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Iosif Pinelis
Iosif Pinelis
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Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $\phi:X\to\mathbb{R}$$f:X\to\mathbb{R}$ be a strictly convex function that is differentiable on the (non-empty) relative interior of $X$.

$\nabla\phi$$\nabla f$ is a bijection, but is it a homeomorphism? This question is in the context of information geometry and Bregman divergences, where $\nabla\phi$$\nabla f$ induces a change of coordinates.

Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $\phi:X\to\mathbb{R}$ be a strictly convex function that is differentiable on the (non-empty) relative interior of $X$.

$\nabla\phi$ is a bijection, but is it a homeomorphism? This question is in the context of information geometry and Bregman divergences, where $\nabla\phi$ induces a change of coordinates.

Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $f:X\to\mathbb{R}$ be a strictly convex function that is differentiable on the (non-empty) relative interior of $X$.

$\nabla f$ is a bijection, but is it a homeomorphism? This question is in the context of information geometry and Bregman divergences, where $\nabla f$ induces a change of coordinates.

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rick
rick
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Is the gradient of a strictly convex, continuously differentiable function a homeomorphism?

Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $\phi:X\to\mathbb{R}$ be a strictly convex function that is differentiable on the (non-empty) relative interior of $X$.

$\nabla\phi$ is a bijection, but is it a homeomorphism? This question is in the context of information geometry and Bregman divergences, where $\nabla\phi$ induces a change of coordinates.