Abstract
We address the problem of detecting a sparse high-dimensional vector against white Gaussian noise. An unknown vector is assumed to have only \(p\) nonzero components, whose positions and sizes are unknown, the number \(p\) being on one hand large but on the other hand small as compared to the dimension. The maximum likelihood (ML) test in this problem has a simple form and, certainly, depends of \(p\). We study statistical properties of overparametrized ML tests, i.e., those constructed based on the assumption that the number of nonzero components of the vector is \(q\) (\(q>p\)) in a situation where the vector actually has only \(p\) nonzero components. We show that in some cases overparametrized tests can be better than standard ML tests.
References
Zhang, C., Bengio, S., Hardt, M., Recht, B., and Vinyals, O., Understanding Deep Learning (Still) Requires Rethinking Generalization, Commun. ACM, 2021, vol. 64, no. 3, pp. 107–115. https://doi.org/10.1145/3446776
Belkin, M., Fit without Fear: Remarkable Mathematical Phenomena of Deep Learning through the Prism of Interpolation, Acta Numer., 2021, vol. 30, pp. 203–248. https://doi.org/10.1017/S0962492921000039
Belkin, M., Hsu, D., and Xu, J., Two Models of Double Descent for Weak Features, SIAM J. Math. Data Sci., 2020, vol. 2, no. 4, pp. 1167–1180. https://doi.org/10.1137/20M1336072
Dar, Y., Muthukumar, V., and Baraniuk, R.G., A Farewell to the Bias-Variance Tradeoff? An Overview of the Theory of Overparameterized Machine Learning, https://arxiv.org/abs/2109.02355 [stat.ML], 2021.
Dobrushin, R.L., A Statistical Problem Arising in the Theory of Detection of Signals in the Presence of Noise in a Multi-Channel System and Leading to Stable Distribution, Teor. Veroyatnost. i Primenen., 1958, vol. 3, no. 2, pp. 173–185 [Theory Probab. Appl. (Engl. Transl.), 1958, vol. 3, no. 2, pp. 161–173]. https://doi.org/10.1137/1103015
Burnashev, M.V. and Begmatov, I.A., On a Problem of Signal Detection Leading to Stable Distributions, Teor. Veroyatnost. i Primenen., 1990, vol. 35, no. 3, pp. 557–560 [Theory Probab. Appl. (Engl. Transl.), 1990, vol. 35, no. 3, pp. 556–560]. https://doi.org/10.1137/1135076
Ingster, Yu.I. and Suslina, I.A., Nonparametric Goodness-of-Fit Testing Under Gaussian Models, Lect. Notes Statist., vol. 169. New York: Springer-Verlag, 2003. https://doi.org/10.1007/978-0-387-21580-8
Bonferroni, C.E., Teoria statistica delle classi e calcolo delle probabilità, Pubbl. del R. Ist. Super. di Sci. Econ. e Commer. di Firenze, vol. 8, Firenze: Seeber, 1936.
Benjamini, Y. and Hochberg, Y., Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing, J. Roy. Statist. Soc. Ser. B, 1995, vol. 57, no. 1, pp. 289–300. https://doi.org/10.1111/j.2517-6161.1995.tb02031.x
Benjamini, Y., Simultaneous and Selective Inference: Current Successes and Future Challenges, Biom. J., 2010, vol. 52, no. 6, pp. 708–721. https://doi.org/10.1002/bimj.200900299
Donoho, D. and Jin, J., Higher Criticism Thresholding: Optimal Feature Selection When Useful Features are Rare and Weak, Proc. Natl. Acad. Sci. U.S.A., 2008, vol. 105, no. 39, pp. 14790–14795. https://doi.org/10.1073/pnas.0807471105
Anderson, T.W., The Integral of a Symmetric Unimodal Function over a Symmetric Convex Set and Some Probability Inequalities, Proc. Amer. Math. Soc., 1955, vol. 6, no. 2, pp. 170–176. https://doi.org/10.1090/S0002-9939-1955-0069229-1
Ibragimov, I.A. and Khas’minskii, R.Z., Asimptoticheskaya teoriya otsenivaniya, Moscow: Nauka, 1979. Translated under the title Statistical Estimation. Asymptotic Theory, New York: Springer, 1981.
Pyke, R., Spacings, J. Roy. Statist. Soc. Ser. B, 1965, vol. 27, no. 3, pp. 395–436; 437–449 (discussion). https://doi.org/10.1111/j.2517-6161.1965.tb00602.x; https://doi.org/10.1111/j.2517-6161.1965.tb00603.x
Fedoryuk, M.V., Asimptotika: Integraly i ryady (Asymptotics: Integrals and Series), Moscow: Nauka, 1987.
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The author is grateful to an anonymous reviewer for his comments, which helped to improve the paper.
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Translated from Problemy Peredachi Informatsii, 2023, Vol. 59, No. 1, pp. 46–63. https://doi.org/10.31857/S0555292323010047
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Golubev, G.K. Overparameterized Maximum Likelihood Tests for Detection of Sparse Vectors. Probl Inf Transm 59, 41–56 (2023). https://doi.org/10.1134/S0032946023010040
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DOI: https://doi.org/10.1134/S0032946023010040