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PreprintCC BY 4.0arXiv

On the Subadditivity of Mutual f-Information

  • Hamidreza Abin
  • Amin Gohari
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Abstract

We address an open problem raised by Polyanskiy and Wu in their textbook concerning the subadditivity of mutual f-information. We study two notions of generalized mutual information: mutual f-information, defined by Ĩf(X;Y)=Df(pXY|pXpY), and variational (G,f)-information, obtained by minimizing an average of transformed f-divergences. For the first notion, we derive a sufficient condition on g(x)=x2f''(x) that guarantees subadditivity under conditionally independent observations. This condition yields explicit families beyond the previously known examples generated by xlog x and (x−1)log x. Our proof relies on the concavity of a generalized variance functional. For the variational notion, we extend existing sufficient conditions beyond the transforms G(x)∈{x,log(1+x),−log(1−x)}. Using a comparison inequality for G−1(G(A)+G(B)), we obtain differential criteria for concave and convex transforms. Explicit constructions illustrate the resulting classes of subadditive information measures.

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Cite

BibTeXRIS
BibTeX
@misc{abin2026subadditivity,
  title = {{On the Subadditivity of Mutual f-Information}},
  author = {Hamidreza Abin and Amin Gohari},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21231},
  eprint = {2610.21231},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21231},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the Subadditivity of Mutual f-Information
AU  - Hamidreza Abin
AU  - Amin Gohari
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21231
UR  - https://arxiv.org/abs/2610.21231
AB  - We address an open problem raised by Polyanskiy and Wu in their textbook concerning the subadditivity of mutual f-information. We study two notions of generalized mutual information: mutual f-information, defined by Ĩ_f(X;Y)=D_f(p_XY|p_Xp_Y), and variational (G,f)-information, obtained by minimizing an average of transformed f-divergences. For the first notion, we derive a sufficient condition on g(x)=x^2f''(x) that guarantees subadditivity under conditionally independent observations. This condition yields explicit families beyond the previously known examples generated by xlog x and (x−1)log x. Our proof relies on the concavity of a generalized variance functional. For the variational notion, we extend existing sufficient conditions beyond the transforms G(x)∈{x,log(1+x),−log(1−x)}. Using a comparison inequality for G^−1(G(A)+G(B)), we obtain differential criteria for concave and convex transforms. Explicit constructions illustrate the resulting classes of subadditive information measures.
ER  -