Manifold Learning

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Manifold learning seeks compact representations of high-dimensional data by modeling it as lying near a lower-dimensional, possibly curved space, enabling visualization, denoising, scientific analysis, and learning with fewer effective degrees of freedom. Common approaches build neighborhood graphs and optimize UMAP- or t-SNE-style embeddings, recover coordinates or operators with spectral and diffusion methods, or use autoencoders with geometric constraints; Riemannian and optimal-transport methods handle structured objects such as covariance matrices and probability distributions. Work increasingly targets properties beyond nearest-neighbor preservation—including density, geodesic distances, topology, tangent spaces, symmetry, and out-of-sample mapping—in settings from images and single-cell data to networks and dynamical systems. A central challenge is balancing local fidelity against global geometry and computational scale, while noise, sparse or anisotropic sampling, and inconsistent evaluation metrics make it difficult to tell when an embedding reflects the underlying manifold.

Latest papers

115 papers10 papers in the last four weeks, up 43% on the four weeks before. 0.1% of all new papers.

Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine
May 16, 2026Wei Huang, Andi Han, Mingyuan Bai +4Latent Diffusion Models Manifold Learning
RIKEN AIP & The Institute of Statistical Mathematics · University of Sydney · Agency for Science, Technology and Research & The Institute of Statistical Mathematics +4
Topo-GS: Continuous Volumetric Embedding of High-Dimensional Data via Topological Gaussian Splatting
May 16, 2026João Paulo Gois, Luis Gustavo Nonato3D Gaussian Splatting Manifold Learning
Universidade Federal do ABC (UFABC), Santo André, Brazil · Instituto de Ciências Matemáticas e de Computação (ICMC), Universidade de São Paulo (USP), São Carlos, Brazil
Neural Point-Forms
May 15, 2026Bruno Trentini, Jacob Hume, Vincenzo Antonio Isoldi +3Point Cloud Learning Geometric Representation Learning
University of Oxford Oxford, UK · School of Engineering and Applied Sciences Harvard University Cambridge, MA, US · Max Planck Institute of Molecular Cell Biology and Genetics Dresden, Germany
Beyond Explained Variance: A Cautionary Tale of PCA
May 13, 2026Gionni MarchettiData Visualization Principal Component Analysis
Barcelona, Spain
FastUMAP: Scalable Dimensionality Reduction via Bipartite Landmark Sampling
May 12, 2026Hongmin LiUnsupervised Learning Nyström Method
School of Life Science and Technology, Institute of Science Tokyo · 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan 2Department of Computational Biology and Medical Sciences Graduate School of Frontier Sciences, The University of Tokyo
Learning Point Cloud Geometry as a Statistical Manifold: Theory and Practice
May 11, 2026Jinwoo Lee, Jiwoo Kim, Woojae Shin +2Geometric Representation Learning Manifold Learning
Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea · Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu, Republic of Korea
SMIXAE: Towards Unsupervised Manifold Discovery in Language Models
May 9, 2026Collin FrancelAutoencoders LLM Interpretability
Department of Arts and Sciences, University of Alabama, Tuscaloosa, AL, United States.
Bilinear autoencoders find interpretable manifolds
May 9, 2026Thomas Dooms, Ward Gauderis, Geraint Wiggins +1Neural Representation Geometry Autoencoders
University of Antwerp · Vrije Universiteit Brussel · University of Antwerp, sqIRL
Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves
May 7, 2026Kartik Tandon, Julian Gould, Tanishq Bhatia +3Graph Neural Networks Geometric Representation Learning
University of Pennsylvania · Sakana AI · Northeastern University +1
Beyond Rigid Alignment: Graph Federated Learning via Dual Manifold Calibration
May 7, 2026Wentao Yu, Bo Han, Jie Yang +1Federated Graph Learning Manifold Learning
School of Computer Science and Engineering Nanjing University of Science and Technology Nanjing, China · Department of Computer Science Hong Kong Baptist University Hong Kong, China · School of Automation and Intelligent Sensing Shanghai Jiao Tong University Shanghai, China
Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior
May 6, 2026Daniel Wurgaft, Can Rager, Matthew Kowal +13Neural Representation Geometry Manifold Learning
Stanford University · University College London · Northeastern University +2
$p$ -adic Manifold Learning and Benchmark Tasks from Impartial Games
May 6, 2026Tomoki MiharaManifold Learning
Complex Diffusion Maps with $ω$ -Parameterized Kernels Revealing Inherent Harmonic Representations
May 3, 2026Tongzhen Dang, Weiyang Ding, Michael K. NgManifold Learning Kernel Methods
Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai, China. · Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai, China; Key Laboratory of Computational Neuroscience and Brain-Inspired Intelligence (Fudan University), Ministry of Education, China. · Department of Mathematics, Hong Kong Baptist University, Hong Kong, China.
Controlled Paraphrase Geometry in Sentence Embedding Space: Local Manifold Modeling and Latent Probing
May 1, 2026Leonid BedratyukRepresentation Learning Neural Representation Geometry
Do Sparse Autoencoders Capture Concept Manifolds?
Apr 30, 2026Usha Bhalla, Thomas Fel, Can Rager +9Neural Representation Geometry Sparse Autoencoders
aHarvard University · bNortheastern University · cTechnion IIT +1
High-Dimensional Noise to Low-Dimensional Manifolds: A Manifold-Space Diffusion Framework for Degraded Hyperspectral Image Classification
Apr 29, 2026Boxiang Yang, Ning Chen, Xia Yue +6Manifold Learning Hyperspectral Image Classification
Institute of Remote Sensing and Geographic Information System, Peking University, Beijing 100871, China · Beijing Key Laboratory of Spatio-temporal Perception and Urban Resilience, Peking University, Beijing 100871, China · School of Computer Science and Engineering, Central South University, Changsha 410083, China +4
DiRe-RAPIDS: Topology-faithful dimensionality reduction at scale
Apr 28, 2026Alexander Kolpakov, Igor RivinData Visualization Manifold Learning
CSTEM, University of Austin, Austin, TX, USA · Department of Mathematics, Temple University, Philadelphia, PA, USA
Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning
Apr 22, 2026Yuhan Peng, Junwen Dong, Yuzhi Zeng +6Graph Neural Networks Geometric Representation Learning
School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore · Chern Institute of Mathematics, Nankai University, Tianjin, China · Department of Mathematics, Faculty of Science, National University of Singapore, Singapore +3
Towards Modality-Agnostic Medical Image Anomaly Detection: A Training-Free Manifold Refinement Approach
Apr 21, 2026Pritam Kar, Gouri Lakshmi S, Saptarshi BejMedical Image Anomaly Detection Manifold Learning
aSchool of Data Science, Indian Institute of Science Education and Research, Thiruvananthapuram, 695551, Kerala, India
S2MAM: Semi-supervised Meta Additive Model for Robust Estimation and Variable Selection
Apr 21, 2026Xuelin Zhang, Hong Chen, Yingjie Wang +2Manifold Learning Feature Selection
Huazhong Agricultural University · China University of Petroleum (East China) · Xi’an Jiaotong University +1
Curvature-Aware PCA with Geodesic Tangent Space Aggregation for Semi-Supervised Learning
Apr 20, 2026Alexandre L. M. LevadaPrincipal Component Analysis Manifold Learning
Computing Department Federal University of São Carlos 13565-905, São Carlos-SP, Brazil
Characterizing AlphaEarth Embedding Geometry for Agentic Environmental Reasoning
Apr 20, 2026Mashrekur Rahman, Samuel J. Barrett, Christina LastGeospatial Foundation Models Geospatial Reasoning
Dartmouth Libraries, Dartmouth College, Hanover, 03755, NH, USA. · LGND AI, Canarias, Spain. · TipplyAI, London, UK.
Neighbor Embedding for High-Dimensional Sparse Poisson Data
Apr 18, 2026Noga Mudrik, Adam S. CharlesManifold Learning Dimensionality Reduction
Department of Biomedical Engineering Center for Imaging Science Kavli Neuroscience Discovery Institute Johns Hopkins University Baltimore, MD 21218, USA
Geometric regularization of autoencoders via observed stochastic dynamics
Apr 17, 2026Sean Hill, Felix X. -F. YeLatent Dynamics Modeling Autoencoders
Department of Mathematics and Statistics, University at Albany, SUNY
Complex Interpolation of Matrices with an application to Multi-Manifold Learning
Apr 15, 2026Adi Arbel, Stefan Steinerberger, Ronen TalmonMulti-View Learning Manifold Learning
Viterbi Faculty of Electrical and Computer Engineering, Technion – Israel Institute of Technology, Haifa, Israel · Department of Mathematics and Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
EigenCoin: sassanid coins classification based on Bhattacharyya distance
Apr 13, 2026Rahele Allahverdi, Mohammad Mahdi Dehshibi, Azam Bastanfard +1Image Classification Class-Imbalanced Learning
Department of Electrical, Computer and IT, Islamic Azad University, Qazvin Branch, Iran · Department of Computer Engineering, Islamic Azad University, Karaj Branch, Iran · National Museum of Iran, Tehran, Iran
MPNet: A Robust and Efficient Manifold Pooling Network for Multi-Rhythm EEG Signal Decoding
Apr 11, 2026Guoqing Cai, Kai Zeng, Shoulin Huang +1Riemannian Neural Networks EEG Decoding
Harbin Institute of Technology (Shenzhen), Shenzhen, China · Guangxi Normal University, Guilin, China · Peng Cheng Laboratory, Shenzhen, China
Diffusion Processes on Implicit Manifolds
Apr 8, 2026Victor Kawasaki-Borruat, Clara Grotehans, Pierre Vandergheynst +1Stochastic Differential Equations Manifold Learning
Signal Processing Laboratory 2 EPFL · Institute of Artificial Intelligence Medical University of Vienna
Energy-based Tissue Manifolds for Longitudinal Multiparametric MRI Analysis
Apr 8, 2026Kartikay Tehlan, Lukas Förner, Sina Wendrich +5Longitudinal Medical Image Analysis Magnetic Resonance Imaging
Dept. of diagnostic and interventional Radiology and Neuroradiology, University Hospital Augsburg, Germany · Digital Medicine, University Hospital Augsburg, Germany · Chair for Computer Aided Medical Procedures and Augmented Reality, Technical University of Munich, Germany +3
Bi-Lipschitz Autoencoder With Injectivity Guarantee
Apr 8, 2026Qipeng Zhan, Zhuoping Zhou, Zexuan Wang +2Autoencoders Representation Learning
University of Pennsylvania

May 16, 2026J/K move · Enter open · S save

Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine

Wei Huang, Andi Han, Mingyuan Bai, Huanjian Zhou+3

RIKEN AIP & The Institute of Statistical Mathematics · University of Sydney · Agency for Science, Technology and Research & The Institute of Statistical Mathematics+4

Latent Diffusion Models Manifold Learning Score-Based Generative Modeling Diffusion Models

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Diffusion models generate high-dimensional data with remarkable quality, yet how their training efficiently learns the score function, bypassing the curse of dimensionality when data is supported on low-dimensional manifolds, remains theoretically unexplained. We identify a collapse-and-refine mechanism driven by the geometry of the score function itself: at small noise scales, the diverging singularity of the score drives a rapid dimensional collapse of the induced denoising map onto the data manifold projection; at moderate noise scales, training refines the intrinsic density on the learned manifold. We instantiate this principle as Score-induced Latent Diffusion (SiLD), a two-stage framework in which both manifold learning and density estimation emerge from a single denoising score matching objective, replacing the heuristic KL regularization of VAE-based latent diffusion models. We prove that the resulting sample complexity depends on the intrinsic dimension rather than the ambient dimension. Experiments on Stacked MNIST, CelebA variants, and molecular generation benchmarks show that SiLD matches or outperforms VAE-based LDMs in generation quality and consistently improves reconstruction, validating our theoretical predictions.