Electronic Journal of Statistics

Minimax lower bounds for function estimation on graphs

Alisa Kirichenko and Harry van Zanten

Full-text: Open access


We study minimax lower bounds for function estimation problems on large graph when the target function is smoothly varying over the graph. We derive minimax rates in the context of regression and classification problems on graphs that satisfy an asymptotic shape assumption and with a smoothness condition on the target function, both formulated in terms of the graph Laplacian.

Article information

Electron. J. Statist., Volume 12, Number 1 (2018), 651-666.

Received: September 2017
First available in Project Euclid: 27 February 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Function estimation on graphs minimax lower bounds

Creative Commons Attribution 4.0 International License.


Kirichenko, Alisa; van Zanten, Harry. Minimax lower bounds for function estimation on graphs. Electron. J. Statist. 12 (2018), no. 1, 651--666. doi:10.1214/18-EJS1407. https://projecteuclid.org/euclid.ejs/1519700497

Export citation


  • Ando, R. K. and Zhang, T. (2007). Learning on graph with laplacian regularization., Advances in neural information processing systems 19, 25.
  • Belkin, M., Matveeva, I. and Niyogi, P. (2004). Regularization and semi-supervised learning on large graphs. In, COLT, volume 3120, pp. 624–638. Springer.
  • Bertozzi, A. L., Luo, X., Stuart, A. M. and Zygalakis, K. C. (2017). Uncertainty Quantification in the Classification of High Dimensional Data., ArXiv e-prints.
  • Cvetković, D., Rowlinson, P. and Simić, S. (2010). An introduction to the theory of graph spectra., Cambridge-New York.
  • Hartog, J. and van Zanten, J. H. (2018). Nonparametric Bayesian label prediction on a graph., Computational Statistics & Data Analysis 120, 111–131.
  • Johnson, R. and Zhang, T. (2007). On the effectiveness of Laplacian normalization for graph semi-supervised learning., Journal of Machine Learning Research 8(4).
  • Kirichenko, A. and van Zanten, J. H. (2017). Estimating a smooth function on a large graph by Bayesian Laplacian regularisation., Electron. J. Statist. 11(1), 891–915.
  • Kolaczyk, E. D. (2009)., Statistical analysis of network data. Springer.
  • Merris, R. (1998). Laplacian graph eigenvectors., Linear Algebra Appl. 278(1–3), 221–236.
  • Sadhanala, V., Wang, Y.-X. and Tibshirani, R. J. (2016). Total variation classes beyond 1d: Minimax rates, and the limitations of linear smoothers. In, Advances in Neural Information Processing Systems, pp. 3513–3521.
  • Smola, A. J. and Kondor, R. (2003). Kernels and regularization on graphs. In, Learning theory and kernel machines, pp. 144–158. Springer.
  • Tsybakov, A. B. (2009)., Introduction to nonparametric estimation. Springer Series in Statistics. Springer, New York.
  • Zhu, J. and Hastie, T. (2005). Kernel logistic regression and the import vector machine., Journal of Computational and Graphical Statistics 14(1), 185–205.