In this paper, we propose HiPoNet, an end-to-end differentiable neural network for regression, classification, and representation learning on high-dimensional point clouds. Our work is motivated by single-cell data which can have very high-dimensionality --exceeding the capabilities of existing methods for point clouds which are mostly tailored for 3D data. Moreover, modern single-cell and spatial experiments now yield entire cohorts of datasets (i.e., one data set for every patient), necessitating models that can process large, high-dimensional point-clouds at scale. Most current approaches build a single nearest-neighbor graph, discarding important geometric and topological information. In contrast, HiPoNet models the point-cloud as a set of higher-order simplicial complexes, with each particular complex being created using a reweighting of features. This method thus generates multiple constructs corresponding to different views of high-dimensional data, which in biology offers the possibility of disentangling distinct cellular processes. It then employs simplicial wavelet transforms to extract multiscale features, capturing both local and global topology from each view. We show that geometric and topological information is preserved in this framework both theoretically and empirically.

Graph Neural Networks (GNNs) experience catastrophic forgetting in continual learning setups, where they tend to lose previously acquired knowledge and perform poorly on old tasks. Rehearsal-based methods, which consolidate old knowledge with a replay memory buffer, are a de facto solution due to their straightforward workflow. However, these methods often fail to adequately capture topological information, leading to incorrect input-label mappings in replay samples. To address this, we propose TACO, a topology-aware graph coarsening and continual learning framework that stores information from previous tasks as a reduced graph. Throughout each learning period, this reduced graph expands by integrating with a new graph and aligning shared nodes, followed by a zoom-out reduction process to maintain a stable size. We have developed a graph coarsening algorithm based on node representation proximities to efficiently reduce a graph while preserving essential topological information. We empirically demonstrate that the learning process on the reduced graph can closely approximate that on the original graph. We compare TACO with a wide range of state-of-the-art baselines, proving its superiority and the necessity of preserving high-quality topological information for effective replaying.

The problem of (point) forecastingunivariate time series is considered.

We further design an effective mechanism to inject PH information into GP at both feature and topology levels, with a novel topology-preserving loss function.

Probabilistic Graphical Models (PGMs) encode conditional dependencies among random variables using a graph -nodes for variables, links for dependencies- and factorize the joint distribution into lower-dimensional components. This makes PGMs well-suited for analyzing complex systems and supporting decision-making. Recent advances in topological signal processing highlight the importance of variables defined on topological spaces in several application domains. In such cases, the underlying topology shapes statistical relationships, limiting the expressiveness of canonical PGMs. To overcome this limitation, we introduce Colored Markov Random Fields (CMRFs), which model both conditional and marginal dependencies among Gaussian edge variables on topological spaces, with a theoretical foundation in Hodge theory. CMRFs extend classical Gaussian Markov Random Fields by including link coloring: connectivity encodes conditional independence, while color encodes marginal independence. We quantify the benefits of CMRFs through a distributed estimation case study over a physical network, comparing it with baselines with different levels of topological prior.

Graph contrastive learning (GCL) aims to learn discriminative semantic invariance by contrasting different views of the same graph that share critical topological patterns. However, existing GCL approaches with structural augmentations often struggle to identify task-relevant topological structures, let alone adapt to the varying coarse-to-fine topological granularities required across different downstream tasks. To remedy this issue, we introduce Hierarchical Topological Granularity Graph Contrastive Learning (HTG-GCL), a novel framework that leverages transformations of the same graph to generate multi-scale ring-based cellular complexes, embodying the concept of topological granularity, thereby generating diverse topological views. Recognizing that a certain granularity may contain misleading semantics, we propose a multi-granularity decoupled contrast and apply a granularity-specific weighting mechanism based on uncertainty estimation. Comprehensive experiments on various benchmarks demonstrate the effectiveness of HTG-GCL, highlighting its superior performance in capturing meaningful graph representations through hierarchical topological information.

We further design an effective mechanism to inject PH information into GP at both feature and topology levels, with a novel topology-preserving loss function.

Integrated circuits (ICs) are extensively used in modern electronic products like computers, smart-phones, and cars. Electronic Design Automation (EDA) includes a set of tools for circuit design in different development stages especially logic synthesis stage and placement stage (Fig.1).