580
Views
4
CrossRef citations to date
0
Altmetric
Theory and Methods

Optimal Design of Experiments on Riemannian Manifolds

& ORCID Icon
Pages 875-886 | Received 04 Feb 2021, Accepted 05 Nov 2022, Published online: 12 Dec 2022

References

  • Alaeddini, A., Craft, E., Meka, R., and Martinez, S. (2019), “Sequential Laplacian Regularized v-optimal Design of Experiments for Response Surface Modeling of Expensive Tests: An Application in Wind Tunnel Testing,” IISE Transactions, 51, 559–576. DOI: 10.1080/24725854.2018.1508928.
  • Aronszajn, N. (1950), “Theory of Reproducing Kernels,” Transactions of the American Mathematical Society, 68, 337–404. DOI: 10.1090/S0002-9947-1950-0051437-7.
  • Belkin, M. (2003), “Problems of Learning on Manifolds,” Ph.D. thesis, The University of Chicago.
  • Belkin, M., and Niyogi, P. (2005), “Towards a Theoretical Foundation for Laplacian-based Manifold Methods,” in Proceedings of Conference on Learning Theory.
  • Belkin, M., Niyogi, P., and Sindhwani, V. (2006), “Manifold Regularization: A Geometric Framework for Learning from Labeled and Unlabeled Examples,” Journal of Machine Learning Research, 7, 2399–2434.
  • Box, G., and Draper, N. (2007), Response Surfaces, Mixtures, and Ridge Analyses. Wiley Series in Probability and Statistics, Hoboken, NJ: Wiley.
  • Cai, D., and He, X. (2012), “Manifold Adaptive Experimental Design for Text Categorization,” IEEE Transactions on Knowledge and Data Engineering, 24, 707–719. DOI: 10.1109/TKDE.2011.104.
  • Chen, C., Chen, Z., Bu, J., Wang, C., Zhang, L., and Zhang, C. (2010), “G-optimal Design with Laplacian Regularization,” Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence, 1, 413–418. DOI: 10.1609/aaai.v24i1.7672.
  • Cheng, M., and Wu, H. (2013), “Local Linear Regression on Manifolds and its Geometric Interpretation,” Journal of the American Statistical Association, 108, 1421–1434. DOI: 10.1080/01621459.2013.827984.
  • Coifman, R., Lafon, S., Lee, A., Maggioni, M., Nadler, B., Warner, F., and Zuker, S. (2005), “Geometric Diffusions as a Tool for Harmonic Analysis and Structure Definition of Data: Diffusion Maps,” Proceedings of the National Academy of Sciences, 102, 7426–7431. DOI: 10.1073/pnas.0500334102.
  • Donoho, D., and Grimes, C. (2003), “Hessian Eigenmaps: Locally Linear Embedding Techniques for High Dimensional Data,” Proceedings of the National Academy of Sciences, 100, 5591–5596. DOI: 10.1073/pnas.1031596100.
  • Ettinger, B., Sangalli, L. M., and Perotto, S. (2016), “Spatial Regression Models over Two-Dimensional Manifolds,” Biometrika, 103, 71–88. DOI: 10.1093/biomet/asv069.
  • Fang, K., Li, R., and Sudjianto, A. (2006), Design and Modeling for Computer Experiments. Computer Sicence and Data Analysis Series, Boca Raton, FL: Chapman and Hall/CRC.
  • Fedorov, V. V. (1972), Theory of Optimal Experiments. New York: Academic Press.
  • Federov, V. V., and Hackle, P. (1997), Model-Oriented Design of Experiments, New York: Springer.
  • Fedorov, V. V., and Leonov, S. (2013a), Optimal Design for Nonlinear Response Models, Boca Raton, FL: CRC Press.
  • Fedorov, V. V., and Leonov, S. (2013b), Optimal Design for Nonlinear Response Models, Boca Raton, FL: CRC Press.
  • Gray, A., Abbena, E., and Salamon, S. (2006), Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition (Studies in Advanced Mathematics), Boca Raton, FL: Chapman & Hall/CRC.
  • He, X. (2010), “Laplacian Regularized d-optimal Design for Active Learning and its Application to Image Retrieval,” IEEE Transactions on Image Processing, 19, 254–263.
  • Hein, M., Audibert, J. Y., and von Luxburg, U. (2005), “From Graphs to Manifolds-Weak and Strong Pointwise Consistency of Graph Laplacians,” in Proceedings of the 18th Conference on Learning Theory.
  • Kiefer, J. (1974), “General Equivalence Theory for Optimum Designs (Approximate Theory),” The Annals of Statistics, 2, 849–879. DOI: 10.1214/aos/1176342810.
  • Kiefer, J., and Wolfowitz, J. (1960), “The Equivalence of Two Extremum Problems,” Canadian Journal of Mathematics, 12, 363–366. DOI: 10.4153/CJM-1960-030-4.
  • Kimeldorf, G. S., and Wahba, G. (1970), “A Correspondence between Bayesian Estimation on Stochastic Processes and Smoothing by Splines,” Annals of Mathematical Statistics, 41, 495–502. DOI: 10.1214/aoms/1177697089.
  • Lafon, S. (2004), “Diffusion Maps and Geometric Harmonics,” Ph. D. thesis, Yale University.
  • Li, H., Del Castillo, E., and Runger, G. (2020), “On Active Learning Methods for Manifold Data,” TEST, 29, 1–33. DOI: 10.1007/s11749-019-00694-y.
  • Lin, L., Thomas, B. S., Zhu, H., and Dunson, D. B. (2017), “Extrinsic Local Regression on Manifold-Valued Data,” Journal of the American Statistical Association, 112, 1261–1273. DOI: 10.1080/01621459.2016.1208615.
  • Marzio, M. D., Panzera, A., and Taylor, C. C. (2014), “Nonparametric Regression for Spherical Data,” Journal of the American Statistical Association, 109, 748–763. DOI: 10.1080/01621459.2013.866567.
  • Pronzato, L., and Pazman, A. (2013), Design of Experiments in Nonlinear Models: Asymtpotic Normality, Optimality Criteria and Small-sample Properties, New York: Springer.
  • Pukelsheim, F. (2006), Optimal Design of Experiments, Philadelphia, PA: Society for Industrial and Applied Mathematics.
  • Roweis, S. T., and Saul, L. K. (2000), “Nonlinear Dimensionality Reduction by Locally Linear Embedding,” Science, 290, 2323–2326. DOI: 10.1126/science.290.5500.2323.
  • Tenenbaum, J. B., de Silva, V., and Langford, J. C. (2000), “A Global Geometric Framework for Nonlinear Dimensionality Reduction,” Science, 290, 2319–2323. DOI: 10.1126/science.290.5500.2319.
  • Vuchkov, I. (1977), “A Ridge-Type Procedure for Design of Experiments,” Biometrika, 64, 147–150. DOI: 10.2307/2335787.
  • Wahba, G. (1990), Spline Models for Observational Data, Philadelphia, PA: Society for Industrial and Applied Mathematics.
  • Wu, C.-F. J., and Hamada, M. S. (2009), Experiments: Planning, Analysis, and Parameter Design Optimization, (2nd ed.), New York: Wiley.
  • Wynn, H. P. (1970), “The Sequential Generation of d-optimum Experimental Designs,” The Annals of Mathematical Statistics, 41, 1655–1664. DOI: 10.1214/aoms/1177696809.
  • Yao, Z., and Zhang, Z. (2020), “Principal Boundary on Riemannian Manifolds,” Journal of the American Statistical Association, 115, 1435–1448. DOI: 10.1080/01621459.2019.1610660.
  • Yu, K., Zhu, S., Xu, W., and Gong, Y. (2008), “Non-greedy Active Learning for Text Categorization using Convex Transductive Experimental Design,” in Proceedings of the 31st annual international ACM SIGIR conference on Research and development in information retrieval, Singapore, pp. 635–642.
  • Zhu, B., Liu, J. Z., Cauley, S. F., Rosen, B. R., and Rosen, M. S. (2018), “Image Reconstruction by Domain-Transform Manifold Learning,” Nature, 555, 487–492. DOI: 10.1038/nature25988.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.