Optimization on a Manifold

All optimization methods discussed so far have been designed to operate in a “flat” Euclidean space. When one wants to optimize a state vector containing one or more variables for which Euclidean space is not meaningful (e.g., rotations or matrices), every parameterization leads to suboptimal solutions and singularities. In recent years, techniques using an overparameterized version of the state vector (Her08) have become widespread; the problem is then optimized directly on the manifold (manifold), which can locally be approximated by a Euclidean tangent vector space.

The idea is to transform the classical minimization of $S \in \mathcal{M}$, where $\mathcal{M}$ is an n-dimensional manifold,

$$\boldsymbol\delta \Leftarrow \left. \dfrac{\partial S(\math... ... 0 \qquad \mathbf{x} \Leftarrow \mathbf{x} + \boldsymbol\delta$$ (4.74)

into

$$: \boldsymbol\epsilon \Leftarrow \left. \dfrac{\partial S(\m... ... \mathbf{x} \Leftarrow \mathbf{x} \boxplus \boldsymbol\epsilon$$ (4.75)

with $\boldsymbol\epsilon \in \mathbb{R}^{n}$, assuming that in the neighborhood $\boldsymbol\epsilon = 0$ the function operates in a Euclidean space. The operator $\boxplus$ allows elements of the manifold space to be added to elements of the Euclidean space $\mathbb{R}^{n}$.

A classic example is the optimization of an orientation expressed in three dimensions using a quaternion in four dimensions.