Graph neural tangent kernels give a principled infinite-width theory for graph neural networks, but inherit a basic limitation of graph models: they see only pairwise structure. Many relational systems contain higher-order interactions that are more naturally represented by simplicial complexes. We introduce the Topological Neural Tangent Kernel (TopoNTK), an infinite-width kernel for simplicial message passing on edge features. TopoNTK combines lower Hodge interactions, capturing graph-like coupling through shared vertices, with upper Hodge interactions, capturing coupling through filled simplices. This makes the kernel sensitive to topology invisible to graph kernels, allowing complexes with the same graph but different filled simplices to induce different kernels. Beyond expressivity, the Hodge structure gives the kernel an interpretable learning geometry. Edge signals decompose into gradient-like, harmonic, and local circulation components, and the spectrum of the TopoNTK determines how quickly each component is learned. This yields a topological form of spectral bias: components aligned with large-eigenvalue modes are learned quickly, while global harmonic modes, retained through the residual channel, often lie at smaller eigenvalues and are learned more slowly. We prove expressivity, Hodge-alignment, spectral learning, and stability properties, and validate them on synthetic simplicial tasks and DBLP higher-order link prediction. The results show that topology is not merely extra structure; it can provide coordinates that make relational learning more faithful, interpretable, and effective.

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.

Persistent homology (PH) encodes global information, such as cycles, and is thus increasingly integrated into graph neural networks (GNNs). PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first expose limitations of this inclusion procedure. To remedy these shortcomings, we analyze contractions as a principled topological operation, in particular, for graph representation learning. We study the persistence of contraction sequences, which we call Contraction Homology (CH). We establish that forward PH and CH differ in expressivity. We then introduce Hourglass Persistence, a class of topological descriptors that interleave a sequence of inclusions and contractions to boost expressivity, learnability, and stability. We also study related families parametrized by two paradigms. We also discuss how our framework extends to simplicial and cellular networks. We further design efficient algorithms that are pluggable into end-to-end differentiable GNN pipelines, enabling consistent empirical improvements over many PH methods across standard real-world graph datasets. Code is available at \href{https://github.com/Aalto-QuML/Hourglass}{this https URL}.

Single-cell RNA-seq data analysis typically requires representations that capture heterogeneous local structure across multiple scales while remaining stable and interpretable. In this work, we propose a hierarchical sheaf spectral embedding (HSSE) framework that constructs informative cell-level features based on persistent sheaf Laplacian analysis. Starting from scale-dependent low-dimensional embeddings, we define cell-centered local neighborhoods at multiple resolutions. For each local neighborhood, we construct a data-driven cellular sheaf that encodes local relationships among cells. We then compute persistent sheaf Laplacians over sampled filtration intervals and extract spectral statistics that summarize the evolution of local relational structure across scales. These spectral descriptors are aggregated into a unified feature vector for each cell and can be directly used in downstream learning tasks without additional model training. We evaluate HSSE on twelve benchmark single-cell RNA-seq datasets covering diverse biological systems and data scales. Under a consistent classification protocol, HSSE achieves competitive or improved performance compared with existing multiscale and classical embedding-based methods across multiple evaluation metrics. The results demonstrate that sheaf spectral representations provide a robust and interpretable approach for single-cell RNA-seq data representation learning.

Theorem 1. Minimizing the expected loss Suppose we have T -dimensional features. Anchor nodes serve as fixed reference points within a simplicial complex, anchoring its structure and providing stability. Furthermore, anchor nodes can also represent important entities. Figure S2: Comparison of TSNE plots of representations learned by various encoders. CCA-SSG methods can not capture higher-order information and show similar artifacts. For example, the two clusters on the bottom and one from the right (corresponding to classes 1,2,3) are students from the same year but in different divisions.

Topological data analysis (TDA) delivers invaluable and complementary information on the intrinsic properties of data inaccessible to conventional methods.