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.

Graph Regularized PCA

High-dimensional data often exhibit dependencies among variables that violate the isotropic-noise assumption under which principal component analysis (PCA) is optimal. For cases where the noise is not independent and identically distributed across features (i.e., the covariance is not spherical) we introduce Graph Regularized PCA (GR-PCA). It is a graph-based regularization of PCA that incorporates the dependency structure of the data features by learning a sparse precision graph and biasing loadings toward the low-frequency Fourier modes of the corresponding graph Laplacian. Consequently, high-frequency signals are suppressed, while graph-coherent low-frequency ones are preserved, yielding interpretable principal components aligned with conditional relationships. We evaluate GR-PCA on synthetic data spanning diverse graph topologies, signal-to-noise ratios, and sparsity levels. Compared to mainstream alternatives, it concentrates variance on the intended support, produces loadings with lower graph-Laplacian energy, and remains competitive in out-of-sample reconstruction. When high-frequency signals are present, the graph Laplacian penalty prevents overfitting, reducing the reconstruction accuracy but improving structural fidelity. The advantage over PCA is most pronounced when high-frequency signals are graph-correlated, whereas PCA remains competitive when such signals are nearly rotationally invariant. The procedure is simple to implement, modular with respect to the precision estimator, and scalable, providing a practical route to structure-aware dimensionality reduction that improves structural fidelity without sacrificing predictive performance.

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.

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.

Revisiting the Ordering of Channel and Spatial Attention: A Comprehensive Study on Sequential and Parallel Designs

Attention mechanisms have become a core component of deep learning models, with Channel Attention and Spatial Attention being the two most representative architectures. Current research on their fusion strategies primarily bifurcates into sequential and parallel paradigms, yet the selection process remains largely empirical, lacking systematic analysis and unified principles. We systematically compare channel-spatial attention combinations under a unified framework, building an evaluation suite of 18 topologies across four classes: sequential, parallel, multi-scale, and residual. Across two vision and nine medical datasets, we uncover a "data scale-method-performance" coupling law: (1) in few-shot tasks, the "Channel-Multi-scale Spatial" cascaded structure achieves optimal performance; (2) in medium-scale tasks, parallel learnable fusion architectures demonstrate superior results; (3) in large-scale tasks, parallel structures with dynamic gating yield the best performance. Additionally, experiments indicate that the "Spatial-Channel" order is more stable and effective for fine-grained classification, while residual connections mitigate vanishing gradient problems across varying data scales. We thus propose scenario-based guidelines for building future attention modules. Code is open-sourced at https://github.com/DWlzm.

CutisAI: Deep Learning Framework for Automated Dermatology and Cancer Screening

The rapid growth of dermatological imaging and mobile diagnostic tools calls for systems that not only demonstrate empirical performance but also provide strong theoretical guarantees. Deep learning models have shown high predictive accuracy; however, they are often criticized for lacking well, calibrated uncertainty estimates without which these models are hardly deployable in a clinical setting. To this end, we present the Conformal Bayesian Dermatological Classifier (CBDC), a well, founded framework that combines Statistical Learning Theory, Topological Data Analysis (TDA), and Bayesian Conformal Inference. CBDC offers distribution, dependent generalization bounds that reflect dermatological variability, proves a topological stability theorem that guarantees the invariance of convolutional neural network embeddings under photometric and morphological perturbations and provides finite conformal coverage guarantees for trustworthy uncertainty quantification. Through exhaustive experiments on the HAM10000, PH2, and ISIC 2020 datasets, we show that CBDC not only attains classification accuracy but also generates calibrated predictions that are interpretable from a clinical perspective. This research constitutes a theoretical and practical leap for deep dermatological diagnostics, thereby opening the machine learning theory clinical applicability interface.

HalluZig: Hallucination Detection using Zigzag Persistence

The factual reliability of Large Language Models (LLMs) remains a critical barrier to their adoption in high-stakes domains due to their propensity to hallucinate. Current detection methods often rely on surface-level signals from the model's output, overlooking the failures that occur within the model's internal reasoning process. In this paper, we introduce a new paradigm for hallucination detection by analyzing the dynamic topology of the evolution of model's layer-wise attention. We model the sequence of attention matrices as a zigzag graph filtration and use zigzag persistence, a tool from Topological Data Analysis, to extract a topological signature. Our core hypothesis is that factual and hallucinated generations exhibit distinct topological signatures. We validate our framework, HalluZig, on multiple benchmarks, demonstrating that it outperforms strong baselines. Furthermore, our analysis reveals that these topological signatures are generalizable across different models and hallucination detection is possible only using structural signatures from partial network depth.

Efficient Cover Construction for Ball Mapper via Accelerated Range Queries

Ball Mapper is an widely used tool in topological data analysis for summarizing the structure of high-dimensional data through metric-based coverings and graph representations. A central computational bottleneck in Ball Mapper is the construction of the underlying cover, which requires repeated range queries to identify data points within a fixed distance of selected landmarks. As data sets grow in size and dimensionality, naive implementations of this step become increasingly inefficient. In this work, we study practical strategies for accelerating cover construction in Ball Mapper by improving the efficiency of range queries. We integrate two complementary approaches into the Ball Mapper pipeline: hierarchical geometric pruning using ball tree data structures, and hardware-aware distance computation using Facebook AI Similarity Search. We describe the underlying algorithms, discuss their trade-offs with respect to metric flexibility and dimensionality, and provide implementation details relevant to large-scale data analysis. Empirical benchmarks demonstrate that both approaches yield substantial speedups over the baseline implementation, with performance gains depending on data set size, dimensionality, and choice of distance function. These results improve the practical scalability of Ball Mapper without modifying its theoretical formulation and provide guidance for the efficient implementation of metric-based exploratory tools in modern data analysis workflows.

Warp-Cortex: An Asynchronous, Memory-Efficient Architecture for Million-Agent Cognitive Scaling on Consumer Hardware

Current multi-agent Large Language Model (LLM) frameworks suffer from linear memory scaling, rendering "System 2" parallel reasoning impractical on consumer hardware. We present Warp Cortex, an asynchronous architecture that theoretically enables million-agent cognitive scaling by decoupling agent logic from physical memory. Through Singleton Weight Sharing and a novel Topological Synapse--inspired by hybrid landmarking techniques from Topological Data Analysis (TDA)--we reduce memory complexity from O(N * L) to O(1) for weights and O(N * k) for context, where k << L. By treating the KV-cache as a point cloud in latent space, we apply witness-complex-inspired sparsification to preserve persistent homological features of the context manifold. On a single NVIDIA RTX 4090, we empirically demonstrate 100 concurrent agents at 2.2 GB total VRAM, with theoretical capacity exceeding 1,000 agents before compute latency becomes the bottleneck. We further introduce Referential Injection, a non-intrusive KV-cache update mechanism that allows asynchronous sub-agents to influence primary generation without stream disruption.

Four-Stage Alzheimer's Disease Classification from MRI Using Topological Feature Extraction, Feature Selection, and Ensemble Learning

Accurate and efficient classification of Alzheimer's disease (AD) severity from brain magnetic resonance imaging (MRI) remains a critical challenge, particularly when limited data and model interpretability are of concern. In this work, we propose TDA-Alz, a novel framework for four-stage Alzheimer's disease severity classification (non-demented, moderate dementia, mild, and very mild) using topological data analysis (TDA) and ensemble learning. Instead of relying on deep convolutional architectures or extensive data augmentation, our approach extracts topological descriptors that capture intrinsic structural patterns of brain MRI, followed by feature selection to retain the most discriminative topological features. These features are then classified using an ensemble learning strategy to achieve robust multiclass discrimination. Experiments conducted on the OASIS-1 MRI dataset demonstrate that the proposed method achieves an accuracy of 98.19% and an AUC of 99.75%, outperforming or matching state-of-the-art deep learning--based methods reported on OASIS and OASIS-derived datasets. Notably, the proposed framework does not require data augmentation, pretrained networks, or large-scale computational resources, making it computationally efficient and fast compared to deep neural network approaches. Furthermore, the use of topological descriptors provides greater interpretability, as the extracted features are directly linked to the underlying structural characteristics of brain MRI rather than opaque latent representations. These results indicate that TDA-Alz offers a powerful, lightweight, and interpretable alternative to deep learning models for MRI-based Alzheimer's disease severity classification, with strong potential for real-world clinical decision-support systems.