of LLMs to enhance their capability for protein sequence comprehension. Specifically, ProteinLMDataset includes 17.46 billion tokens for pretraining and 893,000 instructions for SFT. Additionally, we present ProteinLMBench, the first benchmark dataset consisting of 944 manually verified multiple-choice questions for assessing the protein understanding capabilities of LLMs. ProteinLMBench incorporates protein-related details and sequences in multiple languages, establishing a new standard for evaluating LLMs' abilities in protein comprehension. The large language model InternLM2-7B, pretrained and fine-tuned on the ProteinLMDataset, outperforms GPT-4 on ProteinLMBench, achieving the highest accuracy score.

Spectral Graph Neural Networks (GNNs), alternatively known as graph filters, have gained increasing prevalence for heterophily graphs. Optimal graph filters rely on Laplacian eigendecomposition for Fourier transform. In an attempt to avert prohibitive computations, numerous polynomial filters have been proposed. However, polynomials in the majority of these filters are predefined and remain fixed across different graphs, failing to accommodate the varying degrees of heterophily. Addressing this gap, we demystify the intrinsic correlation between the spectral property of desired polynomial bases and the heterophily degrees via thorough theoretical analyses. Subsequently, we develop a novel adaptive heterophily basis wherein the basis vectors mutually form angles reflecting the heterophily degree of the graph. We integrate this heterophily basis with the homophily basis to construct a universal polynomial basis UniBasis, which devises a polynomial filter based graph neural network - UniFilter. It optimizes the convolution and propagation in GNN, thus effectively limiting over-smoothing and alleviating over-squashing. Our extensive experiments, conducted on a diverse range of real-world and synthetic datasets with varying degrees of heterophily, support the superiority of UniFilter. These results not only demonstrate the universality of UniBasis but also highlight its proficiency in graph explanation., Comment: arXiv admin note: substantial text overlap with arXiv:2311.18177

Large language models (LLMs) have garnered considerable attention for their proficiency in tackling intricate tasks, particularly leveraging their capacities for zero-shot and in-context learning. However, their utility has been predominantly restricted to general tasks due to an absence of domain-specific knowledge. This constraint becomes particularly pertinent in the realm of protein engineering, where specialized expertise is required for tasks such as protein function prediction, protein evolution analysis, and protein design, with a level of specialization that existing LLMs cannot furnish. In response to this challenge, we introduce \textsc{ProteinEngine}, a human-centered platform aimed at amplifying the capabilities of LLMs in protein engineering by seamlessly integrating a comprehensive range of relevant tools, packages, and software via API calls. Uniquely, \textsc{ProteinEngine} assigns three distinct roles to LLMs, facilitating efficient task delegation, specialized task resolution, and effective communication of results. This design fosters high extensibility and promotes the smooth incorporation of new algorithms, models, and features for future development. Extensive user studies, involving participants from both the AI and protein engineering communities across academia and industry, consistently validate the superiority of \textsc{ProteinEngine} in augmenting the reliability and precision of deep learning in protein engineering tasks. Consequently, our findings highlight the potential of \textsc{ProteinEngine} to bride the disconnected tools for future research in the protein engineering domain.

Graph neural networks (GNNs) are a powerful solution for various structure learning applications due to their strong representation capabilities for graph data. However, traditional GNNs, relying on message-passing mechanisms that gather information exclusively from first-order neighbours (known as positive samples), can lead to issues such as over-smoothing and over-squashing. To mitigate these issues, we propose a layer-diverse negative sampling method for message-passing propagation. This method employs a sampling matrix within a determinantal point process, which transforms the candidate set into a space and selectively samples from this space to generate negative samples. To further enhance the diversity of the negative samples during each forward pass, we develop a space-squeezing method to achieve layer-wise diversity in multi-layer GNNs. Experiments on various real-world graph datasets demonstrate the effectiveness of our approach in improving the diversity of negative samples and overall learning performance. Moreover, adding negative samples dynamically changes the graph's topology, thus with the strong potential to improve the expressiveness of GNNs and reduce the risk of over-squashing., Comment: Published in Transactions on Machine Learning Research (03/2024)

Graph convolutions have been a pivotal element in learning graph representations. However, recursively aggregating neighboring information with graph convolutions leads to indistinguishable node features in deep layers, which is known as the over-smoothing issue. The performance of graph neural networks decays fast as the number of stacked layers increases, and the Dirichlet energy associated with the graph decreases to zero as well. In this work, we introduce a framelet system into the analysis of Dirichlet energy and take a multi-scale perspective to leverage the Dirichlet energy and alleviate the over-smoothing issue. Specifically, we develop a Framelet Augmentation strategy by adjusting the update rules with positive and negative increments for low-pass and high-passes respectively. Based on that, we design the Energy Enhanced Convolution (EEConv), which is an effective and practical operation that is proved to strictly enhance Dirichlet energy. From a message-passing perspective, EEConv inherits multi-hop aggregation property from the framelet transform and takes into account all hops in the multi-scale representation, which benefits the node classification tasks over heterophilous graphs. Experiments show that deep GNNs with EEConv achieve state-of-the-art performance over various node classification datasets, especially for heterophilous graphs, while also lifting the Dirichlet energy as the network goes deeper.

Inverse protein folding is challenging due to its inherent one-to-many mapping characteristic, where numerous possible amino acid sequences can fold into a single, identical protein backbone. This task involves not only identifying viable sequences but also representing the sheer diversity of potential solutions. However, existing discriminative models, such as transformer-based auto-regressive models, struggle to encapsulate the diverse range of plausible solutions. In contrast, diffusion probabilistic models, as an emerging genre of generative approaches, offer the potential to generate a diverse set of sequence candidates for determined protein backbones. We propose a novel graph denoising diffusion model for inverse protein folding, where a given protein backbone guides the diffusion process on the corresponding amino acid residue types. The model infers the joint distribution of amino acids conditioned on the nodes' physiochemical properties and local environment. Moreover, we utilize amino acid replacement matrices for the diffusion forward process, encoding the biologically-meaningful prior knowledge of amino acids from their spatial and sequential neighbors as well as themselves, which reduces the sampling space of the generative process. Our model achieves state-of-the-art performance over a set of popular baseline methods in sequence recovery and exhibits great potential in generating diverse protein sequences for a determined protein backbone structure., Comment: NeurIPS 2023

Directed evolution plays an indispensable role in protein engineering that revises existing protein sequences to attain new or enhanced functions. Accurately predicting the effects of protein variants necessitates an in-depth understanding of protein structure and function. Although large self-supervised language models have demonstrated remarkable performance in zero-shot inference using only protein sequences, these models inherently do not interpret the spatial characteristics of protein structures, which are crucial for comprehending protein folding stability and internal molecular interactions. This paper introduces a novel pre-training framework that cascades sequential and geometric analyzers for protein primary and tertiary structures. It guides mutational directions toward desired traits by simulating natural selection on wild-type proteins and evaluates the effects of variants based on their fitness to perform the function. We assess the proposed approach using a public database and two new databases for a variety of variant effect prediction tasks, which encompass a diverse set of proteins and assays from different taxa. The prediction results achieve state-of-the-art performance over other zero-shot learning methods for both single-site mutations and deep mutations.

Directed evolution as a widely-used engineering strategy faces obstacles in finding desired mutants from the massive size of candidate modifications. While deep learning methods learn protein contexts to establish feasible searching space, many existing models are computationally demanding and fail to predict how specific mutational tests will affect a protein's sequence or function. This research introduces a lightweight graph representation learning scheme that efficiently analyzes the microenvironment of wild-type proteins and recommends practical higher-order mutations exclusive to the user-specified protein and function of interest. Our method enables continuous improvement of the inference model by limited computational resources and a few hundred mutational training samples, resulting in accurate prediction of variant effects that exhibit near-perfect correlation with the ground truth across deep mutational scanning assays of 19 proteins. With its affordability and applicability to both computer scientists and biochemical laboratories, our solution offers a wide range of benefits that make it an ideal choice for the community.

Advances in deep learning models have revolutionized the study of biomolecule systems and their mechanisms. Graph representation learning, in particular, is important for accurately capturing the geometric information of biomolecules at different levels. This paper presents a comprehensive review of the methodologies used to represent biological molecules and systems as computer-recognizable objects, such as sequences, graphs, and surfaces. Moreover, it examines how geometric deep learning models, with an emphasis on graph-based techniques, can analyze biomolecule data to enable drug discovery, protein characterization, and biological system analysis. The study concludes with an overview of the current state of the field, highlighting the challenges that exist and the potential future research directions.

Learning to predict agent motions with relationship reasoning is important for many applications. In motion prediction tasks, maintaining motion equivariance under Euclidean geometric transformations and invariance of agent interaction is a critical and fundamental principle. However, such equivariance and invariance properties are overlooked by most existing methods. To fill this gap, we propose EqMotion, an efficient equivariant motion prediction model with invariant interaction reasoning. To achieve motion equivariance, we propose an equivariant geometric feature learning module to learn a Euclidean transformable feature through dedicated designs of equivariant operations. To reason agent's interactions, we propose an invariant interaction reasoning module to achieve a more stable interaction modeling. To further promote more comprehensive motion features, we propose an invariant pattern feature learning module to learn an invariant pattern feature, which cooperates with the equivariant geometric feature to enhance network expressiveness. We conduct experiments for the proposed model on four distinct scenarios: particle dynamics, molecule dynamics, human skeleton motion prediction and pedestrian trajectory prediction. Experimental results show that our method is not only generally applicable, but also achieves state-of-the-art prediction performances on all the four tasks, improving by 24.0/30.1/8.6/9.2%. Code is available at https://github.com/MediaBrain-SJTU/EqMotion., Comment: Accepted to CVPR 2023

In this paper, we propose an efficient simulation method based on adaptive importance sampling, which can automatically find the optimal proposal within the Gaussian family based on previous samples, to evaluate the probability of bit error rate (BER) or word error rate (WER). These two measures, which involve high-dimensional black-box integration and rare-event sampling, can characterize the performance of coded modulation. We further integrate the quasi-Monte Carlo method within our framework to improve the convergence speed. The proposed importance sampling algorithm is demonstrated to have much higher efficiency than the standard Monte Carlo method in the AWGN scenario., Comment: importance sampling for system model

Graph neural networks (GNNs) have achieved champion in wide applications. Neural message passing is a typical key module for feature propagation by aggregating neighboring features. In this work, we propose a new message passing based on multiscale framelet transforms, called Framelet Message Passing. Different from traditional spatial methods, it integrates framelet representation of neighbor nodes from multiple hops away in node message update. We also propose a continuous message passing using neural ODE solvers. It turns both discrete and continuous cases can provably achieve network stability and limit oversmoothing due to the multiscale property of framelets. Numerical experiments on real graph datasets show that the continuous version of the framelet message passing significantly outperforms existing methods when learning heterogeneous graphs and achieves state-of-the-art performance on classic node classification tasks with low computational costs.

Deep learning has been widely used for protein engineering. However, it is limited by the lack of sufficient experimental data to train an accurate model for predicting the functional fitness of high-order mutants. Here, we develop SESNet, a supervised deep-learning model to predict the fitness for protein mutants by leveraging both sequence and structure information, and exploiting attention mechanism. Our model integrates local evolutionary context from homologous sequences, the global evolutionary context encoding rich semantic from the universal protein sequence space and the structure information accounting for the microenvironment around each residue in a protein. We show that SESNet outperforms state-of-the-art models for predicting the sequence-function relationship on 26 deep mutational scanning datasets. More importantly, we propose a data augmentation strategy by leveraging the data from unsupervised models to pre-train our model. After that, our model can achieve strikingly high accuracy in prediction of the fitness of protein mutants, especially for the higher order variants (> 4 mutation sites), when finetuned by using only a small number of experimental mutation data (<50). The strategy proposed is of great practical value as the required experimental effort, i.e., producing a few tens of experimental mutation data on a given protein, is generally affordable by an ordinary biochemical group and can be applied on almost any protein.

The quality of signal propagation in message-passing graph neural networks (GNNs) strongly influences their expressivity as has been observed in recent works. In particular, for prediction tasks relying on long-range interactions, recursive aggregation of node features can lead to an undesired phenomenon called "oversquashing". We present a framework for analyzing oversquashing based on information contraction. Our analysis is guided by a model of reliable computation due to von Neumann that lends a new insight into oversquashing as signal quenching in noisy computation graphs. Building on this, we propose a graph rewiring algorithm aimed at alleviating oversquashing. Our algorithm employs a random local edge flip primitive motivated by an expander graph construction. We compare the spectral expansion properties of our algorithm with that of an existing curvature-based non-local rewiring strategy. Synthetic experiments show that while our algorithm in general has a slower rate of expansion, it is overall computationally cheaper, preserves the node degrees exactly and never disconnects the graph., Comment: 8 pages, 5 figures; Accepted at the 58th Annual Allerton Conference on Communication, Control, and Computing

This paper develops a rotation-invariant needlet convolution for rotation group SO(3) to distill multiscale information of spherical signals. The spherical needlet transform is generalized from $\mathbb{S}^2$ onto the SO(3) group, which decomposes a spherical signal to approximate and detailed spectral coefficients by a set of tight framelet operators. The spherical signal during the decomposition and reconstruction achieves rotation invariance. Based on needlet transforms, we form a Needlet approximate Equivariance Spherical CNN (NES) with multiple SO(3) needlet convolutional layers. The network establishes a powerful tool to extract geometric-invariant features of spherical signals. The model allows sufficient network scalability with multi-resolution representation. A robust signal embedding is learned with wavelet shrinkage activation function, which filters out redundant high-pass representation while maintaining approximate rotation invariance. The NES achieves state-of-the-art performance for quantum chemistry regression and Cosmic Microwave Background (CMB) delensing reconstruction, which shows great potential for solving scientific challenges with high-resolution and multi-scale spherical signal representation.

Gigapixel medical images provide massive data, both morphological textures and spatial information, to be mined. Due to the large data scale in histology, deep learning methods play an increasingly significant role as feature extractors. Existing solutions heavily rely on convolutional neural networks (CNNs) for global pixel-level analysis, leaving the underlying local geometric structure such as the interaction between cells in the tumor microenvironment unexplored. The topological structure in medical images, as proven to be closely related to tumor evolution, can be well characterized by graphs. To obtain a more comprehensive representation for downstream oncology tasks, we propose a fusion framework for enhancing the global image-level representation captured by CNNs with the geometry of cell-level spatial information learned by graph neural networks (GNN). The fusion layer optimizes an integration between collaborative features of global images and cell graphs. Two fusion strategies have been developed: one with MLP which is simple but turns out efficient through fine-tuning, and the other with Transformer gains a champion in fusing multiple networks. We evaluate our fusion strategies on histology datasets curated from large patient cohorts of colorectal and gastric cancers for three biomarker prediction tasks. Both two models outperform plain CNNs or GNNs, reaching a consistent AUC improvement of more than 5% on various network backbones. The experimental results yield the necessity for combining image-level morphological features with cell spatial relations in medical image analysis. Codes are available at https://github.com/yiqings/HEGnnEnhanceCnn., Comment: 21 pages

Neural message passing is a basic feature extraction unit for graph-structured data considering neighboring node features in network propagation from one layer to the next. We model such process by an interacting particle system with attractive and repulsive forces and the Allen-Cahn force arising in the modeling of phase transition. The dynamics of the system is a reaction-diffusion process which can separate particles without blowing up. This induces an Allen-Cahn message passing (ACMP) for graph neural networks where the numerical iteration for the particle system solution constitutes the message passing propagation. ACMP which has a simple implementation with a neural ODE solver can propel the network depth up to one hundred of layers with theoretically proven strictly positive lower bound of the Dirichlet energy. It thus provides a deep model of GNNs circumventing the common GNN problem of oversmoothing. GNNs with ACMP achieve state of the art performance for real-world node classification tasks on both homophilic and heterophilic datasets., Comment: 26 pages, 5 figures. NeurIPS 2022 Workshop on GLFrontiers (Oral). ICLR 2023 (Spotlight)

The research for characterizing GNN expressiveness attracts much attention as graph neural networks achieve a champion in the last five years. The number of linear regions has been considered a good measure for the expressivity of neural networks with piecewise linear activation. In this paper, we present some estimates for the number of linear regions of the classic graph convolutional networks (GCNs) with one layer and multiple-layer scenarios. In particular, we obtain an optimal upper bound for the maximum number of linear regions for one-layer GCNs, and the upper and lower bounds for multi-layer GCNs. The simulated estimate shows that the true maximum number of linear regions is possibly closer to our estimated lower bound. These results imply that the number of linear regions of multi-layer GCNs is exponentially greater than one-layer GCNs per parameter in general. This suggests that deeper GCNs have more expressivity than shallow GCNs.

Learning efficient graph representation is the key to favorably addressing downstream tasks on graphs, such as node or graph property prediction. Given the non-Euclidean structural property of graphs, preserving the original graph data's similarity relationship in the embedded space needs specific tools and a similarity metric. This paper develops a new graph representation learning scheme, namely EGG, which embeds approximated second-order graph characteristics into a Grassmann manifold. The proposed strategy leverages graph convolutions to learn hidden representations of the corresponding subspace of the graph, which is then mapped to a Grassmann point of a low dimensional manifold through truncated singular value decomposition (SVD). The established graph embedding approximates denoised correlationship of node attributes, as implemented in the form of a symmetric matrix space for Euclidean calculation. The effectiveness of EGG is demonstrated using both clustering and classification tasks at the node level and graph level. It outperforms baseline models on various benchmarks.

The performance of graph representation learning is affected by the quality of graph input. While existing research usually pursues a globally smoothed graph embedding, we believe the rarely observed anomalies are as well harmful to an accurate prediction. This work establishes a graph learning scheme that automatically detects (locally) corrupted feature attributes and recovers robust embedding for prediction tasks. The detection operation leverages a graph autoencoder, which does not make any assumptions about the distribution of the local corruptions. It pinpoints the positions of the anomalous node attributes in an unbiased mask matrix, where robust estimations are recovered with sparsity promoting regularizer. The optimizer approaches a new embedding that is sparse in the framelet domain and conditionally close to input observations. Extensive experiments are provided to validate our proposed model can recover a robust graph representation from black-box poisoning and achieve excellent performance., Comment: WWW '23: Proceedings of the ACM Web Conference 2023

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 multiple layers without the concern of over-smoothing, and it guarantees convergence to the optimal solutions. Empirical study proves that our model achieves significantly better performance compared with popular graph convolutions even when the graph is heavily contaminated.

This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the L\'{e}vy-type behaviour of the spatial solution, a fractional derivative in time to depict the intermittency of its temporal solution, and is driven by vector-valued fractional Brownian motion on the unit sphere to characterize its temporal long-range dependence. The solution to the SPDE is presented in the form of the Karhunen-Lo\`{e}ve expansion in terms of vector spherical harmonics. Its covariance matrix function is established as a tensor field on the unit sphere that is an expansion of Legendre tensor kernels. Approximations to the solutions are studied and convergence rates of the approximation errors are given. It is demonstrated how these convergence rates depend on the decay of the power spectrum and variances of the fractional Brownian motion., Comment: 21 pages

Graph Neural Networks (GNNs) are limited in their expressive power, struggle with long-range interactions and lack a principled way to model higher-order structures. These problems can be attributed to the strong coupling between the computational graph and the input graph structure. The recently proposed Message Passing Simplicial Networks naturally decouple these elements by performing message passing on the clique complex of the graph. Nevertheless, these models can be severely constrained by the rigid combinatorial structure of Simplicial Complexes (SCs). In this work, we extend recent theoretical results on SCs to regular Cell Complexes, topological objects that flexibly subsume SCs and graphs. We show that this generalisation provides a powerful set of graph "lifting" transformations, each leading to a unique hierarchical message passing procedure. The resulting methods, which we collectively call CW Networks (CWNs), are strictly more powerful than the WL test and not less powerful than the 3-WL test. In particular, we demonstrate the effectiveness of one such scheme, based on rings, when applied to molecular graph problems. The proposed architecture benefits from provably larger expressivity than commonly used GNNs, principled modelling of higher-order signals and from compressing the distances between nodes. We demonstrate that our model achieves state-of-the-art results on a variety of molecular datasets., Comment: NeurIPS 2021. Contains 28 pages, 9 figures

Detecting anomalies for dynamic graphs has drawn increasing attention due to their wide applications in social networks, e-commerce, and cybersecurity. Recent deep learning-based approaches have shown promising results over shallow methods. However, they fail to address two core challenges of anomaly detection in dynamic graphs: the lack of informative encoding for unattributed nodes and the difficulty of learning discriminate knowledge from coupled spatial-temporal dynamic graphs. To overcome these challenges, in this paper, we present a novel Transformer-based Anomaly Detection framework for DYnamic graphs (TADDY). Our framework constructs a comprehensive node encoding strategy to better represent each node's structural and temporal roles in an evolving graphs stream. Meanwhile, TADDY captures informative representation from dynamic graphs with coupled spatial-temporal patterns via a dynamic graph transformer model. The extensive experimental results demonstrate that our proposed TADDY framework outperforms the state-of-the-art methods by a large margin on six real-world datasets., Comment: 13 pages, 5 figures

The pairwise interaction paradigm of graph machine learning has predominantly governed the modelling of relational systems. However, graphs alone cannot capture the multi-level interactions present in many complex systems and the expressive power of such schemes was proven to be limited. To overcome these limitations, we propose Message Passing Simplicial Networks (MPSNs), a class of models that perform message passing on simplicial complexes (SCs). To theoretically analyse the expressivity of our model we introduce a Simplicial Weisfeiler-Lehman (SWL) colouring procedure for distinguishing non-isomorphic SCs. We relate the power of SWL to the problem of distinguishing non-isomorphic graphs and show that SWL and MPSNs are strictly more powerful than the WL test and not less powerful than the 3-WL test. We deepen the analysis by comparing our model with traditional graph neural networks (GNNs) with ReLU activations in terms of the number of linear regions of the functions they can represent. We empirically support our theoretical claims by showing that MPSNs can distinguish challenging strongly regular graphs for which GNNs fail and, when equipped with orientation equivariant layers, they can improve classification accuracy in oriented SCs compared to a GNN baseline., Comment: ICML 2021. Contains 27 pages, 9 figures

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 networks with the proposed framelet convolution and pooling achieve state-of-the-art performance in many node and graph prediction tasks. Moreover, we propose shrinkage as a new activation for the framelet convolution, which thresholds high-frequency information at different scales. Compared to ReLU, shrinkage activation improves model performance on denoising and signal compression: noises in both node and structure can be significantly reduced by accurately cutting off the high-pass coefficients from framelet decomposition, and the signal can be compressed to less than half its original size with well-preserved prediction performance., Comment: 24 pages, 17 figures, 8 tables, ICML2021 (fix typos)

Graph representation learning has many real-world applications, from super-resolution imaging, 3D computer vision to drug repurposing, protein classification, social networks analysis. An adequate representation of graph data is vital to the learning performance of a statistical or machine learning model for graph-structured data. In this paper, we propose a novel multiscale representation system for graph data, called decimated framelets, which form a localized tight frame on the graph. The decimated framelet system allows storage of the graph data representation on a coarse-grained chain and processes the graph data at multi scales where at each scale, the data is stored at a subgraph. Based on this, we then establish decimated G-framelet transforms for the decomposition and reconstruction of the graph data at multi resolutions via a constructive data-driven filter bank. The graph framelets are built on a chain-based orthonormal basis that supports fast graph Fourier transforms. From this, we give a fast algorithm for the decimated G-framelet transforms, or FGT, that has linear computational complexity O(N) for a graph of size N. The theory of decimated framelets and FGT is verified with numerical examples for random graphs. The effectiveness is demonstrated by real-world applications, including multiresolution analysis for traffic network, and graph neural networks for graph classification tasks., Comment: 69 pages, 10 figures, Published in JMLR

We investigate the expressive power of depth-2 bandlimited random neural networks. A random net is a neural network where the hidden layer parameters are frozen with random assignment, and only the output layer parameters are trained by loss minimization. Using random weights for a hidden layer is an effective method to avoid non-convex optimization in standard gradient descent learning. It has also been adopted in recent deep learning theories. Despite the well-known fact that a neural network is a universal approximator, in this study, we mathematically show that when hidden parameters are distributed in a bounded domain, the network may not achieve zero approximation error. In particular, we derive a new nontrivial approximation error lower bound. The proof utilizes the technique of ridgelet analysis, a harmonic analysis method designed for neural networks. This method is inspired by fundamental principles in classical signal processing, specifically the idea that signals with limited bandwidth may not always be able to perfectly recreate the original signal. We corroborate our theoretical results with various simulation studies, and generally, two main take-home messages are offered: (i) Not any distribution for selecting random weights is feasible to build a universal approximator; (ii) A suitable assignment of random weights exists but to some degree is associated with the complexity of the target function., Comment: Published as a conference paper at ICML 2023

Graph Neural Networks (GNNs) have recently caught great attention and achieved significant progress in graph-level applications. In this paper, we propose a framework for graph neural networks with multiresolution Haar-like wavelets, or MathNet, with interrelated convolution and pooling strategies. The underlying method takes graphs in different structures as input and assembles consistent graph representations for readout layers, which then accomplishes label prediction. To achieve this, the multiresolution graph representations are first constructed and fed into graph convolutional layers for processing. The hierarchical graph pooling layers are then involved to downsample graph resolution while simultaneously remove redundancy within graph signals. The whole workflow could be formed with a multi-level graph analysis, which not only helps embed the intrinsic topological information of each graph into the GNN, but also supports fast computation of forward and adjoint graph transforms. We show by extensive experiments that the proposed framework obtains notable accuracy gains on graph classification and regression tasks with performance stability. The proposed MathNet outperforms various existing GNN models, especially on big data sets., Comment: 32 pages, 6 figures, 6 tables

Learning mappings of data on manifolds is an important topic in contemporary machine learning, with applications in astrophysics, geophysics, statistical physics, medical diagnosis, biochemistry, 3D object analysis. This paper studies the problem of learning real-valued functions on manifolds through filtered hyperinterpolation of input-output data pairs where the inputs may be sampled deterministically or at random and the outputs may be clean or noisy. Motivated by the problem of handling large data sets, it presents a parallel data processing approach which distributes the data-fitting task among multiple servers and synthesizes the fitted sub-models into a global estimator. We prove quantitative relations between the approximation quality of the learned function over the entire manifold, the type of target function, the number of servers, and the number and type of available samples. We obtain the approximation rates of convergence for distributed and non-distributed approaches. For the non-distributed case, the approximation order is optimal., Comment: 50 pages, 4 figures, 2 tables

Graph neural networks (GNNs) extends the functionality of traditional neural networks to graph-structured data. Similar to CNNs, an optimized design of graph convolution and pooling is key to success. Borrowing ideas from physics, we propose a path integral based graph neural networks (PAN) for classification and regression tasks on graphs. Specifically, we consider a convolution operation that involves every path linking the message sender and receiver with learnable weights depending on the path length, which corresponds to the maximal entropy random walk. It generalizes the graph Laplacian to a new transition matrix we call maximal entropy transition (MET) matrix derived from a path integral formalism. Importantly, the diagonal entries of the MET matrix are directly related to the subgraph centrality, thus providing a natural and adaptive pooling mechanism. PAN provides a versatile framework that can be tailored for different graph data with varying sizes and structures. We can view most existing GNN architectures as special cases of PAN. Experimental results show that PAN achieves state-of-the-art performance on various graph classification/regression tasks, including a new benchmark dataset from statistical mechanics we propose to boost applications of GNN in physical sciences., Comment: 15 pages, 4 figures, 6 tables. arXiv admin note: text overlap with arXiv:1904.10996

The transparent cornea is the window of the eye, facilitating the entry of light rays and controlling focusing the movement of the light within the eye. The cornea is critical, contributing to 75% of the refractive power of the eye. Keratoconus is a progressive and multifactorial corneal degenerative disease affecting 1 in 2000 individuals worldwide. Currently, there is no cure for keratoconus other than corneal transplantation for advanced stage keratoconus or corneal cross-linking, which can only halt KC progression. The ability to accurately identify subtle KC or KC progression is of vital clinical significance. To date, there has been little consensus on a useful model to classify KC patients, which therefore inhibits the ability to predict disease progression accurately. In this paper, we utilised machine learning to analyse data from 124 KC patients, including topographical and clinical variables. Both supervised multilayer perceptron and unsupervised variational autoencoder models were used to classify KC patients with reference to the existing Amsler-Krumeich (A-K) classification system. Both methods result in high accuracy, with the unsupervised method showing better performance. The result showed that the unsupervised method with a selection of 29 variables could be a powerful tool to provide an automatic classification tool for clinicians. These outcomes provide a platform for additional analysis for the progression and treatment of keratoconus., Comment: 7 pages, 8 figures, 3 tables

Cosmic microwave background radiation (CMB) is critical to the understanding of the early universe and precise estimation of cosmological constants. Due to the contamination of thermal dust noise in the galaxy, the CMB map that is an image on the two-dimensional sphere has missing observations, mainly concentrated on the equatorial region. The noise of the CMB map has a significant impact on the estimation precision for cosmological parameters. Inpainting the CMB map can effectively reduce the uncertainty of parametric estimation. In this paper, we propose a deep learning-based variational autoencoder --- CosmoVAE, to restoring the missing observations of the CMB map. The input and output of CosmoVAE are square images. To generate training, validation, and test data sets, we segment the full-sky CMB map into many small images by Cartesian projection. CosmoVAE assigns physical quantities to the parameters of the VAE network by using the angular power spectrum of the Gaussian random field as latent variables. CosmoVAE adopts a new loss function to improve the learning performance of the model, which consists of $\ell_1$ reconstruction loss, Kullback-Leibler divergence between the posterior distribution of encoder network and the prior distribution of latent variables, perceptual loss, and total-variation regularizer. The proposed model achieves state of the art performance for Planck \texttt{Commander} 2018 CMB map inpainting., Comment: 7 pages, 6 figures