The DOI matches its Crossref record; 1 of the 1 references it cites were found in a registry or an index.
Laplacian Eigenmaps for Dimensionality Reduction and Data Representation
Abstract
One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low-dimensional manifold embedded in a high-dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high-dimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
A Global Geometric Framework for Nonlinear Dimensionality Reduction
Tenenbaum, J. B., Silva, V. D., & Langford, J. C. (2000) · Science · doi:10.1126/science.290.5500.2319 · Cited by 13,585
Cited by
Representation Learning: A Review and New Perspectives
Bengio, Y., Courville, A., & Vincent, P. (2013) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2013.50 · Cited by 13,001
UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction
McInnes, L., Healy, J., & Melville, J. (2018) · arXiv · doi:10.48550/arxiv.1802.03426 · Cited by 7,101
Dictionary learning for integrative, multimodal and scalable single-cell analysis
Hao, Y., Stuart, T., Kowalski, M. H., Choudhary, S., Hoffman, P., Hartman, A., Srivastava, A., Molla, G., Madad, S., Fernandez-Granda, C., et al. (2024) · Nature Biotechnology · doi:10.1038/s41587-023-01767-y · Cited by 5,631
Geometric Deep Learning: Going beyond Euclidean data
Bronstein, M. M., Bruna, J., LeCun, Y., Szlam, A., & Vandergheynst, P. (2017) · IEEE Signal Processing Magazine · doi:10.1109/msp.2017.2693418 · Cited by 3,661
(2006) · Cited by 3,624
Slingshot: cell lineage and pseudotime inference for single-cell transcriptomics
Street, K., Risso, D., Fletcher, R. B., Das, D., Ngai, J., Yosef, N., Purdom, E., & Dudoit, S. (2018) · BMC Genomics · doi:10.1186/s12864-018-4772-0 · Cited by 3,159
Structural Deep Network Embedding
Wang, D., Cui, P., & Zhu, W. (2016) · Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining · doi:10.1145/2939672.2939753 · Cited by 2,845
Unsupervised Learning of Visual Representations by Solving Jigsaw Puzzles
Noroozi, M., & Favaro, P. (2016) · Lecture Notes in Computer Science · doi:10.1007/978-3-319-46466-4_5 · Cited by 2,775
Coifman, R. R., Lafon, S., Lee, A. B., Maggioni, M., Nadler, B., Warner, F., & Zucker, S. W. (2005) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0500334102 · Cited by 1,675
Discrete Signal Processing on Graphs
Sandryhaila, A., & Moura, J. M. F. (2013) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2013.2238935 · Cited by 1,531
Generative Modeling by Estimating Gradients of the Data Distribution
Song, Y., & Ermon, S. (2019) · arXiv · doi:10.48550/arxiv.1907.05600 · Cited by 1,453
Fouss, F., Pirotte, A., Renders, J.-M., & Saerens, M. (2007) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/tkde.2007.46 · Cited by 1,265
Cui, P., Wang, X., Pei, J., & Zhu, W. (2019) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/tkde.2018.2849727 · Cited by 1,274
DISFA: A Spontaneous Facial Action Intensity Database
Mavadati, S. M., Mahoor, M. H., Bartlett, K., Trinh, P., & Cohn, J. F. (2013) · IEEE Transactions on Affective Computing · doi:10.1109/t-affc.2013.4 · Cited by 793
Variations and extension of the convex–concave procedure
Lipp, T., & Boyd, S. (2016) · Optimization and Engineering · doi:10.1007/s11081-015-9294-x · Cited by 744
A general and flexible method for signal extraction from single-cell RNA-seq data
Risso, D., Perraudeau, F., Gribkova, S., Dudoit, S., & Vert, J.-P. (2018) · Nature Communications · doi:10.1038/s41467-017-02554-5 · Cited by 718
Criminisi, A., Shotton, J., & Konukoglu, E. (2012) · Foundations and Trends® in Computer Graphics and Vision · doi:10.1561/0600000035 · Cited by 677
Machine learning in acoustics: Theory and applications
Bianco, M. J., Gerstoft, P., Traer, J., Ozanich, E., Roch, M. A., Gannot, S., & Deledalle, C.-A. (2019) · The Journal of the Acoustical Society of America · doi:10.1121/1.5133944 · Cited by 609
Consistency of spectral clustering
von Luxburg, U., Belkin, M., & Bousquet, O. (2008) · The Annals of Statistics · doi:10.1214/009053607000000640 · Cited by 540
Self-citationCoarse master equations for peptide folding dynamics
Buchete, N.-V., & Hummer, G. (2008) · The Journal of Physical Chemistry B · doi:10.1021/jp0761665 · Cited by 477
Lin, T., & Zha, H. (2008) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2007.70735 · Cited by 446
Methodologies for Cross-Domain Data Fusion: An Overview
Zheng, Y. (2015) · IEEE Transactions on Big Data · doi:10.1109/tbdata.2015.2465959 · Cited by 454
Visualizing high-dimensional data: advances in the past decade
Liu, S., Maljovec, D., Wang, B., Bremer, P.-T., & Pascucci, V. (2017) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2016.2640960 · Cited by 427
What are higher-order networks?
Bick, C., Gross, E., Harrington, H. A., & Schaub, M. T. (2023) · SIAM Review · doi:10.1137/21m1414024 · Cited by 420
A Generalization of Transformer Networks to Graphs
Dwivedi, V. P., & Bresson, X. (2020) · arXiv · doi:10.48550/arxiv.2012.09699 · Cited by 398
Simplifying the representation of complex free-energy landscapes using sketch-map
Ceriotti, M., Tribello, G. A., & Parrinello, M. (2011) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1108486108 · Cited by 340
Das, P., Moll, M., Stamati, H., Kavraki, L. E., & Clementi, C. (2006) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0603553103 · Cited by 331
Data Fusion and Multicue Data Matching by Diffusion Maps
Lafon, S., Keller, Y., & Coifman, R. R. (2006) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2006.223 · Cited by 312
Tutorial on practical tips of the most influential data preprocessing algorithms in data mining
García, S., Luengo, J., & Herrera, F. (2016) · Knowledge-Based Systems · doi:10.1016/j.knosys.2015.12.006 · Cited by 310
Causes and consequences of representational drift
Rule, M. E., O’Leary, T., & Harvey, C. D. (2019) · Current Opinion in Neurobiology · doi:10.1016/j.conb.2019.08.005 · Cited by 227
Bronstein, A. M., Bronstein, M. M., Kimmel, R., Mahmoudi, M., & Sapiro, G. (2010) · International Journal of Computer Vision · doi:10.1007/s11263-009-0301-6 · Cited by 267
Zhi, R., Flierl, M., Ruan, Q., & Kleijn, W. B. (2011) · IEEE Transactions on Systems Man and Cybernetics Part B (Cybernetics) · doi:10.1109/tsmcb.2010.2044788 · Cited by 260
Nonverbal social withdrawal in depression: Evidence from manual and automatic analyses
Girard, J. M., Cohn, J. F., Mahoor, M. H., Mavadati, S. M., Hammal, Z., & Rosenwald, D. P. (2014) · Image and Vision Computing · doi:10.1016/j.imavis.2013.12.007 · Cited by 255
Integrating Machine Learning with Human Knowledge
Deng, C., Ji, X., Rainey, C., Zhang, J., & Lu, W. (2020) · iScience · doi:10.1016/j.isci.2020.101656 · Cited by 243
Using sketch-map coordinates to analyze and bias molecular dynamics simulations
Tribello, G. A., Ceriotti, M., & Parrinello, M. (2012) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1201152109 · Cited by 202
A fast, scalable and versatile tool for analysis of single-cell omics data
Zhang, K., Zemke, N. R., Armand, E. J., & Ren, B. (2024) · Nature Methods · doi:10.1038/s41592-023-02139-9 · Cited by 185
Systematic determination of order parameters for chain dynamics using diffusion maps
Ferguson, A. L., Panagiotopoulos, A. Z., Debenedetti, P. G., & Kevrekidis, I. G. (2010) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1003293107 · Cited by 165
Li, Z., Yan, X., Tian, Z., Yuan, C., Peng, Z., & Li, L. (2013) · Measurement · doi:10.1016/j.measurement.2012.06.013 · Cited by 154
Generalized extreme learning machine autoencoder and a new deep neural network
Sun, K., Zhang, J., Zhang, C., & Hu, J. (2017) · Neurocomputing · doi:10.1016/j.neucom.2016.12.027 · Cited by 154
Classification of Hyperspectral Images by Gabor Filtering Based Deep Network
Kang, X., Li, C., Li, S., & Lin, H. (2018) · IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing · doi:10.1109/jstars.2017.2767185 · Cited by 151
Variational autoencoders for cancer data integration: design principles and computational practice
Simidjievski, N., Bodnar, C., Tariq, I., Scherer, P., Andres Terre, H., Shams, Z., Jamnik, M., & Liò, P. (2019) · Frontiers in Genetics · doi:10.3389/fgene.2019.01205 · Cited by 145
Topological data analysis of contagion maps for examining spreading processes on networks
Taylor, D., Klimm, F., Harrington, H. A., Kramár, M., Mischaikow, K., Porter, M. A., & Mucha, P. J. (2015) · Nature Communications · doi:10.1038/ncomms8723 · Cited by 148
Demonstrating the Transferability and the Descriptive Power of Sketch-Map
Ceriotti, M., Tribello, G. A., & Parrinello, M. (2013) · Journal of Chemical Theory and Computation · doi:10.1021/ct3010563 · Cited by 126
Decoding lifespan changes of the human brain using resting-state functional connectivity MRI
Wang, L., Su, L., Shen, H., & Hu, D. (2012) · PLoS ONE · doi:10.1371/journal.pone.0044530 · Cited by 115
Learning to Hash With Optimized Anchor Embedding for Scalable Retrieval
Guo, Y., Ding, G., Liu, L., Han, J., & Shao, L. (2017) · IEEE Transactions on Image Processing · doi:10.1109/tip.2017.2652730 · Cited by 108
Learning in High Dimension Always Amounts to Extrapolation
Balestriero, R., Pesenti, J., & LeCun, Y. (2021) · arXiv · doi:10.48550/arxiv.2110.09485 · Cited by 93
Multiscale PHATE identifies multimodal signatures of COVID-19
Kuchroo, M., Huang, J., Wong, P., Grenier, J.-C., Shung, D., Tong, A., Lucas, C., Klein, J., Burkhardt, D. B., Gigante, S., et al. (2022) · Nature Biotechnology · doi:10.1038/s41587-021-01186-x · Cited by 72
Computational methods for single-cell multi-omics integration and alignment
Stanojevic, S., Li, Y., Ristivojevic, A., & Garmire, L. X. (2022) · Genomics, Proteomics & Bioinformatics · doi:10.1016/j.gpb.2022.11.013 · Cited by 74
Nonlinear dimensionality reduction and mapping of compound libraries for drug discovery
Reutlinger, M., & Schneider, G. (2012) · Journal of Molecular Graphics and Modelling · doi:10.1016/j.jmgm.2011.12.006 · Cited by 72
Machine Learning and Imaging Informatics in Oncology
Tseng, H.-H., Wei, L., Cui, S., Luo, Y., Ten Haken, R. K., & El Naqa, I. (2020) · Oncology · doi:10.1159/000493575 · Cited by 64
On the Intrinsic Dimensionality of Image Representations
Gong, S., Boddeti, V. N., & Jain, A. K. (2019) · 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) · doi:10.1109/cvpr.2019.00411 · Cited by 59
(2017) · doi:10.3389/fncom.2016.00131 · Cited by 59
Multi-Modal Medical Image Registration with Full or Partial Data: A Manifold Learning Approach
Bashiri, F. S., Baghaie, A., Rostami, R., Yu, Z., & D’Souza, R. M. (2018) · Journal of Imaging · doi:10.3390/jimaging5010005 · Cited by 46
Estimating smile intensity: A better way
Girard, J. M., Cohn, J. F., & De la Torre, F. (2015) · Pattern Recognition Letters · doi:10.1016/j.patrec.2014.10.004 · Cited by 44
Mansbach, R. A., & Ferguson, A. L. (2015) · The Journal of Chemical Physics · doi:10.1063/1.4914144 · Cited by 39
A unified probabilistic framework for measuring the intensity of spontaneous facial action units
Li, Y., Mavadati, S. M., Mahoor, M. H., & Ji, Q. (2013) · 2013 10th IEEE International Conference and Workshops on Automatic Face and Gesture Recognition (FG) · doi:10.1109/fg.2013.6553757 · Cited by 36
Magnetic Eigenmaps for the visualization of directed networks
Fanuel, M., Alaíz, C. M., Fernández, Á., & Suykens, J. A. K. (2018) · Applied and Computational Harmonic Analysis · doi:10.1016/j.acha.2017.01.004 · Cited by 35
Learning Weighted Submanifolds With Variational Autoencoders and Riemannian Variational Autoencoders
Miolane, N., & Holmes, S. (2020) · 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) · doi:10.1109/cvpr42600.2020.01451 · Cited by 16
Mining Strengths and Weaknesses of Cricket Players Using Short Text Commentary
Behera, S. R., Agrawal, P., Awekar, A., & Vedula, V. S. (2019) · 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA) · doi:10.1109/icmla.2019.00122 · Cited by 11
Cui, D., & Xia, K. (2018) · Arabian Journal for Science and Engineering · doi:10.1007/s13369-017-2825-3 · Cited by 8
Metadata
- Authors
- Belkin, M., & Niyogi, P.
- Venue
- Neural Computation
- Publisher
- MIT Press - Journals
- Published
- 2003-06-01
- Volume
- 15
- Issue
- 6
- Pages
- 1373-1396
- ISSN
- 0899-7667 (ISSN-L)
- 1530-888X
- OpenAlex
- W2097308346
- Cited by
- 7,621
- References
- 1
- FWCI
- 97.98
- Type
- journal article
- Language
- en
- Open access
- closed
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
