Optimization on a Manifold

All optimization methods considered 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 (for example, rotations or matrices), every parametrization leads to suboptimal solutions affected by singularities. In recent years, techniques using an overparameterized version of the state vector (Her08) have become increasingly popular; 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 $S \in \mathcal{M}$, with $\mathcal{M}$ an $n$-dimensional manifold,

$$\boldsymbol\delta \Leftarrow \left. \dfrac{\partial S(\math... ... 0 \qquad \mathbf{x} \Leftarrow \mathbf{x} + \boldsymbol\delta$$ (4.69)

into

$$: \boldsymbol\epsilon \Leftarrow \left. \dfrac{\partial S(\m... ... \mathbf{x} \Leftarrow \mathbf{x} \boxplus \boldsymbol\epsilon$$ (4.70)

with $\boldsymbol\epsilon \in \mathbb{R}^{n}$, assuming that in the neighborhood $\boldsymbol\epsilon = 0$ the function operates in a Euclidean space. The operator $\boxplus$ enables addition between elements of the manifold space and 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.