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Tensor Bernstein concentration inequalities with an application to sample estimators for high-order moments

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Abstract

This paper develops the Bernstein tensor concentration inequality for random tensors of general order, based on the use of Einstein products for tensors. This establishes a strong link between these and matrices, which in turn allows exploitation of existing results for the latter. An interesting application to sample estimators of high-order moments is presented as an illustration.

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Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11771038) and the Beijing Natural Science Foundation (Grant No. Z190002), and the Hong Kong Research Grant Council (Grant Nos. PolyU 15300715, 15301716, 15300717). The third author gratefully acknowledges the support from Prof. Xiaojun Chen in Hong Kong Polytechnic University for the visit during which this research was initiated.

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Science, Beijing Jiaotong University, Beijing, 100044, China

    Ziyan Luo

  2. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

    Liqun Qi

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong, China

    Liqun Qi

  4. Namur Center for Complex Systems (naXys), University of Namur, 61, rue de Bruxelles, B-5000, Namur, Belgium

    Philippe L. Toint

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  1. Ziyan Luo
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  2. Liqun Qi
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  3. Philippe L. Toint
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Correspondence to Liqun Qi.

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Luo, Z., Qi, L. & Toint, P.L. Tensor Bernstein concentration inequalities with an application to sample estimators for high-order moments. Front. Math. China 15, 367–384 (2020). https://doi.org/10.1007/s11464-020-0830-4

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  • DOI: https://doi.org/10.1007/s11464-020-0830-4

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