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<p>Log-sum-exp function (LSE) is of great interest in optimization, machine learning and statistics. In this note, we establish a general inequality which holds true in a general Hilbert space, and use it to show that the smallest modulus of Lipschitz smoothness of log-sum-exp function on <i>n</i>-dimensional Euclidean space is 1/2 instead of 1 when <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\ge 2\)</EquationSource> </InlineEquation>. Proofs based on the Gerschgorin circle theorem or Popoviciu’s inequality on variances are also given.</p>

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On the modulus of Lipschitz smoothness of log-sum-exp function

  • 期刊论文

  • 发表日期：

    2026年1月28日

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  • Honglin Luo,
  • Xianfu Wang,
  • Matthew Werenski,
  • Xinmin Yang

摘要

Log-sum-exp function (LSE) is of great interest in optimization, machine learning and statistics. In this note, we establish a general inequality which holds true in a general Hilbert space, and use it to show that the smallest modulus of Lipschitz smoothness of log-sum-exp function on n-dimensional Euclidean space is 1/2 instead of 1 when \(n\ge 2\) . Proofs based on the Gerschgorin circle theorem or Popoviciu’s inequality on variances are also given.