This work develops a framework for learning over Mapper induced structured representations, and studies mathematical properties of these representations, including invariance under relabeling, a distance functional on the space of Mapper representations, structural complexity of multiscale decompositions, and learning oriented stability under representation perturbations.
Abstract
Modern machine learning (ML) methods are highly effective for prediction tasks, but many commonly used representations reduce complex data to fixed dimensional embeddings that may suppress multiscale structural organization. The Mapper algorithm from topological data analysis (TDA) provides a different perspective by decomposing data into overlapping local regions connected through a nerve construction, producing a structured representation that captures geometric organization, local statistical behavior, and relational connectivity simultaneously. In this work, we develop a framework for learning over Mapper induced structured representations. Rather than treating Mapper as a preprocessing step that produces a graph for downstream learning, we treat the full Mapper construction as part of the representation itself. We study mathematical properties of these representations, including invariance under relabeling, a distance functional on the space of Mapper representations, structural complexity of multiscale decompositions, and learning oriented stability under representation perturbations. Experiments on time series and graph classification datasets validate the proposed framework through controlled studies of representation ablation, Mapper parameter sensitivity, and the geometry of the induced representation space. Together, these results demonstrate how the proposed mathematical framework enables systematic comparison, interpretation, and analysis of Mapper representations, providing practical tools for studying representation geometry, structural complexity, and learning stability in learning tasks.
Graph Autoencoder Uncovering Descriptive Information (GAUDI), an unsupervised graph deep learning framework designed to capture both local details and global structure, is introduced, providing new insights into emergent phenomena across diverse scientific domains.
Machine learning models continue to face challenges in out-of-distribution (OOD) generalization, where domain generalization (DG) aims to improve performance on unseen domains under distributional shifts. A prevalent paradigm in DG focuses on learning domain-invariant feature representations. However, feature representations from existing methods often exhibit weak interpretability. To bridge this gap, we propose Sparse Additive Domain Generalization (SpADG). We incorporate an additive structure into the DG framework and employ ℓq,1 -norm regularization to induce sparsity, thereby enabling structured feature selection and enhancing interpretability. We present two distinct realizations: an additive kernel-based formulation and a neural additive model-based approach. The former leverages the representer theorem for flexible data adaptation, while the latter learns nonlinear shape functions. Theoretically, we derive generalization error bounds for both realizations and prove the feature selection consistency of our method under rate-scaled regularization condition. Empirical evaluations on synthetic and real-world datasets validate the effectiveness of SpADG, particularly its robustness in high-dimensional settings.
Jiayi Wang, Han Li· Proceedings of the 32nd ACM...· 0 citations
High dimensional biomedical data often exhibit nonlinear, heterogeneous, and manifold driven structures that challenge global parametric and tree-based models. We propose MAPLE (mapper-based Adaptive Prediction via Local Estimation), a localized prediction framework grounded in topological data analysis. The method is formulated as a nonparametric estimator of conditional class probabilities that adapts to the intrinsic geometry of the predictor space. Neighborhoods are defined through connectivity in a data-adaptive Mapper graph, enabling localized averaging within graph induced regions that capture complex structures such as branching and multi-scale heterogeneity. We introduce a statistically principled, data driven procedure for cover selection based on a bias-variance trade off, yielding optimal asymptotic scaling for interval widths and overlaps. The framework accommodates binary, nominal, and ordinal outcomes and incorporates a permutation-based variable importance measure to quantify covariate contributions in prediction. We establish theoretical guarantees, including pointwise consistency and Bayes risk consistency under standard regularity conditions. Simulations show that MAPLE consistently outperforms or matches multinomial regression, ordinal regression, and random forest, with the largest gains observed under heterogeneous and high-noise settings. Applications to Parkinson's disease progression (PPMI) and glioma classification (TCGA RNA sequencing) demonstrate strong predictive accuracy and interpretable, topology-aware summaries of underlying data structure.
Muhammad Ahsan, Priyam Das, Nitai D. Mukhopadhyay· 0 citations
Over eight diverse protein foundational models trained on 550,120 SwissProt proteins with AlphaFold structures, enriched embeddings improved zero-shot remote homology retrieval, increasing Precision@10 and MRR by up to 0.13 and 0.11, respectively.
Gabriel Bianchin de Oliveira, Fahad Saeed· bioRxiv· 0 citations
Search Trajectory Networks (STNs) are a graph-based tool for visualizing and characterizing the behavior of optimization algorithms. STNs'reliance on discretization of the search space has largely confined them to low-dimensional or combinatorial settings. We introduce a methodology for constructing STNs in semantic spaces, defined as the space of a model's predictions on a fixed sample set. Our approach discretizes semantic vectors and aggregates them into network nodes via agglomerative clustering with complete linkage under a normalized Hamming distance. Since any predictor can be summarized by its semantic vector, this method enables comparison of learning dynamics across otherwise incomparable algorithm families. We apply semantic space STNs to classification and regression tasks solved using different machine learning algorithms, recovering known qualitative differences between them. Additionally, we use semantic space STNs to study neural network generalization by contrasting standard training with the label randomization regime of Zhang et al. (2017). The resulting STNs exhibit consistent structural differences, training on real labels produces denser, more efficient and more centralized graphs than training on shuffled labels. Together, our results show that semantic space STNs capture functional training dynamics arising from the interaction between learning algorithms and data, providing a tool for analyzing and comparing learning dynamics across machine learning models and training regimes.
J. Agudelo, A. Tonda, Gabriela Ochoa et al.· 0 citations
Foundation models have emerged as powerful tools for learning transferable representations of biological systems, yet their latent spaces are typically optimized to capture cellular state rather than the effects of perturbations. Here, we demonstrate that a biological foundation model can be repurposed to learn a fundamentally different representation by changing its learning objective. We fine-tuned scGPT, a transformer pre-trained on over 30 million single-cell transcriptomes, on more than three million LINCS L1000 perturbation profiles using a supervised objective that predicts perturbation identity. This transformed the latent space into a perturbation-centric representation that aligned transcriptional responses induced by the same chemical or genetic perturbation across heterogeneous experimental conditions. Fine-tuned embeddings substantially outperformed both gene expression profiles and the original pre-trained model, recovering 85–100% of perturbations within the top 100 nearest neighbors and increasing perturbation classification accuracy from 10–19% to 25–49%. Remarkably, although the model was trained exclusively to recognize perturbation identity, the learned representation spontaneously captured orthogonal biological relationships never provided during training, including chemical similarity (AUROC up to 0.81), mechanisms of action (Hit@10 up to 100%), compound–target relationships (AUROC up to 0.74), and functional relationships between genetic perturbations. The resulting embedding space enabled mechanism-of-action annotation of nearly 12,000 previously uncharacterized compounds, prioritization of target-related chemical–genetic associations, and contextualization of unseen perturbations and external transcriptomic datasets. Together, our results establish objective-driven adaptation as a general strategy for repurposing biological foundation models to learn reusable representations of complex biological phenomena.
Elena Pareja-Lorente, Patrick Aloy· bioRxiv· 0 citations