What is an Euclidean retraction (Riemannian Optimization)?

Question

I am working on an optimisation algorithm using the results from this paper. On page 16 it says "For Riemannian SGD, we report the result of using Euclidean retraction", but up to that point in the paper they used exponential maps instead and never clarify what exactly is an Euclidean retraction. I could not find anything online and this is not my area so maybe I'm missing something basic.

Edit: They are using this Euclidean retraction as a substitute for the exponential map, which they define as $Exp_{\Sigma}(\xi) = \Sigma *\text{exp}(\Sigma^{-1} \xi)$ (for a symmetric matrix $\xi$ in the tangent space of $\Sigma$, which is an element of the manifold of symmetric positive definite matrices $\mathbb{M}$), and they use this to project a gradient descent step back into the mainifold, so I'm trying to find the formula they were referring to by "Euclidean retraction", which hopefully is less computationally expensive.

Answer 1

Here are the classical definitions:

Definition 1. Let $X$ be a topological space and $A\subset X$, then $A$ is a deformation retract of $X$ if there exists a continuous map $r\colon X\rightarrow A$ such that $r_{\vert A}=\operatorname{id}_A$ (such a map is called a retraction) and a continuous map $H\colon X\times[0,1]\rightarrow X$ such that $H(\cdot,0)=\operatorname{id}_X$ and $H(\cdot,1)=r$ (such a map is called a homotopy from $\operatorname{id}_X$ to $r$).

Remark. Some authors required that for all $t\in [0,1]$, $H(\cdot,t)_{\vert A}\colon A\rightarrow A$, the others called such a data a strong deformation retract.

Definition 2. Let $X$ be a topological space and $A\subset X$, then $A$ is a Euclidean retract if and only if there exists an embedding $j\colon A\rightarrow\mathbb{R}^n$ for some $n$ such that there exists $U$ an open neighborhood of $j(A)$ such that $j(A)$ is a deformation retract of $U$.

Remark. Some authors will only require that $j(A)$ is a retraction of $U$, they ask only for the existence of the map $r$ in definition $1$, not a deformation retract.

Informally, a Euclidean retract is homotopical to a Euclidean space.

Does it help?

Answer 2

I found the answer on page 10 of Suvrit Sra, Reshad Hosseini, Conic geometric optimisation on the manifold of positive definite matrices, arXiv:1312.1039:

If the inner product is the induced inner product of the manifold, then the retraction is normal retraction on the Euclidean space which is obtained by summing the point on the manifold and the vector on the tangent space