This paper introduces an active learning framework for manifold Gaussian Process (GP) regression, combining manifold learning with strategic data selection to improve accuracy in high-dimensional spaces. Our method jointly optimizes a neural network for dimensionality reduction and a Gaussian process regressor in the latent space, supervised by an active learning criterion that minimizes global prediction error. Experiments on synthetic data demonstrate superior performance over randomly sequential learning. The framework efficiently handles complex, discontinuous functions while preserving computational tractability, offering practical value for scientific and engineering applications. Future work will focus on scalability and uncertainty-aware manifold learning.
Nonlinear networks are often multistable, exhibiting coexisting stable states with competing regions of attraction (ROAs). As a result, ROAs can have complex "tentacle-like" morphologies that are challenging to characterize analytically or computationally. In addition, the high dimensionality of the state space prohibits the automated construction of Lyapunov functions using state-of-the-art optimization methods, such as sum-of-squares (SOS) programming. In this letter, we propose a distributed approach for the construction of Lyapunov functions based solely on local information. To this end, we establish an augmented comparison lemma that characterizes the existence conditions of partial Lyapunov functions, while also accounting for residual effects caused by the associated dimensionality reduction. These theoretical results allow us to formulate an SOS optimization that iteratively constructs such partial functions, whose aggregation forms a composite Lyapunov function. The resulting composite function provides accurate convex approximations of both the volumes and shapes of the ROAs. We validate our method on networks of van der Pol and Ising oscillators, demonstrating its effectiveness in characterizing high-dimensional systems with non-convex ROAs.
Deep autoencoders have become a fundamental tool in various machine learning applications, ranging from dimensionality reduction and reduced order modeling of partial differential equations to anomaly detection and neural machine translation. Despite their empirical success, a solid theoretical foundation for their expressiveness remains elusive, particularly when compared to classical projection-based techniques. In this work, we aim to take a step forward in this direction by presenting a comprehensive analysis of what we refer to as symmetric autoencoders, a broad class of deep learning architectures ubiquitous in the literature. Specifically, we introduce a formal distinction between different classes of symmetric architectures, analyzing their strengths and limitations from a mathematical perspective. For instance, we show that the reconstruction error of symmetric autoencoders with orthonormality constraints can be understood by leveraging the well-renowned Eckart-Young-Schmidt (EYS) theorem. As a byproduct of our analysis, we end up developing the EYS initialization strategy for symmetric autoencoders, which is based on an iterated application of the Singular Value Decomposition (SVD). To validate our findings, we conduct a series of numerical experiments where we benchmark our proposal against conventional deep autoencoders, discussing the importance of model design and initialization.
Computational inverse problems for biomedical simulators suffer from limited data and relatively high parameter dimensionality. This often requires sensitivity analysis, where parameters of the model are ranked based on their influence on the specific quantities of interest. This is especially important for simulators used to build medical digital twins, as the amount of data is typically limited. For expensive models, such as blood flow models, emulation is employed to expedite the simulation time. Parameter ranking and fixing using sensitivity analysis are often heuristic, though, and vary with the specific application or simulator used. The present study provides an innovative solution to this problem by leveraging polynomial chaos expansions (PCEs) for both multioutput global sensitivity analysis and formal parameter identifiability. For the former, we use dimension reduction to efficiently quantify time-series sensitivity of a one-dimensional pulmonary hemodynamics model. We consider both Windkessel and structured tree boundary conditions. We then use PCEs to construct profile-likelihood confidence intervals to formally assess parameter identifiability, and show how changes in experimental design improve identifiability. Our work presents a novel approach to determining parameter identifiability and leverages a common emulation strategy for enabling profile-likelihood analysis in problems governed by partial differential equations.
We consider the problem of propagating the uncertainty from a possibly large number of random inputs through a computationally expensive model. Stratified sampling is a well-known variance reduction strategy, but its application, thus far, has focused on models with a limited number of inputs due to the challenges of creating uniform partitions in high dimensions. To overcome these challenges, we perform stratification with respect to the uniform distribution defined over the unit interval, and then derive the corresponding strata in the original space using nonlinear dimensionality reduction. We show that our approach is effective in high dimensions and can be used to further reduce the variance of multifidelity Monte Carlo estimators.
Gene selection in high-dimensional genomic data is essential for understanding disease mechanisms and improving therapeutic outcomes. Traditional feature selection methods effectively identify predictive genes but often ignore complex biological pathways and regulatory networks, leading to unstable and biologically irrelevant signatures. Prior approaches, such as Lasso-based methods and statistical filtering, either focus solely on individual gene-outcome associations or fail to capture pathway-level interactions, presenting a key challenge: how to integrate biological pathway knowledge while maintaining statistical rigor in gene selection? To address this gap, we propose a novel two-stage framework that integrates statistical selection with biological pathway knowledge using multi-agent reinforcement learning (MARL). First, we introduce a pathway-guided pre-filtering strategy that leverages multiple statistical methods alongside KEGG pathway information for initial dimensionality reduction. Next, for refined selection, we model genes as collaborative agents in a MARL framework, where each agent optimizes both predictive power and biological relevance. Our framework incorporates pathway knowledge through Graph Neural Network-based state representations, a reward mechanism combining prediction performance with gene centrality and pathway coverage, and collaborative learning strategies using shared memory and a centralized critic component. Extensive experiments on multiple gene expression datasets demonstrate that our approach significantly improves both prediction accuracy and biological interpretability compared to traditional methods.
Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction. Various methods have been proposed to extend PCA to the union of subspace (UoS) setting for clustering data that come from multiple subspaces like K-Subspaces (KSS). However, some applications involve heterogeneous data that vary in quality due to noise characteristics associated with each data sample. Heteroscedastic methods aim to deal with such mixed data quality. This paper develops a heteroscedastic-focused subspace clustering method, named ALPCAHUS, that can estimate the sample-wise noise variances and use this information to improve the estimate of the subspace bases associated with the low-rank structure of the data. This clustering algorithm builds on K-Subspaces (KSS) principles by extending the recently proposed heteroscedastic PCA method, named LR-ALPCAH, for clusters with heteroscedastic noise in the UoS setting. Simulations and real-data experiments show the effectiveness of accounting for data heteroscedasticity compared to existing clustering algorithms. Code available at https://github.com/javiersc1/ALPCAHUS.
Cross-modal embeddings form the foundation for multi-modal models. However, visualization methods for interpreting cross-modal embeddings have been primarily confined to traditional dimensionality reduction (DR) techniques like PCA and t-SNE. These DR methods primarily focus on feature distributions within a single modality, whilst failing to incorporate metrics (e.g., CLIPScore) across multiple modalities. This paper introduces AKRMap, a new DR technique designed to visualize cross-modal embeddings metric with enhanced accuracy by learning kernel regression of the metric landscape in the projection space. Specifically, AKRMap constructs a supervised projection network guided by a post-projection kernel regression loss, and employs adaptive generalized kernels that can be jointly optimized with the projection. This approach enables AKRMap to efficiently generate visualizations that capture complex metric distributions, while also supporting interactive features such as zoom and overlay for deeper exploration. Quantitative experiments demonstrate that AKRMap outperforms existing DR methods in generating more accurate and trustworthy visualizations. We further showcase the effectiveness of AKRMap in visualizing and comparing cross-modal embeddings for text-to-image models. Code and demo are available at https://github.com/yilinye/AKRMap.
Motivation: Externalizing behaviors in children, such as aggression, hyperactivity, and defiance, are influenced by complex interplays between genetic predispositions and environmental factors, particularly parental behaviors. Unraveling these intricate causal relationships can benefit from the use of robust data-driven methods. Methods: We developed a method called Hillclimb-Causal Inference, a causal discovery approach that integrates the Hill Climb Search algorithm with a customized Linear Gaussian Bayesian Information Criterion (BIC). This method was applied to data from the Adolescent Brain Cognitive Development (ABCD) Study, which included parental behavior assessments, children's genotypes, and externalizing behavior measures. We performed dimensionality reduction to address multicollinearity among parental behaviors and assessed children's genetic risk for externalizing disorders using polygenic risk scores (PRS), which were computed based on GWAS summary statistics from independent cohorts. Once the causal pathways were identified, we employed structural equation modeling (SEM) to quantify the relationships within the model. Results: We identified prominent causal pathways linking parental behaviors to children's externalizing outcomes. Parental alcohol misuse and broader behavioral issues exhibited notably stronger direct effects (0.33 and 0.20, respectively) compared to children's polygenic risk scores (0.07). Moreover, when considering both direct and indirect paths, parental substance misuse (alcohol, drug, and tobacco) collectively resulted in a total effect exceeding 1.1 on externalizing behaviors. Bootstrap and sensitivity analyses further validated the robustness of these findings.
Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data through learning Riemannian geometry algorithms can achieve improved performance and interpretability in tasks like clustering, dimensionality reduction, and interpolation. In particular, learned pullback geometry has recently undergone transformative developments that now make it scalable to learn and scalable to evaluate, which further opens the door for principled non-linear data analysis and interpretable machine learning. However, there are still steps to be taken when considering real-world multi-modal data. This work focuses on addressing distortions and modeling errors that can arise in the multi-modal setting and proposes to alleviate both challenges through isometrizing the learned Riemannian structure and balancing regularity and expressivity of the diffeomorphism parametrization. We showcase the effectiveness of the synergy of the proposed approaches in several numerical experiments with both synthetic and real data.
Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction. However, some applications involve heterogeneous data that vary in quality due to noise characteristics associated with each data sample. Heteroscedastic methods aim to deal with such mixed data quality. This paper develops a subspace learning method, named ALPCAH, that can estimate the sample-wise noise variances and use this information to improve the estimate of the subspace basis associated with the low-rank structure of the data. Our method makes no distributional assumptions of the low-rank component and does not assume that the noise variances are known. Further, this method uses a soft rank constraint that does not require subspace dimension to be known. Additionally, this paper develops a matrix factorized version of ALPCAH, named LR-ALPCAH, that is much faster and more memory efficient at the cost of requiring subspace dimension to be known or estimated. Simulations and real data experiments show the effectiveness of accounting for data heteroscedasticity compared to existing algorithms. Code available at https://github.com/javiersc1/ALPCAH.
One of the fundamental problems within the field of machine learning is dimensionality reduction. Dimensionality reduction methods make it possible to combat the so-called curse of dimensionality, visualize high-dimensional data and, in general, improve the efficiency of storing and processing large data sets. One of the best-known nonlinear dimensionality reduction methods is Diffusion Maps. However, despite their virtues, both Diffusion Maps and many other manifold learning methods based on the spectral decomposition of kernel matrices have drawbacks such as the inability to apply them to data outside the initial set, their computational complexity, and high memory costs for large data sets. In this work, we propose to alleviate these problems by resorting to deep learning. Specifically, a new formulation of Diffusion Maps embedding is offered as a solution to a certain unconstrained minimization problem and, based on it, a cost function to train a neural network which computes Diffusion Maps embedding -- both inside and outside the training sample -- without the need to perform any spectral decomposition. The capabilities of this approach are compared on different data sets, both real and synthetic, with those of Diffusion Maps and the Nystrom method.
Manifold learning aims to discover and represent low-dimensional structures underlying high-dimensional data while preserving critical topological and geometric properties. Existing methods often fail to capture local details with global topological integrity from noisy data or construct a balanced dimensionality reduction, resulting in distorted or fractured embeddings. We present an AutoEncoder-based method that integrates a manifold reconstruction layer, which uncovers latent manifold structures from noisy point clouds, and further provides regularizations on topological and geometric properties during dimensionality reduction, whereas the two components promote each other during training. Experiments on point cloud datasets demonstrate that our method outperforms baselines like t-SNE, UMAP, and Topological AutoEncoders in discovering manifold structures from noisy data and preserving them through dimensionality reduction, as validated by visualization and quantitative metrics. This work demonstrates the significance of combining manifold reconstruction with manifold learning to achieve reliable representation of the latent manifold, particularly when dealing with noisy real-world data. Code repository: https://github.com/Thanatorika/mrtg.
Phylogenetic profiles visualize the presence-absence pattern of genes across taxa and are essential for delineating the evolutionary fate of genes and gene families. Integrating phylogenetic profiles across many genes and taxa reveals patterns of coevolution, aiding the predictions of gene functions and interactions. The surge of genome sequences generated by biodiversity genomics projects allows to compile phylogenetic profiles at an unprecedented scale. PhyloProfile v2 was designed to cope with the novel challenges of visualizing and analyzing phylogenetic profiles comprising millions of pairwise orthology relationships. By providing the ability to interact with the visualization and dynamically filter the data, PhyloProfile v2 facilitates a seamless transition from survey analyses across thousands of genes and taxa down to the feature architecture comparison of two ortholog pairs within the same analysis. As one key innovation, PhyloProfile v2 allows the display of phylogenetic profiles in 2D or 3D using dimensionality reduction techniques. This novel perspective eases, for example, the identification of taxa with similar presence/absence patterns of genes irrespective of their phylogenetic relationships. PhyloProfile v2 is available as an R package at Bioconductor https://doi.org/doi:10.18129/B9.bioc.PhyloProfile. The open-source code and documentation are provided under MIT license at https://github.com/BIONF/PhyloProfile
This study proposes a computationally efficient method for optimizing multi-zone thermostatically controlled loads (TCLs) by leveraging dimensionality reduction through an auto-encoder. We develop a multi-task learning framework to jointly represent latent variables and formulate a state-space model based on observed TCL operation data. This significantly reduces the dimensionality of TCL variables and states while preserving critical nonlinear interdependencies in TCL control. To address various application scenarios, we introduce optimization algorithms based on system identification (OptIden) and system simulation (OptSim) tailored to the latent variable representation. These approaches employ automatic differentiation and zeroth-order techniques, respectively, for efficient implementation. We evaluate the proposed method using a 90-zone apartment prototype, comparing its performance to traditional high-dimensional optimization. Results demonstrate that our approach effectively reduces control costs while achieving significantly higher computational efficiency.
Solar flares are among the most powerful and dynamic events in the solar system, resulting from the sudden release of magnetic energy stored in the Sun's atmosphere. These energetic bursts of electromagnetic radiation can release up to 10^32 erg of energy, impacting space weather and posing risks to technological infrastructure and therefore require accurate forecasting of solar flare occurrences and intensities. This study evaluates the predictive performance of three machine learning algorithms: Random Forest, k-Nearest Neighbors (KNN), and Extreme Gradient Boosting (XGBoost) for classifying solar flares into 4 categories (B, C, M, X). Using the dataset of 13 SHARP parameters, the effectiveness of the models was evaluated in binary and multiclass classification tasks. The analysis utilized 8 principal components (PC), capturing 95% of data variance, and 100 PCs, capturing 97.5% of variance. Our approach uniquely combines binary and multiclass classification with different levels of dimensionality reduction, an innovative methodology not previously explored in the context of solar flare prediction. Employing a 10-fold stratified cross-validation and grid search for hyperparameter tuning ensured robust model evaluation. Our findings indicate that Random Forest and XGBoost consistently demonstrate strong performance across all metrics, benefiting significantly from increased dimensionality. The insights of this study enhance future research by optimizing dimensionality reduction techniques and informing model selection for astrophysical tasks. By integrating this newly acquired knowledge into future research, more accurate space weather forecasting systems can be developed, along with a deeper understanding of solar physics.
We propose a generalization of modern representation learning objectives by reframing them as recursive divergence alignment processes over localized conditional distributions While recent frameworks like Information Contrastive Learning I-Con unify multiple learning paradigms through KL divergence between fixed neighborhood conditionals we argue this view underplays a crucial recursive structure inherent in the learning process. We introduce Recursive KL Divergence Optimization RKDO a dynamic formalism where representation learning is framed as the evolution of KL divergences across data neighborhoods. This formulation captures contrastive clustering and dimensionality reduction methods as static slices while offering a new path to model stability and local adaptation. Our experiments demonstrate that RKDO offers dual efficiency advantages approximately 30 percent lower loss values compared to static approaches across three different datasets and 60 to 80 percent reduction in computational resources needed to achieve comparable results. This suggests that RKDOs recursive updating mechanism provides a fundamentally more efficient optimization landscape for representation learning with significant implications for resource constrained applications.
Dimensionality reduction techniques are fundamental for analyzing and visualizing high-dimensional data. With established methods like t-SNE and PCA presenting a trade-off between representational power and interpretability. This paper introduces a novel approach that bridges this gap by combining the interpretability of linear methods with the expressiveness of non-linear transformations. The proposed algorithm constructs a non-linear mapping between high-dimensional and low-dimensional spaces through a combination of linear transformations, each weighted by Gaussian functions. This architecture enables complex non-linear transformations while preserving the interpretability advantages of linear methods, as each transformation can be analyzed independently. The resulting model provides both powerful dimensionality reduction and transparent insights into the transformed space. Techniques for interpreting the learned transformations are presented, including methods for identifying suppressed dimensions and how space is expanded and contracted. These tools enable practitioners to understand how the algorithm preserves and modifies geometric relationships during dimensionality reduction. To ensure the practical utility of this algorithm, the creation of user-friendly software packages is emphasized, facilitating its adoption in both academia and industry.
Background: The integration and analysis of multi-modal data are increasingly essential across various domains including bioinformatics. As the volume and complexity of such data grow, there is a pressing need for computational models that not only integrate diverse modalities but also leverage their complementary information to improve clustering accuracy and insights, especially when dealing with partial observations with missing data. Results: We propose Generalized Probabilistic Canonical Correlation Analysis (GPCCA), an unsupervised method for the integration and joint dimensionality reduction of multi-modal data. GPCCA addresses key challenges in multi-modal data analysis by handling missing values within the model, enabling the integration of more than two modalities, and identifying informative features while accounting for correlations within individual modalities. The model demonstrates robustness to various missing data patterns and provides low-dimensional embeddings that facilitate downstream clustering and analysis. In a range of simulation settings, GPCCA outperforms existing methods in capturing essential patterns across modalities. Additionally, we demonstrate its applicability to multi-omics data from TCGA cancer datasets and a multi-view image dataset. Conclusion: GPCCA offers a useful framework for multi-modal data integration, effectively handling missing data and providing informative low-dimensional embeddings. Its performance across cancer genomics and multi-view image data highlights its robustness and potential for broad application. To make the method accessible to the wider research community, we have released an R package, GPCCA, which is available at https://github.com/Kaversoniano/GPCCA.
Speech foundation models have significantly advanced various speech-related tasks by providing exceptional representation capabilities. However, their high-dimensional output features often create a mismatch with downstream task models, which typically require lower-dimensional inputs. A common solution is to apply a dimensionality reduction (DR) layer, but this approach increases parameter overhead, computational costs, and risks losing valuable information. To address these issues, we propose Nested Res2Net (Nes2Net), a lightweight back-end architecture designed to directly process high-dimensional features without DR layers. The nested structure enhances multi-scale feature extraction, improves feature interaction, and preserves high-dimensional information. We first validate Nes2Net on CtrSVDD, a singing voice deepfake detection dataset, and report a 22% performance improvement and an 87% back-end computational cost reduction over the state-of-the-art baseline. Additionally, extensive testing across four diverse datasets: ASVspoof 2021, ASVspoof 5, PartialSpoof, and In-the-Wild, covering fully spoofed speech, adversarial attacks, partial spoofing, and real-world scenarios, consistently highlights Nes2Net's superior robustness and generalization capabilities. The code package and pre-trained models are available at https://github.com/Liu-Tianchi/Nes2Net.