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Results tagged with convex-analysis
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user 36687
1
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0
answers
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Uniqueness of (generalized) Moreau decomposition
Let $H$ be some Hilbert space, which we can take to be the usual finite-dimensional Euclidean space if needed. For a function $f : H \to \mathbb{R}$, let $f^* : H \to \mathbb{R}$ be its conjugate dual …
- 1,731
1
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1
answer
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Bounds on the curvature of a sequence of convex functions
Let $\{f_n\}$ be a sequence of (real-valued) smooth convex functions on $[0,1]$, with $f_n(0) = f_n(1) = 0$ for all $n$.
Let $t_n \in [0,1]$ be the minimizer of $f_n$ and assume that $M_n:= f_n(t_n) …
- 1,731
4
votes
Accepted
Fenchel-Rockafellar Duality in Villani's Book
More details based on Steve's comment: We have
\begin{align}
-\Theta^*(-z^*) &= - \sup_{x \in E} \big[ \langle-z^*,x \rangle - \Theta(x) \big] \\
&=\inf_{x \in E} \big[ \langle z^*,x \rangle + \Thet …
- 1,731
4
votes
1
answer
778
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Convex support of an exponential family and its mean parameter space $\mathcal{M}$
This question comes up in studying mean parametrization of exponential families of distributions. (See Brown's 1986 book on the subject.)
Let $\nu$ be a (Borel) measure on $\mathbb R^d$. Let $p(\cd …
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