Sparsity is Combinatorial Depth: Quantifying MoE Expressivity via Tropical Geometry

While Mixture-of-Experts (MoE) architectures define the state-of-the-art, their theoretical success is often attributed to heuristic efficiency rather than geometric expressivity. In this work, we present the first analysis of MoE through the lens of tropical geometry, establishing that the Top-$k$ routing mechanism is algebraically isomorphic to the $k$-th elementary symmetric tropical polynomial. This isomorphism partitions the input space into the Normal Fan of a Hypersimplex, revealing that \textbf{sparsity is combinatorial depth} which scales geometric capacity by the binomial coefficient $\binom{N}{k}$. Moving beyond ambient bounds, we introduce the concept of \textit{Effective Capacity} under the Manifold Hypothesis. We prove that while dense networks suffer from capacity collapse on low-dimensional data, MoE architectures exhibit \textit{Combinatorial Resilience}, maintaining high expressivity via the transversality of routing cones. In this study, our framework unifies the discrete geometry of the Hypersimplex with the continuous geometry of neural functions, offering a rigorous theoretical justification for the topological supremacy of conditional computation.

Hybrid Topological and Deep Feature Fusion for Accurate MRI-Based Alzheimer's Disease Severity Classification

Early and accurate diagnosis of Alzheimer's disease (AD) remains a critical challenge in neuroimaging-based clinical decision support systems. In this work, we propose a novel hybrid deep learning framework that integrates Topological Data Analysis (TDA) with a DenseNet121 backbone for four-class Alzheimer's disease classification using structural MRI data from the OASIS dataset. TDA is employed to capture complementary topological characteristics of brain structures that are often overlooked by conventional neural networks, while DenseNet121 efficiently learns hierarchical spatial features from MRI slices. The extracted deep and topological features are fused to enhance class separability across the four AD stages. Extensive experiments conducted on the OASIS-1 Kaggle MRI dataset demonstrate that the proposed TDA+DenseNet121 model significantly outperforms existing state-of-the-art approaches. The model achieves an accuracy of 99.93% and an AUC of 100%, surpassing recently published CNN-based, transfer learning, ensemble, and multi-scale architectures. These results confirm the effectiveness of incorporating topological insights into deep learning pipelines and highlight the potential of the proposed framework as a robust and highly accurate tool for automated Alzheimer's disease diagnosis.

Learning Topology-Aware Implicit Field for Unified Pulmonary Tree Modeling with Incomplete Topological Supervision

Pulmonary trees extracted from CT images frequently exhibit topological incompleteness, such as missing or disconnected branches, which substantially degrades downstream anatomical analysis and limits the applicability of existing pulmonary tree modeling pipelines. Current approaches typically rely on dense volumetric processing or explicit graph reasoning, leading to limited efficiency and reduced robustness under realistic structural corruption. We propose TopoField, a topology-aware implicit modeling framework that treats topology repair as a first-class modeling problem and enables unified multi-task inference for pulmonary tree analysis. TopoField represents pulmonary anatomy using sparse surface and skeleton point clouds and learns a continuous implicit field that supports topology repair without relying on complete or explicit disconnection annotations, by training on synthetically introduced structural disruptions over \textit{already} incomplete trees. Building upon the repaired implicit representation, anatomical labeling and lung segment reconstruction are jointly inferred through task-specific implicit functions within a single forward pass.Extensive experiments on the Lung3D+ dataset demonstrate that TopoField consistently improves topological completeness and achieves accurate anatomical labeling and lung segment reconstruction under challenging incomplete scenarios. Owing to its implicit formulation, TopoField attains high computational efficiency, completing all tasks in just over one second per case, highlighting its practicality for large-scale and time-sensitive clinical applications. Code and data will be available at https://github.com/HINTLab/TopoField.

A Sheaf-Theoretic and Topological Perspective on Complex Network Modeling and Attention Mechanisms in Graph Neural Models

Combinatorial and topological structures, such as graphs, simplicial complexes, and cell complexes, form the foundation of geometric and topological deep learning (GDL and TDL) architectures. These models aggregate signals over such domains, integrate local features, and generate representations for diverse real-world applications. However, the distribution and diffusion behavior of GDL and TDL features during training remains an open and underexplored problem. Motivated by this gap, we introduce a cellular sheaf theoretic framework for modeling and analyzing the local consistency and harmonicity of node features and edge weights in graph-based architectures. By tracking local feature alignments and agreements through sheaf structures, the framework offers a topological perspective on feature diffusion and aggregation. Furthermore, a multiscale extension inspired by topological data analysis (TDA) is proposed to capture hierarchical feature interactions in graph models. This approach enables a joint characterization of GDL and TDL architectures based on their underlying geometric and topological structures and the learned signals defined on them, providing insights for future studies on conventional tasks such as node classification, substructure detection, and community detection.

Test-Time Adaptation for Anomaly Segmentation via Topology-Aware Optimal Transport Chaining

Deep topological data analysis (TDA) offers a principled framework for capturing structural invariants such as connectivity and cycles that persist across scales, making it a natural fit for anomaly segmentation (AS). Unlike thresholdbased binarisation, which produces brittle masks under distribution shift, TDA allows anomalies to be characterised as disruptions to global structure rather than local fluctuations. We introduce TopoOT, a topology-aware optimal transport (OT) framework that integrates multi-filtration persistence diagrams (PDs) with test-time adaptation (TTA). Our key innovation is Optimal Transport Chaining, which sequentially aligns PDs across thresholds and filtrations, yielding geodesic stability scores that identify features consistently preserved across scales. These stabilityaware pseudo-labels supervise a lightweight head trained online with OT-consistency and contrastive objectives, ensuring robust adaptation under domain shift. Across standard 2D and 3D anomaly detection benchmarks, TopoOT achieves state-of-the-art performance, outperforming the most competitive methods by up to +24.1% mean F1 on 2D datasets and +10.2% on 3D AS benchmarks.

Synthetic Pattern Generation and Detection of Financial Activities using Graph Autoencoders

Illicit financial activities such as money laundering often manifest through recurrent topological patterns in transaction networks. Detecting these patterns automatically remains challenging due to the scarcity of labeled real-world data and strict privacy constraints. To address this, we investigate whether Graph Autoencoders (GAEs) can effectively learn and distinguish topological patterns that mimic money laundering operations when trained on synthetic data. The analysis consists of two phases: (i) data generation, where synthetic samples are created for seven well-known illicit activity patterns using parametrized generators that preserve structural consistency while introducing realistic variability; and (ii) model training and validation, where separate GAEs are trained on each pattern without explicit labels, relying solely on reconstruction error as an indicator of learned structure. We compare three GAE implementations based on three distinct convolutional layers: Graph Convolutional (GAE-GCN), GraphSAGE (GAE-SAGE), and Graph Attention Network (GAE-GAT). Experimental results show that GAE-GCN achieves the most consistent reconstruction performance across patterns, while GAE-SAGE and GAE-GAT exhibit competitive results only in few specific patterns. These findings suggest that graph-based representation learning on synthetic data provides a viable path toward developing AI-driven tools for detecting illicit behaviors, overcoming the limitations of financial datasets.

MMSF: Multitask and Multimodal Supervised Framework for WSI Classification and Survival Analysis

Multimodal evidence is critical in computational pathology: gigapixel whole slide images capture tumor morphology, while patient-level clinical descriptors preserve complementary context for prognosis. Integrating such heterogeneous signals remains challenging because feature spaces exhibit distinct statistics and scales. We introduce MMSF, a multitask and multimodal supervised framework built on a linear-complexity MIL backbone that explicitly decomposes and fuses cross-modal information. MMSF comprises a graph feature extraction module embedding tissue topology at the patch level, a clinical data embedding module standardizing patient attributes, a feature fusion module aligning modality-shared and modality-specific representations, and a Mamba-based MIL encoder with multitask prediction heads. Experiments on CAMELYON16 and TCGA-NSCLC demonstrate 2.1--6.6\% accuracy and 2.2--6.9\% AUC improvements over competitive baselines, while evaluations on five TCGA survival cohorts yield 7.1--9.8\% C-index improvements compared with unimodal methods and 5.6--7.1\% over multimodal alternatives.

Persistent Sheaf Laplacian Analysis of Protein Stability and Solubility Changes upon Mutation

Genetic mutations frequently disrupt protein structure, stability, and solubility, acting as primary drivers for a wide spectrum of diseases. Despite the critical importance of these molecular alterations, existing computational models often lack interpretability, and fail to integrate essential physicochemical interaction. To overcome these limitations, we propose SheafLapNet, a unified predictive framework grounded in the mathematical theory of Topological Deep Learning (TDL) and Persistent Sheaf Laplacian (PSL). Unlike standard Topological Data Analysis (TDA) tools such as persistent homology, which are often insensitive to heterogeneous information, PSL explicitly encodes specific physical and chemical information such as partial charges directly into the topological analysis. SheafLapNet synergizes these sheaf-theoretic invariants with advanced protein transformer features and auxiliary physical descriptors to capture intrinsic molecular interactions in a multiscale and mechanistic manner. To validate our framework, we employ rigorous benchmarks for both regression and classification tasks. For stability prediction, we utilize the comprehensive S2648 and S350 datasets. For solubility prediction, we employ the PON-Sol2 dataset, which provides annotations for increased, decreased, or neutral solubility changes. By integrating these multi-perspective features, SheafLapNet achieves state-of-the-art performance across these diverse benchmarks, demonstrating that sheaf-theoretic modeling significantly enhances both interpretability and generalizability in predicting mutation-induced structural and functional changes.

An Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learning

Geometric and Topological Deep Learning are rapidly growing research areas that enhance machine learning through the use of geometric and topological structures. Within this framework, Group Equivariant Non-Expansive Operators (GENEOs) have emerged as a powerful class of operators for encoding symmetries and designing efficient, interpretable neural architectures. Originally introduced in Topological Data Analysis, GENEOs have since found applications in Deep Learning as tools for constructing equivariant models with reduced parameter complexity. GENEOs provide a unifying framework bridging Geometric and Topological Deep Learning and include the operator computing persistence diagrams as a special case. Their theoretical foundations rely on group actions, equivariance, and compactness properties of operator spaces, grounding them in algebra and geometry while enabling both mathematical rigor and practical relevance. While a previous representation theorem characterized linear GENEOs acting on data of the same type, many real-world applications require operators between heterogeneous data spaces. In this work, we address this limitation by introducing a new representation theorem for linear GENEOs acting between different perception pairs, based on generalized T-permutant measures. Under mild assumptions on the data domains and group actions, our result provides a complete characterization of such operators. We also prove the compactness and convexity of the space of linear GENEOs. We further demonstrate the practical impact of this theory by applying the proposed framework to improve the performance of autoencoders, highlighting the relevance of GENEOs in modern machine learning applications.

CLiMB: A Domain-Informed Novelty Detection Clustering Framework for Scientific Discovery

In data-driven scientific discovery, a challenge lies in classifying well-characterized phenomena while identifying novel anomalies. Current semi-supervised clustering algorithms do not always fully address this duality, often assuming that supervisory signals are globally representative. Consequently, methods often enforce rigid constraints that suppress unanticipated patterns or require a pre-specified number of clusters, rendering them ineffective for genuine novelty detection. To bridge this gap, we introduce CLiMB (CLustering in Multiphase Boundaries), a domain-informed framework decoupling the exploitation of prior knowledge from the exploration of unknown structures. Using a sequential two-phase approach, CLiMB first anchors known clusters using constrained partitioning, and subsequently applies density-based clustering to residual data to reveal arbitrary topologies. We demonstrate this framework on RR Lyrae stars data from the Gaia Data Release 3. CLiMB attains an Adjusted Rand Index of 0.829 with 90% seed coverage in recovering known Milky Way substructures, drastically outperforming heuristic and constraint-based baselines, which stagnate below 0.20. Furthermore, sensitivity analysis confirms CLiMB's superior data efficiency, showing monotonic improvement as knowledge increases. Finally, the framework successfully isolates three dynamical features (Shiva, Shakti, and the Galactic Disk) in the unlabelled field, validating its potential for scientific discovery.