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Publications

Accepted/Published:
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  • N. García Trillos, M. Jacobs, "An Analytical and Geometric Perspective on Adversarial Robustness." Notices of the American Mathematical Society, Vol. 70, Issue 08, p. 1). American Mathematical Society (AMS), 2023 https://doi.org/10.1090/noti2758
     
  • N. García Trillos, P. He, C. Li “Large sample spectral analysis of graph-based multi-manifold clustering”  Journal of Machine Learning Research (JMLR) 24(143):1−71, (2023). https://jmlr.org/papers/v24/21-1254.html
     
  • N. García Trillos, B. Hosseini, D. Sanz-Alonso, "From Optimization to Sampling Through Gradient Flows." Notices of the American Mathematical Society, Vol. 70, Issue 06, p. 1 American Mathematical Society (AMS), 2023. https://doi.org/10.1090/noti2717
     
  • N. García Trillos, M. Jacobs, J. Kim. "The multimarginal optimal transport formulation of adversarial multiclass classification." Journal of Machine Learning Research (JMLR) 24(45):1−56, 2023. https://www.jmlr.org/papers/v24/22-0698.html
     
  • L. Bungert, N. García Trillos, R. Murray, “The geometry of adversarial training in binary classification.” Information and Inference: A Journal of the IMA, 2023;, iaac029, https://doi.org/10.1093/imaiai/iaac029
     
  • N. García Trillos, R. Murray. "Adversarial Classification: Necessary Conditions and Geometric Flows"
    Journal of Machine Learning Research (JMLR) 23(187):1−38, (2022).     https://jmlr.org/papers/v23/21-0222.html
     
  • C.A. García Trillos, N. García Trillos. "On the regularized risk of distributionally robust learning over deep neural networks." Res Math Sci (RMS) 9, 54 (2022). https://doi.org/10.1007/s40687-022-00349-9
     
  • N. García Trillos, D. Sanz-Alonso, R. Yang, “Mathematical foundations of graph-based Bayesian semi-supervised learning.” Notices of the American Mathematical Society, Vol 69, No. 10, 1717-1729, (2022).  https://doi.org/10.1090/noti2568
     
  • N. García Trillos, R. Murray, M. Thorpe. "From Graph Cuts to Isoperimetric Inequalities: Convergence Rates of Cheeger Cuts on Data Clouds." Arch Rational Mech Anal (ARMA) 244, 541–598 (2022). https://doi.org/10.1007/s00205-022-01770-8
     
  • J. Calder, N. García Trillos. "Improved spectral convergence rates for graph Laplacians on ε-graphs and k-NN graphs." Applied and Computational Harmonic Analysis (ACHA). Volume 60, September 2022, Pages 123-175. https://doi.org/10.1016/j.acha.2022.02.004
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  • N. García Trillos, J. Morales. "Semi-discrete Optimization Through Semi-discrete Optimal Transport: A Framework for Neural Architecture Search." J Nonlinear Sci 32, 27 (2022). https://doi.org/10.1007/s00332-022-09780-2
     
  • K. Craig, N. García Trillos, D. Slepčev. (2022). "Clustering Dynamics on Graphs: From Spectral Clustering to Mean Shift Through Fokker–Planck Interpolation." Active Particles, Volume 3. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-93302-9_4
     
  • J. Calder, N. García Trillos, M. Lewicka. "Lipschitz Regularity of Graph Laplacians on Random Data Clouds." SIAM Journal on Mathematical Analysis (2022) 54:1, 1169-1222. https://doi.org/10.1137/20M1356610
     
  • N. García Trillos, F. Morales, J. Morales "Traditional and accelerated gradient descent for neural architecture search." Geometric Science of Information 2021, Springer International Publishing. https://doi.org/10.1007/978-3-030-80209-7_55 ArXiv version at: https://arxiv.org/abs/2006.15218 
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  • N. García Trillos, F. Hoffmann, B. Hosseini, "Geometric structure of graph Laplacian embeddings." The Journal of Machine Learning Research (JMLR) 2021.  http://jmlr.org/papers/v22/19-683.html
     

  • D. Bigoni, Y. Chen, N. García Trillos, Y. Marzouk, D. Sanz-Alonso, "Data-Driven Forward Discretizations for Bayesian Inversion." Inverse Problems 2020. https://doi.org/10.1088/1361-6420/abb2fa

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  • N. García Trillos, R. Murray,  “A maximum principle argument for the uniform convergence of graph Laplacian regressors.”  SIAM Journal on Mathematics of Data Science, 2(3), 705-739. (35 pages). 2020. https://doi.org/10.1137/19M1245372

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  • N. García Trillos, Z. Kaplan, T. Samakhoana and D. Sanz-Alonso "On the consistency of graph-based Bayesian learning and the scalability of sampling algorithms." The Journal of Machine Learning Research (JMLR). 21(28):1−47, 2020. https://jmlr.org/papers/v21/17-698.html

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  • N. García Trillos, M. Gerlach, M. Hein, D. Slepčev "Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator." Foundations of Computational Mathematics 20, pages 827–887(2020). https://doi.org/10.1007/s10208-019-09436-w

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  • N. García Trillos. “Variational limits of k-nn graph-based functionals on data clouds.” SIAM Journal on Mathematics of Data Science, 1(1):93-120, 2019. https://doi.org/10.1137/18M1188999

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  • N. García Trillos, D. Sanz-Alonso, R. Yang, “Local Regularization of Noisy Point Clouds: Improved Global Geometric Estimates and Data Analysis” Journal of Machine Learning Research 20 (2019) 1-37 9; http://jmlr.org/papers/v20/19-261.html

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  • N. García Trillos, Z. Kaplan, D. Sanz-Alonso “Variational Characterizations of Local Entropy and Heat Regularization in Deep Learning” Entropy 2019, 21(5), 511; https://doi.org/10.3390/e21050511

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  • N. García Trillos and D. Sanz-Alonso. “Continuum limits of posteriors in graph bayesian inverse problems.” SIAM Journal on Mathematical Analysis, 50(4):4020-4040, 2018. https://doi.org/10.1137/17M1138005

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  • N. García Trillos and D. Slepčev. “A variational approach to the consistency of spectral clustering.” Applied and Computational Harmonic Analysis (ACHA),  45(2):239-281, 2018. https://doi.org/10.1016/j.acha.2016.09.003

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  • N. García Trillos and R. Murray. “A new analytical approach to consistency and overfitting in regularized empirical risk minimization.” European Journal of Applied Mathematics (EJAM), 28(6):886-921, 2017. https://doi.org/10.1017/S0956792517000201

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  • N. García Trillos, D. Slepčev, J. von Brecht. “Estimating perimeter using graph cuts.” Advances in Applied Probability. Cambridge Press. Volume 49, Issue 4 December 2017 , pp. 1067-1090.  https://doi.org/10.1017/apr.2017.34

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  • N. García Trillos, D. Sanz-Alonso. “The Bayesian formulation and well-posedness of fractional elliptic inverse problems” Inverse Problems, Volume 33, Number 6  Published 24 May 2017. https://doi.org/10.1088/1361-6420/aa711e

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  • A. Ramdas, N. García Trillos, M. Cuturi. On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests. Entropy 2017, 19(2), 47; https://doi.org/10.3390/e19020047

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  • N. García Trillos, D. Slepčev, J. Von Brecht, T. Laurent, and X. Bresson. “Consistency of Cheeger and ratio graph cuts.” The Journal of Machine Learning Research (JMLR), 17(1):6268-6313, 2016. https://www.jmlr.org/papers/volume17/14-490/14-490

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  • N. García Trillos and D. Slepčev. “Continuum limit of total variation on point clouds.” Archive for rational mechanics and analysis (ARMA), 220(1):193-241, 2016. https://doi.org/10.1007/s00205-015-0929-z

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  • N. García Trillos, D. Slepčev. “On the Rate of Convergence of Empirical Measures in ∞-transportation Distance.” Canadian Journal of Mathematics. Volume 67, Issue 6 01 December 2015 , pp. 1358-1383.  https://doi.org/10.4153/CJM-2014-044-6

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Preprints:
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  • N. García Trillos, M. Jacobs, J. Kim, M. Werenski. "An Optimal Transport Approach for Computing Adversarial Training Lower Bounds in Multiclass Classification." (2024) Preprint available at: https://arxiv.org/abs/2401.09191
     

  • N. García Trillos and B. Sen. "A New Perspective On Denoising Based On Optimal Transport." (2023) Preprint available at: https://arxiv.org/abs/2312.08135
     

  • C. Li, R. Sonthalia, and N. García Trillos. "Spectral Neural Networks: Approximation Theory and Optimization Landscape." (2023) Preprint available at: https://arxiv.org/abs/2310.00729
     

  • N. García Trillos, A. Little, D. McKenzie, and J.M. Murphy. "Fermat Distances: Metric Approximation, Spectral Convergence, and Clustering Algorithms." (2023) Preprint available at: https://arxiv.org/pdf/2307.05750
     

  • N. García Trillos and M. Weber "Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower bounds." (2023) Preprint available at: https://arxiv.org/abs/2307.02378
     

  • L. Bungert, N. García Trillos, M. Jacobs, D. McKenzie, D. Nikolic, Q. Wang "It begins with a boundary: A geometric view on probabilistically robust learning." (2023) Preprint available at: https://arxiv.org/abs/2305.18779
     

  • J.A. Carrillo, N. García Trillos, S. Li, Y. Zhu "FedCBO: Reaching Group Consensus in Clustered Federated Learning through Consensus-based Optimization." (2023) Preprint available at: https://arxiv.org/abs/2305.02894 
     

  • N. García Trillos, M. Jacobs, J. Kim "On the existence of solutions to adversarial training in multiclass classification."(2023) Preprint available at:  https://arxiv.org/abs/2305.00075
     

  • C.A. García Trillos and N. García Trillos "On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce it." (2023) Preprint available at https://arxiv.org/abs/2301.03662
     

  • Y. Luo, N. García Trillos "Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective." (2022) Preprint available at: https://arxiv.org/abs/2209.15130
     

  • N. García Trillos, R. Murray, M. Thorpe "Rates of Convergence for Regression with the Graph Poly-Laplacian." (2022)  Preprint available at: https://arxiv.org/abs/2209.02305
     

  • N. García Trillos , R. Murray, and D. Sanz-Alonso. “Spatial extreme values via variational techniques.” 2018. Preprint available in ArXiv: https://arxiv.org/abs/1808.03218


     

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