1 IG of Poincaré inequalities …
15
1.4.3 Otto’s Metric
Let P = p · γ with p in the maximal exponential model of γ . Let f and g be in the
p-fiber of the statistical manifold, that is, f, g ∈ L (cosh −1) ( p) = L (cosh −1) (γ ) and
f (x) γ (x) dx =
g(x) γ (x) dx = 0. The Otto’s inner product (1.2) becomes
∇ f (x) · ∇g(x) p(x) γ (x) dx =
f (x) δ · ( p(x)∇g(x)) γ (x) dx .
The LHS is well defined and regular if we assume ∇ f, ∇g ∈ L
2
(cosh −1) (γ ), because,
in such a case, |∇ f |
2
, |∇g|
2
∈ L (cosh −1) (γ ) = L (cosh −1) ( p). The RHS provides the
representation of the inner product in the inner product defined in L (cosh −1) (γ ). Note
that the mapping g → δ · ( p∇g) is 1-to-1 if g is restricted by
g(x) p(x) γ (x) dx =
0. The inverse of this mapping provides the natural gradient of the Otto’s inner product
in the sense of [2, 12].
1.4.4 Conclusion and Acknowledgments
In this paper we have derived bounds of the Orlicz norms of interest in IG based
on the Orlicz norm of the gradient. The schematic examples above provide, in our
opinion, a motivation for further study of this approach. There is a large literature
on Sobolev spaces with weight that we have, regrettably, not used here. Its study
would surely provide more precise and deep results than those presented here. I like
to thank the Editor and the Referees for the very helpful and detailed review of this
paper.
Disclaimer: Views and opinions expressed are those of the authors and do not necessarily represent
official positions of their respective companies.
References
1. Adams, R.A., Fournier, J.J.F.: Sobolev spaces, Pure and Applied Mathematics (Amsterdam),
vol. 140, 2nd edn. Elsevier/Academic Press, Amsterdam (2003)
2. Amari, S.I.: Natural gradient works efficiently in learning. Neural Comput. 10(2), 251–276
(1998). https://doi.org/10.1162/089976698300017746
3. Bogachev, V.I.: Differentiable measures and the Malliavin calculus, Mathematical Surveys and
Monographs, vol. 164. American Mathematical Society, Providence, RI (2010) https://doi.org/
10.1090/surv/164
4. Buldygin, V.V., Kozachenko, Y.V.: Metric characterization of random variables and random
processes, Translations of Mathematical Monographs, vol. 188. American Mathematical Society, Providence, RI (2000), translated from the 1998 Russian original by V. Zaiats
15
1.4.3 Otto’s Metric
Let P = p · γ with p in the maximal exponential model of γ . Let f and g be in the
p-fiber of the statistical manifold, that is, f, g ∈ L (cosh −1) ( p) = L (cosh −1) (γ ) and
f (x) γ (x) dx =
g(x) γ (x) dx = 0. The Otto’s inner product (1.2) becomes
∇ f (x) · ∇g(x) p(x) γ (x) dx =
f (x) δ · ( p(x)∇g(x)) γ (x) dx .
The LHS is well defined and regular if we assume ∇ f, ∇g ∈ L
2
(cosh −1) (γ ), because,
in such a case, |∇ f |
2
, |∇g|
2
∈ L (cosh −1) (γ ) = L (cosh −1) ( p). The RHS provides the
representation of the inner product in the inner product defined in L (cosh −1) (γ ). Note
that the mapping g → δ · ( p∇g) is 1-to-1 if g is restricted by
g(x) p(x) γ (x) dx =
0. The inverse of this mapping provides the natural gradient of the Otto’s inner product
in the sense of [2, 12].
1.4.4 Conclusion and Acknowledgments
In this paper we have derived bounds of the Orlicz norms of interest in IG based
on the Orlicz norm of the gradient. The schematic examples above provide, in our
opinion, a motivation for further study of this approach. There is a large literature
on Sobolev spaces with weight that we have, regrettably, not used here. Its study
would surely provide more precise and deep results than those presented here. I like
to thank the Editor and the Referees for the very helpful and detailed review of this
paper.
Disclaimer: Views and opinions expressed are those of the authors and do not necessarily represent
official positions of their respective companies.
References
1. Adams, R.A., Fournier, J.J.F.: Sobolev spaces, Pure and Applied Mathematics (Amsterdam),
vol. 140, 2nd edn. Elsevier/Academic Press, Amsterdam (2003)
2. Amari, S.I.: Natural gradient works efficiently in learning. Neural Comput. 10(2), 251–276
(1998). https://doi.org/10.1162/089976698300017746
3. Bogachev, V.I.: Differentiable measures and the Malliavin calculus, Mathematical Surveys and
Monographs, vol. 164. American Mathematical Society, Providence, RI (2010) https://doi.org/
10.1090/surv/164
4. Buldygin, V.V., Kozachenko, Y.V.: Metric characterization of random variables and random
processes, Translations of Mathematical Monographs, vol. 188. American Mathematical Society, Providence, RI (2000), translated from the 1998 Russian original by V. Zaiats