High-dimensional data that exhibit an intrinsic low-dimensional structure are ubiquitous in machine learning and data science. While various approaches allow for learning the corresponding data manifold from finite samples, performing downstream tasks such as optimization directly on these learned manifolds presents a significant challenge. This work introduces a principled framework for optimization on learned data manifolds using iso-Riemannian geometry. Our approach addresses key limitations of classical Riemannian optimization in this setting, specifically, that the Levi-Civita connection fails to yield constant-speed geodesics, and that geodesic convexity assumptions break down under the learned pullback constructions commonly used in practice. To overcome these challenges, we propose new notions of monotonicity and Lipschitz continuity tailored to the iso-Riemannian setting and propose iso-Riemannian descent algorithms for which we provide a detailed convergence analysis. We demonstrate the practical effectiveness of those algorithms on both synthetic and real datasets, including MNIST under a learned pullback structure. Our approach yields interpretable barycentres, improved clustering, and provably efficient solutions to inverse problems, even in high-dimensional settings. These results establish that optimization under iso-Riemannian geometry can overcome distortions inherent to learned manifold mappings.

Labeling or classifying time series is a persistent challenge in the physical sciences, where expert annotations are scarce, costly, and often inconsistent. Yet robust labeling is essential to enable machine learning models for understanding, prediction, and forecasting. We present the \textit{Clustering and Indexation Pipeline with Human Evaluation for Recognition} (CIPHER), a framework designed to accelerate large-scale labeling of complex time series in physics. CIPHER integrates \textit{indexable Symbolic Aggregate approXimation} (iSAX) for interpretable compression and indexing, density-based clustering (HDBSCAN) to group recurring phenomena, and a human-in-the-loop step for efficient expert validation. Representative samples are labeled by domain scientists, and these annotations are propagated across clusters to yield systematic, scalable classifications. We evaluate CIPHER on the task of classifying solar wind phenomena in OMNI data, a central challenge in space weather research, showing that the framework recovers meaningful phenomena such as coronal mass ejections and stream interaction regions. Beyond this case study, CIPHER highlights a general strategy for combining symbolic representations, unsupervised learning, and expert knowledge to address label scarcity in time series across the physical sciences. The code and configuration files used in this study are publicly available to support reproducibility.

This paper establishes the theoretical limits of graph clustering under the Popularity-Adjusted Block Model (PABM), addressing limitations of existing models. In contrast to the Stochastic Block Model (SBM), which assumes uniform vertex degrees, and to the Degree-Corrected Block Model (DCBM), which applies uniform degree corrections across clusters, PABM introduces separate popularity parameters for intra- and inter-cluster connections. Our main contribution is the characterization of the optimal error rate for clustering under PABM, which provides novel insights on clustering hardness: we demonstrate that unlike SBM and DCBM, cluster recovery remains possible in PABM even when traditional edge-density signals vanish, provided intra- and inter-cluster popularity coefficients differ. This highlights a dimension of degree heterogeneity captured by PABM but overlooked by DCBM: local differences in connectivity patterns can enhance cluster separability independently of global edge densities. Finally, because PABM exhibits a richer structure, its expected adjacency matrix has rank between $k$ and $k^2$, where $k$ is the number of clusters. As a result, spectral embeddings based on the top $k$ eigenvectors may fail to capture important structural information. Our numerical experiments on both synthetic and real datasets confirm that spectral clustering algorithms incorporating $k^2$ eigenvectors outperform traditional spectral approaches.

Analyzing multi-featured time series data is critical for space missions making efficient event detection, potentially onboard, essential for automatic analysis. However, limited onboard computational resources and data downlink constraints necessitate robust methods for identifying regions of interest in real time. This work presents an adaptive outlier detection algorithm based on the reconstruction error of Principal Component Analysis (PCA) for feature reduction, designed explicitly for space mission applications. The algorithm adapts dynamically to evolving data distributions by using Incremental PCA, enabling deployment without a predefined model for all possible conditions. A pre-scaling process normalizes each feature's magnitude while preserving relative variance within feature types. We demonstrate the algorithm's effectiveness in detecting space plasma events, such as distinct space environments, dayside and nightside transients phenomena, and transition layers through NASA's MMS mission observations. Additionally, we apply the method to NASA's THEMIS data, successfully identifying a dayside transient using onboard-available measurements.

\kmeans clustering is a fundamental problem in many scientific and engineering domains. The optimization problem associated with \kmeans clustering is nonconvex, for which standard algorithms are only guaranteed to find a local optimum. Leveraging the hidden structure of local solutions, we propose a general algorithmic framework for escaping undesirable local solutions and recovering the global solution or the ground truth clustering. This framework consists of iteratively alternating between two steps: (i) detect mis-specified clusters in a local solution, and (ii) improve the local solution by non-local operations. We discuss specific implementation of these steps, and elucidate how the proposed framework unifies many existing variants of \kmeans algorithms through a geometric perspective. We also present two natural variants of the proposed framework, where the initial number of clusters may be over- or under-specified. We provide theoretical justifications and extensive experiments to demonstrate the efficacy of the proposed approach.

Evaluating the quality of reasoning traces from large language models remains understudied, labor-intensive, and unreliable: current practice relies on expert rubrics, manual annotation, and slow pairwise judgments. Automated efforts are dominated by graph-based proxies that quantify structural connectivity but do not clarify what constitutes high-quality reasoning; such abstractions can be overly simplistic for inherently complex processes. We introduce a topological data analysis (TDA)-based evaluation framework that captures the geometry of reasoning traces and enables label-efficient, automated assessment. In our empirical study, topological features yield substantially higher predictive power for assessing reasoning quality than standard graph metrics, suggesting that effective reasoning is better captured by higher-dimensional geometric structures rather than purely relational graphs. We further show that a compact, stable set of topological features reliably indicates trace quality, offering a practical signal for future reinforcement learning algorithms.

Recent research has focused on mitigating algorithmic bias in clustering by incorporating fairness constraints into algorithmic design. Notions such as disparate impact, community cohesion, and cost per population have been implemented to enforce equitable outcomes. Among these, group fairness (balance) ensures that each protected group is proportionally represented within every cluster. However, incorporating balance as a metric of fairness into spectral clustering algorithms has led to computational times that can be improved. This study aims to enhance the efficiency of spectral clustering algorithms by reformulating the constrained optimization problem using a new formulation derived from the Lagrangian method and the Sherman-Morrison-Woodbury (SMW) identity, resulting in the Fair-SMW algorithm. Fair-SMW employs three alternatives to the Laplacian matrix with different spectral gaps to generate multiple variations of Fair-SMW, achieving clustering solutions with comparable balance to existing algorithms while offering improved runtime performance. We present the results of Fair-SMW, evaluated using the Stochastic Block Model (SBM) to measure both runtime efficiency and balance across real-world network datasets, including LastFM, FacebookNet, Deezer, and German. We achieve an improvement in computation time that is twice as fast as the state-of-the-art, and also flexible enough to achieve twice as much balance.

Medical artificial intelligence systems have achieved remarkable diagnostic capabilities, yet they consistently exhibit performance disparities across demographic groups, causing real-world harm to underrepresented populations. While recent multimodal reasoning foundation models have advanced clinical diagnosis through integrated analysis of diverse medical data, reasoning trainings via reinforcement learning inherit and often amplify biases present in training datasets dominated by majority populations. We introduce Fairness-aware Group Relative Policy Optimization (FairGRPO), a hierarchical reinforcement learning approach that promotes equitable learning across heterogeneous clinical populations. FairGRPO employs adaptive importance weighting of advantages based on representation, task difficulty, and data source. To address the common issue of missing demographic labels in the clinical domain, we further employ unsupervised clustering, which automatically discovers latent demographic groups when labels are unavailable. Through comprehensive experiments across 7 clinical diagnostic datasets spanning 5 clinical modalities across X-ray, CT scan, dermoscropy, mammography and ultrasound, we demonstrate that FairGRPO reduces predictive parity by 27.2% against all vanilla and bias mitigated RL baselines, while improving F1 score by 12.49%. Furthermore, training dynamics analysis reveals that FairGRPO progressively improves fairness throughout optimization, while baseline RL methods exhibit deteriorating fairness as training progresses. Based on FairGRPO, we release FairMedGemma-4B, a fairness-aware clinical VLLM that achieves state-of-the-art performance while demonstrating significantly reduced disparities across demographic groups.

Aging is a highly complex and heterogeneous process that progresses at different rates across individuals, making biological age (BA) a more accurate indicator of physiological decline than chronological age. While previous studies have built aging clocks using single-omics data, they often fail to capture the full molecular complexity of human aging. In this work, we leveraged the Human Phenotype Project, a large-scale cohort of 10,000 adults aged 40-70 years, with extensive longitudinal profiling that includes clinical, behavioral, environmental, and multi-omics datasets spanning transcriptomics, lipidomics, metabolomics, and the microbiome. By employing advanced machine learning frameworks capable of modeling nonlinear biological dynamics, we developed and rigorously validated a multi-omics aging clock that robustly predicts diverse health outcomes and future disease risk. Unsupervised clustering of the integrated molecular profiles from multi-omics uncovered distinct biological subtypes of aging, revealing striking heterogeneity in aging trajectories and pinpointing pathway-specific alterations associated with different aging patterns. These findings demonstrate the power of multi-omics integration to decode the molecular landscape of aging and lay the groundwork for personalized healthspan monitoring and precision strategies to prevent age-related diseases.

AI methods are increasingly shaping pharmaceutical drug discovery. However, their translation to industrial applications remains limited due to their reliance on public datasets, lacking scale and diversity of proprietary pharmaceutical data. Federated learning (FL) offers a promising approach to integrate private data into privacy-preserving, collaborative model training across data silos. This federated data access complicates important data-centric tasks such as estimating dataset diversity, performing informed data splits, and understanding the structure of the combined chemical space. To address this gap, we investigate how well federated clustering methods can disentangle and represent distributed molecular data. We benchmark three approaches, Federated kMeans (Fed-kMeans), Federated Principal Component Analysis combined with Fed-kMeans (Fed-PCA+Fed-kMeans), and Federated Locality-Sensitive Hashing (Fed-LSH), against their centralized counterparts on eight diverse molecular datasets. Our evaluation utilizes both, standard mathematical and a chemistry-informed evaluation metrics, SF-ICF, that we introduce in this work. The large-scale benchmarking combined with an in-depth explainability analysis shows the importance of incorporating domain knowledge through chemistry-informed metrics, and on-client explainability analyses for federated diversity analysis on molecular data.