Bernoulli

  • Bernoulli
  • Volume 24, Number 4A (2018), 2429-2460.

Adaptive confidence sets for matrix completion

Alexandra Carpentier, Olga Klopp, Matthias Löffler, and Richard Nickl

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Abstract

In the present paper, we study the problem of existence of honest and adaptive confidence sets for matrix completion. We consider two statistical models: the trace regression model and the Bernoulli model. In the trace regression model, we show that honest confidence sets that adapt to the unknown rank of the matrix exist even when the error variance is unknown. Contrary to this, we prove that in the Bernoulli model, honest and adaptive confidence sets exist only when the error variance is known a priori. In the course of our proofs, we obtain bounds for the minimax rates of certain composite hypothesis testing problems arising in low rank inference.

Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2429-2460.

Dates
Received: August 2016
Revised: January 2017
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051214

Digital Object Identifier
doi:10.3150/17-BEJ933

Mathematical Reviews number (MathSciNet)
MR3779691

Zentralblatt MATH identifier
06853254

Keywords
adaptivity confidence sets low rank recovery matrix completion minimax hypothesis testing unknown variance

Citation

Carpentier, Alexandra; Klopp, Olga; Löffler, Matthias; Nickl, Richard. Adaptive confidence sets for matrix completion. Bernoulli 24 (2018), no. 4A, 2429--2460. doi:10.3150/17-BEJ933. https://projecteuclid.org/euclid.bj/1522051214


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References

  • [1] Bandeira, A.S. and van Handel, R. (2016). Sharp nonasymptotic bounds on the norm of random matrices with independent entries. Ann. Probab. 44 2479–2506.
  • [2] Baraud, Y. (2004). Confidence balls in Gaussian regression. Ann. Statist. 32 528–551.
  • [3] Bennett, J. and Lanning, S. (2007). The Netflix prize. In Proceedings of KDD Cup and Workshop.
  • [4] Biswas, P., Liang, T., Wang, T. and Ye, Y. (2006). Semidefinite programming based algorithms for sensor network localization. ACM Trans. Sens. Netw. 2 188–220.
  • [5] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities. Oxford: Oxford Univ. Press.
  • [6] Bull, A.D. and Nickl, R. (2013). Adaptive confidence sets in $L^{2}$. Probab. Theory Related Fields 156 889–919.
  • [7] Cai, T.T. and Guo, Z. (2016). Accuracy assessment for high-dimensional linear regression. Preprint. Available at arXiv:1603.03474.
  • [8] Cai, T.T. and Low, M.G. (2004). An adaptation theory for nonparametric confidence intervals. Ann. Statist. 32 1805–1840.
  • [9] Cai, T.T. and Zhou, W.-X. (2016). Matrix completion via max-norm constrained optimization. Electron. J. Stat. 10 1493–1525.
  • [10] Candès, E.J. and Plan, Y. (2009). Matrix completion with noise. Proc. IEEE 98 925–936.
  • [11] Candès, E.J. and Plan, Y. (2011). Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements. IEEE Trans. Inform. Theory 57 2342–2359.
  • [12] Candès, E.J. and Recht, B. (2009). Exact matrix completion via convex optimization. Found. Comput. Math. 9 717–772.
  • [13] Candès, E.J. and Tao, T. (2010). The power of convex relaxation: Near-optimal matrix completion. IEEE Trans. Inform. Theory 56 2053–2080.
  • [14] Carpentier, A. (2013). Honest and adaptive confidence sets in $L_{p}$. Electron. J. Stat. 7 2875–2923.
  • [15] Carpentier, A., Eisert, J., Gross, D. and Nickl, R. (2015). Uncertainty quantification for matrix compressed sensing and quantum tomography problems. Preprint. Available at arXiv:1504.03234.
  • [16] Carpentier, A. and Nickl, R. (2015). On signal detection and confidence sets for low rank inference problems. Electron. J. Stat. 9 2675–2688.
  • [17] Chatterjee, S. (2015). Matrix estimation by universal singular value thresholding. Ann. Statist. 43 177–214.
  • [18] Chi, E.C., Zhou, H., Chen, G.K., Del Vecchyo, D.O. and Lange, K. (2013). Genotype imputation via matrix completion. Genome Res. 23 509–518.
  • [19] Giné, E. and Nickl, R. (2010). Confidence bands in density estimation. Ann. Statist. 38 1122–1170.
  • [20] Giné, E. and Nickl, R. (2016). Mathematical Foundations of Infinite-Dimensional Statistical Models. Cambridge Series in Statistical and Probabilistic Mathematics. New York: Cambridge Univ. Press.
  • [21] Goldberg, D., Nichols, D., Oki, B.M. and Terry, D. (1992). Using collaborative filtering to weave an information tapestry. Commun. ACM 35 61–70.
  • [22] Gross, D. (2011). Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inform. Theory 57 1548–1566.
  • [23] Hoffmann, M. and Nickl, R. (2011). On adaptive inference and confidence bands. Ann. Statist. 39 2383–2409.
  • [24] Jain, P., Netrapalli, P. and Sanghavi, S. (2013). Low-rank matrix completion using alternating minimization. In STOC’13—Proceedings of the 2013 ACM Symposium on Theory of Computing 665–674. New York: ACM.
  • [25] Juditsky, A. and Lambert-Lacroix, S. (2004). Nonparametric confidence set estimation. Math. Methods Statist. 12 410–428.
  • [26] Keshavan, R.H., Montanari, A. and Oh, S. (2010). Matrix completion from noisy entries. J. Mach. Learn. Res. 11 2057–2078.
  • [27] Klopp, O. (2014). Noisy low-rank matrix completion with general sampling distribution. Bernoulli 20 282–303.
  • [28] Klopp, O. (2015). Matrix completion by singular value thresholding: Sharp bounds. Electron. J. Stat. 9 2348–2369.
  • [29] Koltchinskii, V., Lounici, K. and Tsybakov, A.B. (2011). Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion. Ann. Statist. 39 2302–2329.
  • [30] Low, M.G. (1997). On nonparametric confidence intervals. Ann. Statist. 25 2547–2554.
  • [31] Negahban, S. and Wainwright, M.J. (2012). Restricted strong convexity and weighted matrix completion: Optimal bounds with noise. J. Mach. Learn. Res. 13 1665–1697.
  • [32] Nickl, R. and Szabó, B. (2016). A sharp adaptive confidence ball for self-similar functions. Stochastic Process. Appl. 126 3913–3934.
  • [33] Nickl, R. and van de Geer, S. (2013). Confidence sets in sparse regression. Ann. Statist. 41 2852–2876.
  • [34] Recht, B. (2011). A simpler approach to matrix completion. J. Mach. Learn. Res. 12 3413–3430.
  • [35] Robins, J. and van der Vaart, A. (2006). Adaptive nonparametric confidence sets. Ann. Statist. 34 229–253.
  • [36] Rohde, A. and Tsybakov, A.B. (2011). Estimation of high-dimensional low-rank matrices. Ann. Statist. 39 887–930.
  • [37] Singer, A. (2008). A remark on global positioning from local distances. Proc. Natl. Acad. Sci. USA 105 9507–9511.
  • [38] Szabó, B., van der Vaart, A.W. and van Zanten, J.H. (2015). Frequentist coverage of adaptive nonparametric Bayesian credible sets. Ann. Statist. 43 1391–1428.
  • [39] Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505–563.