New Characterizations of Matrix $Φ$-Entropies, Poincaré and Sobolev Inequalities and an Upper Bound to Holevo Quantity

@article{Cheng2015NewCO,
  title={New Characterizations of Matrix \$$\Phi$\$-Entropies, Poincar{\'e} and Sobolev Inequalities and an Upper Bound to Holevo Quantity},
  author={Hao-Chung Cheng and Min-Hsiu Hsieh},
  journal={ArXiv},
  year={2015},
  volume={abs/1506.06801}
}
We derive new characterizations of the matrix $\Phi$-entropies introduced in [Electron.~J.~Probab., 19(20): 1--30, 2014}]. These characterizations help to better understand the properties of matrix $\Phi$-entropies, and are a powerful tool for establishing matrix concentration inequalities for matrix-valued functions of independent random variables. In particular, we use the subadditivity property to prove a Poincar\'e inequality for the matrix $\Phi$-entropies. We also provide a new proof for… 

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