arXiv subjects

Kelin Xia

Publications and source records attributed to Kelin Xia.

At least 19 recordsLinked to original sources

arXiv2026

Recovering topological information of light by topological learning

The evolution of modern-day communication networks towards optical solutions with enhanced capacity and robustness is driving interest in topological light waves, exploiting their stability against perturbations through a topological invariant, e.g., the skyrmion number. However, detecting the underlying topology remains a computationally intense process even under ideal conditions, becoming intractable after passing through strongly disordered channels, where the degradation into unrecognisable speckle appears to destroy the topology. Here, we propose and demonstrate a topology-enhanced artificial intelligence (AI) approach to recover and classify such apparently lost topological information by computationally leveraging topological invariants in the data across many length scales. By aligning the topological classification of information with the topology of light, our topology-enhanced learning protocol, termed TOPO$^{2}$, achieves highly efficient recognition of the topological states of light, even from speckle, without the need for any prior learning. Our approach outperforms benchmark tests against standard computational algorithms and has the benefit of requiring just a single intensity pattern as the input, facilitating single-shot operation. To demonstrate this, we leverage the skyrmion number as a robust data carrier of images through a disordered channel, using TOPO$^{2}$ to accurately reconstruct the transmitted images. This work synergises topological photonics and topological AI for unravelling hidden topological signatures in light, opening a pathway towards robust communications even in extreme disordered environments.

physics.optics↗

arXiv2026

Demystifying Oversmoothing in Sheaf Neural Networks: An Index-Theoretic Criterion

To combat oversmoothing in Graph Convolutional Networks, Sheaf Neural Networks (SNNs) were proposed as a generalization by equipping the graph with a sheaf structure and replacing the graph Laplacian with a sheaf Laplacian $\mathcal{L}$. Existing analyses connect sheaf diffusion to oversmoothing via the harmonic space ($\ker\mathcal{L}$), taking its absolute dimension as an indicator of anti-oversmoothing capacity. However, absolute dimension alone is not a reliable measure: certain sheaf configurations inflate $\dim \ker \mathcal{L}$ while their harmonic sections remain entirely constant, without enriching discriminative capacity. We instead introduce the first relative, geometric approach, yielding a precise characterisation of anti-oversmoothing capacity. Under natural conditions on stalk transportation and global sheaf structure, we establish an index-theoretic comparison criterion showing that one sheaf's harmonic space genuinely contains another's beyond trivial inflation. We illustrate this with a concrete instance and further introduce \textit{GyroSheaf}, a sheaf with curved gyrovector-space stalks, extending the criterion to the non-linear setting via local tangent-space linearization. Experiments across ten models confirm the theoretical criterion: sheaf models violating the criterion collapse despite possessing index jumps, while compliant models maintain depth-stable representations.

cs.LG↗

arXiv2026

Periodic Topological Deep Learning for Polymer Design and Discovery

Polymers underpin applications across energy, healthcare, and materials science, yet their vast chemical space makes systematic discovery challenging. Most machine learning approaches represent polymers as molecular graphs of a single repeating unit, thereby missing both the periodicity of polymer chains and many-body interactions beyond pairwise bonds. We introduce Periodic-TDL, a deep learning framework built on periodic Vietoris-Rips complexes that capture many-body interactions across multiple spatial scales, followed by a hierarchical simplicial message-passing (HSMP) encoder that propagates information from long-range interactions to covalent bonds, yielding representations enriched by higher-order topological features. Periodic-TDL outperforms all state-of-the-art models across polymer property prediction tasks spanning electronic, optical, physical, and thermal targets. Furthermore, we quantitatively validate how ester-to-amide substitution and $α$-methylation enhance thermal stability. Using a computationally synthesized dataset of 48,208 structures-generated via systematic substitution of acrylate and acrylamide polymers-we observed a mean $T_g$ increase of $\sim 55^\circ$C for ester-to-amide substitutions and $\sim 14^\circ$C for backbone $α$-methylation across matched polymer pairs. To verify these predicted trends, we use our Periodic-TDL model to analyze six novel polymer pairs from independent experimental measurements, including three newly synthesized polymers previously unreported in the literature. The experimental data successfully confirmed the model's predictions. Ultimately, these findings demonstrate that Periodic-TDL captures the underlying physical effects of specific functional group modifications, rather than merely optimizing predictive performance on benchmark datasets.

cs.LG↗

arXiv2026

Communicability-Inspired Positional Encoding (CIPE)

Positional encodings (PEs) are essential for Transformers. Yet designing effective PEs for non-Euclidean graphs remains challenging. Such encodings should ideally induce an Attention-Compatible Geometry for self-attention: not merely describing graph structure, but defining a geometry whose inner products reflect meaningful structural relatedness. To realize this geometry, we propose Communicability-Inspired Positional Encoding (CIPE), built from communicability, a measure between pairs of nodes that aggregates contributions from paths of all lengths. By construction, CIPE inner products recover communicability, converting global multi-path connectivity into an attention-ready similarity geometry. For practical Transformer training, we introduce dimensionality alignment, mapping graph-size-dependent CIPE representations to prescribed dimensions while faithfully preserving the induced geometry. Empirically, CIPE improves structure-agnostic Transformers by 35.5% on average across seven benchmarks, outperforming representative PEs; it also consistently improves structure-biased graph Transformers, where competing PEs often yield only marginal benefits. These results position CIPE as a principled framework for attention-compatible graph positional encodings.

cs.LG↗

arXiv2026

Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning

Graph neural networks face two fundamental challenges rooted in the linear structure of Euclidean vector spaces: (1) Current architectures represent geometry through vectors (directions, gradients), yet many tasks require matrix-valued representations that capture relationships between directions-such as how atomic orientations covary in a molecule. These second-order representations are naturally captured by points on the symmetric positive definite matrices (SPD) manifold; (2) Standard message passing applies shared transformations across edges. Sheaf neural networks address this via edge-specific transformations, but existing formulations remain confined to vector spaces and therefore cannot propagate matrix-valued features. We address both challenges by developing the first sheaf neural network operates natively on the SPD manifold. Our key insight is that the SPD manifold admits a Lie group structure, enabling well-posed analogs of sheaf operators without projecting to Euclidean space. Theoretically, we prove that SPD-valued sheaves are strictly more expressive than Euclidean sheaves: they admit consistent configurations (global sections) that vector-valued sheaves cannot represent, directly translating to richer learned representations. Empirically, our sheaf convolution transforms effectively rank-1 directional inputs into full-rank matrices encoding local geometric structure. Our dual-stream architecture achieves SOTA on 6/7 MoleculeNet benchmarks, with the sheaf framework providing consistent depth robustness.

cs.LG↗

arXiv2026

Full-Spectrum Graph Neural Networks: Expressive and Scalable

It is well established that spectral graph neural networks (GNNs) can universally approximate node signals; however, their expressive power remains bounded by the 1-dimensional Weisfeiler-Lehman test, which is mirrored in their lack of universality for higher-order signals. To go beyond this bound, we propose the Full-Spectrum GNNs (FSpecGNNs), a second-order generalization of classical spectral GNNs. FSpecGNN advances spectral filtering from two perspectives: (1) it lifts signals from the node domain to the node-pair domain; and (2) it extends the univariate spectral filter over eigenvalues to a bivariate filter over eigenvalue pairs. We show that classical spectral GNNs arise as a diagonal special case of FSpecGNNs, and prove that FSpecGNNs can be at most as expressive as Local 2-GNN while universally approximating node-pair signals, the latter being particularly beneficial for heterophilic graph learning. Moreover, FSpecGNN admits scalable implementations that avoid explicit node-pair-level computations; combined with a low-rank approximation that reduces full-spectrum convolution to a combination of polynomial spectral filters, it enables learning on large graphs. Empirically, FSpecGNN validates the predicted expressivity and delivers strong performance on heterophilic benchmarks.

cs.LG↗

arXiv2026

Topology-Aware Multiscale Mixture of Experts for Efficient Molecular Property Prediction

Many molecular properties depend on 3D geometry, where non-covalent interactions, stereochemical effects, and medium- to long-range forces are determined by spatial distances and angles that cannot be uniquely captured by a 2D bond graph. Yet most 3D molecular graph neural networks still rely on globally fixed neighborhood heuristics, typically defined by distance cutoffs and maximum neighbor limits, to define local message-passing neighborhoods, leading to rigid, data-agnostic interaction budgets. We propose Multiscale Interaction Mixture of Experts (MI-MoE) to adapt interaction modeling across geometric regimes. Our contributions are threefold: (1) we introduce a distance-cutoff expert ensemble that explicitly captures short-, mid-, and long-range interactions without committing to a single cutoff; (2) we design a topological gating encoder that routes inputs to experts using filtration-based descriptors, including persistent homology features, summarizing how connectivity evolves across radii; and (3) we show that MI-MoE is a plug-in module that consistently improves multiple strong 3D molecular backbones across diverse molecular and polymer property prediction benchmark datasets, covering both regression and classification tasks. These results highlight topology-aware multiscale routing as an effective principle for 3D molecular graph learning.

cs.LG↗

arXiv2025

A roadmap for curvature-based geometric data analysis and learning

Geometric data analysis and learning has emerged as a distinct and rapidly developing research area, increasingly recognized for its effectiveness across diverse applications. At the heart of this field lies curvature, a powerful and interpretable concept that captures intrinsic geometric structure and underpins numerous tasks, from community detection to geometric deep learning. A wide range of discrete curvature models have been proposed for various data representations, including graphs, simplicial complexes, cubical complexes, and point clouds sampled from manifolds. These models not only provide efficient characterizations of data geometry but also constitute essential components in geometric learning frameworks. In this paper, we present the first comprehensive review of existing discrete curvature models, covering their mathematical foundations, computational formulations, and practical applications in data analysis and learning. In particular, we discuss discrete curvature from both Riemannian and metric geometry perspectives and propose a systematic pipeline for curvature-driven data analysis. We further examine the corresponding computational algorithms across different data representations, offering detailed comparisons and insights. Finally, we review state-of-the-art applications of curvature in both supervised and unsupervised learning. This survey provides a conceptual and practical roadmap for researchers to gain a better understanding of discrete curvature as a fundamental tool for geometric understanding and learning.

cs.LG↗

arXiv2025

Molecule Graph Networks with Many-body Equivariant Interactions

Message passing neural networks have demonstrated significant efficacy in predicting molecular interactions. Introducing equivariant vectorial representations augments expressivity by capturing geometric data symmetries, thereby improving model accuracy. However, two-body bond vectors in opposition may cancel each other out during message passing, leading to the loss of directional information on their shared node. In this study, we develop Equivariant N-body Interaction Networks (ENINet) that explicitly integrates l = 1 equivariant many-body interactions to enhance directional symmetric information in the message passing scheme. We provided a mathematical analysis demonstrating the necessity of incorporating many-body equivariant interactions and generalized the formulation to $N$-body interactions. Experiments indicate that integrating many-body equivariant representations enhances prediction accuracy across diverse scalar and tensorial quantum chemical properties.

cs.LG↗

arXiv2025

Quotient Complex Transformer (QCformer) for Perovskite Data Analysis

The discovery of novel functional materials is crucial in addressing the challenges of sustainable energy generation and climate change. Hybrid organic-inorganic perovskites (HOIPs) have gained attention for their exceptional optoelectronic properties in photovoltaics. Recently, geometric deep learning, particularly graph neural networks (GNNs), has shown strong potential in predicting material properties and guiding material design. However, traditional GNNs often struggle to capture the periodic structures and higher-order interactions prevalent in such systems. To address these limitations, we propose a novel representation based on quotient complexes (QCs) and introduce the Quotient Complex Transformer (QCformer) for material property prediction. A material structure is modeled as a quotient complex, which encodes both pairwise and many-body interactions via simplices of varying dimensions and captures material periodicity through a quotient operation. Our model leverages higher-order features defined on simplices and processes them using a simplex-based Transformer module. We pretrain QCformer on benchmark datasets such as the Materials Project and JARVIS, and fine-tune it on HOIP datasets. The results show that QCformer outperforms state-of-the-art models in HOIP property prediction, demonstrating its effectiveness. The quotient complex representation and QCformer model together contribute a powerful new tool for predictive modeling of perovskite materials.

cs.LG↗

arXiv2025

Rhomboid Tiling for Geometric Graph Deep Learning

Graph Neural Networks (GNNs) have proven effective for learning from graph-structured data through their neighborhood-based message passing framework. Many hierarchical graph clustering pooling methods modify this framework by introducing clustering-based strategies, enabling the construction of more expressive and powerful models. However, all of these message passing framework heavily rely on the connectivity structure of graphs, limiting their ability to capture the rich geometric features inherent in geometric graphs. To address this, we propose Rhomboid Tiling (RT) clustering, a novel clustering method based on the rhomboid tiling structure, which performs clustering by leveraging the complex geometric information of the data and effectively extracts its higher-order geometric structures. Moreover, we design RTPool, a hierarchical graph clustering pooling model based on RT clustering for graph classification tasks. The proposed model demonstrates superior performance, outperforming 21 state-of-the-art competitors on all the 7 benchmark datasets.

cs.LG↗

arXiv2025

Topology-enhanced machine learning model (Top-ML) for anticancer peptide prediction

Recently, therapeutic peptides have demonstrated great promise for cancer treatment. To explore powerful anticancer peptides, artificial intelligence (AI)-based approaches have been developed to systematically screen potential candidates. However, the lack of efficient featurization of peptides has become a bottleneck for these machine-learning models. In this paper, we propose a topology-enhanced machine learning model (Top-ML) for anticancer peptides prediction. Our Top-ML employs peptide topological features derived from its sequence "connection" information characterized by vector and spectral descriptors. Our Top-ML model, employing an Extra-Trees classifier, has been validated on the AntiCP 2.0 and mACPpred 2.0 benchmark datasets, achieving state-of-the-art performance or results comparable to existing deep learning models, while providing greater interpretability. Our results highlight the potential of leveraging novel topology-based featurization to accelerate the identification of anticancer peptides.

q-bio.QM↗

arXiv2025

Commutative algebra-enhanced topological data analysis

Topological Data Analysis (TDA) combines computational topology and data science to extract and analyze intrinsic topological and geometric structures in data set in a metric space. While the persistent homology (PH), a widely used tool in TDA, which tracks the lifespan information of topological features through a filtration process, has shown its effectiveness in applications,it is inherently limited in homotopy invariants and overlooks finer geometric and combinatorial details. To bridge this gap, we introduce two novel commutative algebra-based frameworks which extend beyond homology by incorporating tools from computational commutative algebra : (1) \emph{the persistent ideals} derived from the decomposition of algebraic objects associated to simplicial complexes, like those in theory of edge ideals and Stanley--Reisner ideals, which will provide new commutative algebra-based barcodes and offer a richer characterization of topological and geometric structures in filtrations.(2)\emph{persistent chain complex of free modules} associated with traditional persistent simplicial complex by labelling each chain in the chain complex of the persistent simplicial complex with elements in a commutative ring, which will enable us to detect local information of the topology via some pure algebraic operations. \emph{Crucially, both of the two newly-established framework can recover topological information got from conventional PH and will give us more information.} Therefore, they provide new insights in computational topology, computational algebra and data science.

cs.CG↗

arXiv2025

A cohomology-based Gromov-Hausdorff metric approach for quantifying molecular similarity

We introduce, for the first time, a cohomology-based Gromov-Hausdorff ultrametric method to analyze 1-dimensional and higher-dimensional (co)homology groups, focusing on loops, voids, and higher-dimensional cavity structures in simplicial complexes, to address typical clustering questions arising in molecular data analysis. The Gromov-Hausdorff distance quantifies the dissimilarity between two metric spaces. In this framework, molecules are represented as simplicial complexes, and their cohomology vector spaces are computed to capture intrinsic topological invariants encoding loop and cavity structures. These vector spaces are equipped with a suitable distance measure, enabling the computation of the Gromov-Hausdorff ultrametric to evaluate structural dissimilarities. We demonstrate the methodology using organic-inorganic halide perovskite (OIHP) structures. The results highlight the effectiveness of this approach in clustering various molecular structures. By incorporating geometric information, our method provides deeper insights compared to traditional persistent homology techniques.

math.AT↗

arXiv2024

A Physics-informed Sheaf Model

Normal mode analysis (NMA) provides a mathematical framework for exploring the intrinsic global dynamics of molecules through the definition of an energy function, where normal modes correspond to the eigenvectors of the Hessian matrix derived from the second derivatives of this function. The energy required to 'trigger' each normal mode is proportional to the square of its eigenvalue, with six zero-eigenvalue modes representing universal translation and rotation, common to all molecular systems. In contrast, modes associated with small non-zero eigenvalues are more easily excited by external forces and are thus closely related to molecular functions. Inspired by the anisotropic network model (ANM), this work establishes a novel connection between normal mode analysis and sheaf theory by introducing a cellular sheaf structure, termed the anisotropic sheaf, defined on undirected, simple graphs, and identifying the conventional Hessian matrix as the sheaf Laplacian. By interpreting the global section space of the anisotropic sheaf as the kernel of the Laplacian matrix, we demonstrate a one-to-one correspondence between the zero-eigenvalue-related normal modes and a basis for the global section space. We further analyze the dimension of this global section space, representing the space of harmonic signals, under conditions typically considered in normal mode analysis. Additionally, we propose a systematic method to streamline the Delaunay triangulation-based construction for more efficient graph generation while preserving the ideal number of normal modes with zero eigenvalues in ANM analysis.

physics.chem-ph↗

arXiv2024

KA-GNN: Kolmogorov-Arnold Graph Neural Networks for Molecular Property Prediction

As key models in geometric deep learning, graph neural networks have demonstrated enormous power in molecular data analysis. Recently, a specially-designed learning scheme, known as Kolmogorov-Arnold Network (KAN), shows unique potential for the improvement of model accuracy, efficiency, and explainability. Here we propose the first non-trivial Kolmogorov-Arnold Network-based Graph Neural Networks (KA-GNNs), including KAN-based graph convolutional networks(KA-GCN) and KAN-based graph attention network (KA-GAT). The essential idea is to utilizes KAN's unique power to optimize GNN architectures at three major levels, including node embedding, message passing, and readout. Further, with the strong approximation capability of Fourier series, we develop Fourier series-based KAN model and provide a rigorous mathematical prove of the robust approximation capability of this Fourier KAN architecture. To validate our KA-GNNs, we consider seven most-widely-used benchmark datasets for molecular property prediction and extensively compare with existing state-of-the-art models. It has been found that our KA-GNNs can outperform traditional GNN models. More importantly, our Fourier KAN module can not only increase the model accuracy but also reduce the computational time. This work not only highlights the great power of KA-GNNs in molecular property prediction but also provides a novel geometric deep learning framework for the general non-Euclidean data analysis.

cs.LG↗

arXiv2024

TopoQA: a topological deep learning-based approach for protein complex structure interface quality assessment

Even with the significant advances of AlphaFold-Multimer (AF-Multimer) and AlphaFold3 (AF3) in protein complex structure prediction, their accuracy is still not comparable with monomer structure prediction. Efficient quality assessment (QA) or estimation of model accuracy (EMA) models that can evaluate the quality of the predicted protein-complexes without knowing their native structures, are of key importance for protein structure generation and model selection. In this paper, we leverage persistent homology (PH) to capture the atomic-level topological information around residues and design a topological deep learning-based QA method, TopoQA, to assess the accuracy of protein complex interfaces. We integrate PH from topological data analysis into graph neural networks (GNNs) to characterize complex higher-order structures that GNNs might overlook, enhancing the learning of the relationship between the topological structure of complex interfaces and quality scores. Our TopoQA model is extensively validated based on the two most-widely used datasets, DBM55-AF2 and HAF2, along with our newly constructed ABAG-AF3 dataset to facilitate comparisons with AF3. For all three datasets, TopoQA outperforms AF-Multimer-based AF2Rank and shows an advantage over AF3 in nearly half of the targets. In particular, in the DBM55-AF2 dataset, a ranking loss of 73.6% lower than AF-Multimer-based AF2Rank is obtained. Further, other than AF-Multimer and AF3, we have also extensively compared with nearly-all the state-of-the-art models (as far as we know), it has been found that our TopoQA can achieve the highest Top 10 Hit-rate on the DBM55-AF2 dataset and the lowest ranking loss on the HAF2 dataset. Ablation experiments show that our topological features significantly improve the model performance. At the same time, our method also provides a new paradigm for protein structure representation learning.

q-bio.BM↗

arXiv2024

Molecular topological deep learning for polymer property prediction

Accurate and efficient prediction of polymer properties is of key importance for polymer design. Traditional experimental tools and density function theory (DFT)-based simulations for polymer property evaluation, are both expensive and time-consuming. Recently, a gigantic amount of graph-based molecular models have emerged and demonstrated huge potential in molecular data analysis. Even with the great progresses, these models tend to ignore the high-order and mutliscale information within the data. In this paper, we develop molecular topological deep learning (Mol-TDL) for polymer property analysis. Our Mol-TDL incorporates both high-order interactions and multiscale properties into topological deep learning architecture. The key idea is to represent polymer molecules as a series of simplicial complices at different scales and build up simplical neural networks accordingly. The aggregated information from different scales provides a more accurate prediction of polymer molecular properties.

cond-mat.mtrl-sci↗