Geometry of matrix decompositions seen through optimal transport and information geometry

Figure 1.  Illustration of the geometry of the polar decomposition of diffeomorphisms. The element $\nabla\phi$ in the factorization $\varphi=\nabla\phi\circ\psi$ is obtained at the intersection of the polar cone and the fiber of $\mu_1=\pi(\varphi)$. To compute $\nabla\phi$, one may start at $\varphi$ and follow a gradient flow constrained to the fiber of $\mu_1$ (vertical gradient flow, see $\S 2.2.1$), or one may take a gradient flow of a functional on the space of densities that approaches $\mu_1$ (entropy gradient flow, see $\S 2.2.2$) and lift it to a corresponding gradient flow on the polar cone (lifted gradient flow, see $\S 2.2.3$).

Figure 2.  Evolution of the matrix elements of $B(t)$ for the vertical gradient flow in Example 1. Notice that $B(0)=A$ and that $B(t)$ converges towards $P_{\infty}$ in (43) as $t\to\infty$.

Figure 3.  Convergence towards the limit $P_{\infty}$ of the vertical gradient flow in Example 1.

Figure 4.  Evolution of the lifted gradient flow in Example 2. Notice that $P(0)$ is the identity and that $P(t)$ converges towards $P_{\infty}$ in (54) as $t\to\infty$.

Figure 5.  Convergence towards the limit $P_{\infty}$ of the lifted gradient flow in Example 2. Notice that the convergence of both $-F(P(t))$ and $d^{2}(P(t),P_{\infty})$ as $t\to\infty$ is exponential, as fully explained by Theorem 2.14.

Figure 6.  Evolution of the lifted gradient flow in Example 3. Notice that $R(0)$ is the identity and that $R(t)$ converges towards $R_{\infty}$ in (80) as $t\to\infty$.

Figure 7.  Convergence towards the limit $R_{\infty}$ of the lifted gradient flow in Example 3. The convergence of both $-F(R(t))$ and $d^{2}(R(t),R_{\infty})$ is exponential with rate $\exp(-2t)$, as ensured by Theorem 3.10.

Figure 8.  Phase diagram of equation (86) for geodesics on ${\text{D}}(n)$. For every $l>0$ there is a unique integral curve $\lambda(t)$ such that $\lambda(0)=1$ and $\lambda(1)=l$. In consequence, every $\Lambda\in{\text{D}}(n)$ is connected to the identity $I$ by a unique horizontal geodesic.

Figure 9.  Evolution of the horizontal gradient flow in Example 4. $\Lambda(t)$ appears to converge to the ordered sequence of eigenvalues $(1/2,1,5)$.

Figure 10.  Convergence of $F(\Lambda(t))$ towards the minimum for the horizontal gradient flow in Example 4. The convergence appears to be exponential.