Overparameterized maximum likelihood tests for detection of sparse vectors

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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.

About the authors

G. K Golubev

Kharkevich Institute for Information Transmission Problems of the Russian Academy of Sciences

Email: golubev.yuri@gmail.com
Moscow, Russia

References

  1. Zhang C., Bengio S., Hardt M., Recht B., Vinyals O. Understanding Deep Learning (Still) Requires Rethinking Generalization // Commun. ACM. 2021. V. 64. № 3. P. 107-115. https://doi.org/10.1145/3446776
  2. Belkin M. Fit without Fear: Remarkable Mathematical Phenomena of Deep Learning through the Prism of Interpolation // Acta Numer. 2021. V. 30. P. 203-248. https://doi.org/10.1017/S0962492921000039
  3. Belkin M., Hsu D., Xu J. Two Models of Double Descent for Weak Features // SIAM J. Math. Data Sci. 2020. V. 2. № 4. P. 1167-1180. https://doi.org/10.1137/20M1336072
  4. Dar Y., Muthukumar V., 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
  5. Добрушин Р.Л. Одна статистическая задача теории обнаружения сигнала на фоне шума в многоканальной системе, приводящая к устойчивым законам распределения // Теория вероятн. и ее примен. 1958. Т. 3. № 2. С. 173-185. https://www.mathnet.ru/rus/tvp4928
  6. Бурнашев М.В., Бегматов И.А. Об одной задаче обнаружения сигнала, приводящей к устойчивым распределениям // Теория вероятн. и ее примен. 1990. Т. 35. № 3. С. 557-560. https://www.mathnet.ru/rus/tvp1261
  7. Ingster Yu.I., Suslina I.A. Nonparametric Goodness-of-Fit Testing Under Gaussian Models // Lect. Notes Statist. V. 169. New York: Springer-Verlag, 2003. https://doi.org/10.1007/978-0-387-21580-8
  8. Bonferroni C.E. Teoria statistica delle classi e calcolo delle probabilità // Pubbl. del R. Ist. Super. di Sci. Econ. e Commer. di Firenze. V. 8. Firenze: Seeber, 1936
  9. Benjamini Y., Hochberg Y. Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing //j. Roy. Statist. Soc. Ser. B. 1995. V. 57. № 1. P. 289-300. https://doi.org/10.1111/j.2517-6161.1995.tb02031.x
  10. Benjamini Y. Simultaneous and Selective Inference: Current Successes and Future Challenges // Biom. J. 2010. V. 52. № 6. P. 708-721. https://doi.org/10.1002/bimj.200900299
  11. Donoho D., Jin J. Higher Criticism Thresholding: Optimal Feature Selection When Useful Features are Rare and Weak // Proc. Natl. Acad. Sci. U.S.A. 2008. V. 105. № 39. P. 14790-14795. https://doi.org/10.1073/pnas.0807471105
  12. Anderson T.W. The Integral of a Symmetric Unimodal Function over a Symmetric Convex Set and Some Probability Inequalities // Proc. Amer. Math. Soc. 1955. V. 6. № 2. P. 170-176. https://doi.org/10.1090/S0002-9939-1955-0069229-1
  13. Ибрагимов И.А., Хасьминский Р.З. Асимптотическая теория оценивания. М.: Наука, 1 979
  14. Pyke R. Spacings // J. Roy. Statist. Soc. Ser. B. 1965. V. 27. № 3. P. 395-436; 37-449 (discussion). https://doi.org/10.1111/j.2517-6161.1965.tb00602.x; https://doi.org/10.1111/j.2517-6161.1965.tb00603.x
  15. Федорюк М.В. Асимптотика: интегралы и ряды. М.: Наука, 1987

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