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Los transformados de Legendre generalizados tienen raíces en la geometría de la información

Frank Nielsen1

  • 1Sony Computer Science Laboratories, 3-14-13 Higashi Gotanda, Shinagawa Ku, Tokyo 141-0022, Japan.

Entropy (Basel, Switzerland)
|January 28, 2026
PubMed
Resumen
Este resumen es generado por máquina.

Este estudio revela que los transformados de Legendre generalizados son equivalentes al transformado de Legendre estándar de funciones dualmente deformadas afines. Estos transformados se derivan de las estructuras duales de Hessiana dentro de la geometría de la información.

Palabras clave:
divergencias de Bregman y Fenchel-Youngsistemas de coordenadas afines y curvilíneosespacio dual plano y variedad de Hessianageometría de la informacióntransformado de Legendreorden inverso y dualidad convexa

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Área de la Ciencia:

  • Análisis Convexo
  • Geometría de la Información
  • Análisis Funcional

Sus antecedentes:

  • El transformado de Legendre es una herramienta fundamental en el análisis convexo y sus aplicaciones.
  • Artstein-Avidan y Milman caracterizaron previamente transformaciones invertibles específicas como deformaciones afines del transformado de Legendre.

Objetivo del estudio:

  • Establecer una correspondencia directa entre los transformados de Legendre generalizados y el transformado de Legendre ordinario.
  • Demostrar la derivación de estos transformados generalizados a partir de las estructuras duales de Hessiana de la geometría de la información.

Principales métodos:

  • Demostrar la equivalencia de los transformados de Legendre generalizados al transformado de Legendre ordinario de funciones dualmente deformadas afines.
  • Utilizar las estructuras duales de Hessiana inherentes a la geometría de la información.

Principales resultados:

  • Todos los transformados de Legendre generalizados estudiados corresponden al transformado de Legendre ordinario de funciones dualmente deformadas afines.
  • Se demuestra que los conjugados convexos generalizados son conjugados convexos ordinarios de funciones dualmente deformadas afines.
  • Se presenta un método para derivar estos transformados generalizados a partir de la geometría de la información.

Conclusiones:

  • Los hallazgos proporcionan una perspectiva unificada sobre los transformados de Legendre generalizados.
  • Este trabajo conecta el análisis convexo con la geometría de la información a través de la lente de los transformados de Legendre.