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Límites estrechos entre la Divergencia Jensen-Shannon y la Divergencia Minmax
Arseniy Akopyan1, Herbert Edelsbrunner2, Žiga Virk3,4
1Fora Capital, Miami, FL 33131, USA.
Entropy (Basel, Switzerland)
|August 28, 2025
Resumen
Demostramos que la divergencia mínima, una medida compleja para distribuciones categóricas, puede ser aproximada por la divergencia de Jensen-Shannon. Este hallazgo simplifica el análisis en geometría de la información.
Área de la Ciencia:
- Teoría de la información
- Análisis geométrico
- Probabilidad y Estadísticas
Sus antecedentes:
- La comparación de distribuciones de probabilidad es crucial en varios campos científicos.
- La divergencia de Jensen-Shannon (JSD) es una métrica computable ampliamente utilizada para la comparación de la distribución.
- La divergencia mínima ofrece una medida alternativa con posibles interpretaciones geométricas, pero es un desafío computacional.
Objetivo del estudio:
- Para comparar la divergencia de Jensen-Shannon y la divergencia mínima para distribuciones categóricas finitas.
- Establecer un vínculo teórico entre estas dos medidas de disimilaridad.
- Para investigar las propiedades métricas de la divergencia minmax.
Principales métodos:
- Aprovechando la divergencia Kullback-Leibler como base para ambas medidas.
- Desarrollo de límites teóricos para aproximar la divergencia mínima utilizando la divergencia de Jensen-Shannon.
- Analizando las propiedades métricas de la raíz cuadrada de la divergencia minmax.
Principales resultados:
- La divergencia mínima se puede aproximar estrechamente por la divergencia de Jensen-Shannon.
- Los límites derivados apoyan la hipótesis de que la raíz cuadrada de la divergencia mínima es una métrica.
- Demostró que la raíz cuadrada de la divergencia mínima es una métrica en el caso unidimensional.
Conclusiones:
- La divergencia de Jensen-Shannon proporciona una aproximación práctica y precisa para la divergencia minmax computacionalmente intensiva.
- El estudio avanza en la comprensión de las medidas de disimilaridad en la geometría de la información.
- Se necesita más investigación para demostrar la propiedad métrica de la divergencia mínima en el caso general.
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