Polar Decomposition as an Entropy Gradient Flow
Klas Modin
Outline
- Polar decompositions and optimal transport
- Wasserstein geometry
- Optimal transport in linear category
- Lifted entropy gradient flow
- Convexity and convergence
- Numerical example
- Outlook: infinite-dimensional flow
Polar decomposition
A\in\mathrm{GL}(n) \quad\Rightarrow\quad A=PQ
Q=
P=
Proof techniques
- Elementary linear algebra
- Toy example of Brenier's factorization of maps
- Limit of entropy gradient flow on
\mathrm{P}(n) = \{ P \in \mathbb{R}^n \mid P^\top = P, P>0 \}
Optimal transport
\mu_0
\mu_1
\eta_*\mu_0
(L^2)
\min_{\eta_*\mu_0=\mu_1} \int_{\mathbb{R}^n} \left|\eta(x)-x \right|^2 d\mu_0
\mathbb{R^n}
Monge
Wasserstein distance
d_W^2(\mu_0,\mu_1) = \inf_{\eta_*\mu_0=\mu_1} \int_{\mathbb{R}^n} \left|\eta(x)-x \right|^2 d\mu_0
Symmetric by change of variables
Riemannian structure
\mathrm{Diff}(\mathbb{R}^n)
\mathrm{Dens}(\mathbb{R}^n)
\mathrm{Id}
\mu_0
\mu_1
\pi(\eta)=\eta_*\mu_0
Moser 1965:
Principal bundle
\mathrm{Diff}(\mathbb{R}^n)/\mathrm{Diff}_{\mu_0}(\mathbb{R}^n)
Otto 2001:
\mathcal{G}_\eta(\dot\eta,\dot\eta) = \int_{\mathbb{R}^n}\left\vert \dot\eta \right\vert^2 d \mu_0
Induces metric
\overline{\mathcal{G}}_\mu(\dot\mu,\dot\mu) \Rightarrow d_W^2(\mu_0,\mu_1)
\mathrm{Hor}
polar cone
densities
\mathrm{id}
\nabla\phi
fiber
\pi
fiber
\mu_0
\mu_1
K = \{ \nabla\phi \mid \nabla^2\phi > 0 \}
\mathrm{Dens}(\mathbb{R}^n)
Central Lemma: The mapping
is an isomorphism (section of principal bundle)
\pi|_{K}\colon K \to \mathrm{Dens}(\mathbb{R}^n)
\eta
= \nabla\phi \circ \psi
\psi
Linear/Gaussian category
\mu_\Sigma = \frac{1}{\sqrt{\det(\Sigma)(2\pi)^n}}\exp(-\frac{1}{2}x^\top \Sigma^{-1}x) dx
Multivariate zero-mean Gaussian densities
\mathrm{GL}(n) \simeq \{ x\mapsto Ax \mid A \in \mathrm{GL}(n) \} \subset \mathrm{Diff}(\mathbb{R}^n)
Linear transformations are totally geodesic
\mathcal{N}_n =\{ \mu_\Sigma \mid \Sigma\in \mathrm{P}(n) \} \subset \mathrm{Dens}(\mathbb{R}^n)
Action map:
\eta \mapsto \eta_*\mu \quad\Leftrightarrow\quad A \mapsto A\Sigma A^\top
polar cone
covar. matrices
I
P
fiber
\pi
fiber
\Sigma_0
\Sigma_1
K = \mathrm{P}(n)
\mathrm{P}(n)
Central Lemma: The mapping
is an isomorphism (section of principal bundle)
K \ni P \mapsto P\Sigma_0 P^\top \in \mathrm{P}(n)
A
= P Q
Q
polar cone
covar. matrices
I
P
\Sigma_0
\Sigma_1
A
Lifted gradient flow
K = \mathrm{P}(n)
\mathrm{P}(n)
- Convex functional on with
- Lifted functional on
- Consider gradient flow on polar cone
\bar F
\mathrm{P}(n)
\nabla_{\bar{\mathcal{G}}}\bar{F}(\Sigma_1) = 0
\mathrm{GL}(n)
F(A) = \bar F(\pi(A)) = \bar F(A\Sigma_0 A^\top)
\dot P = -\Pi \nabla_{\mathcal G} F(P)
Lifted relative entropy
\bar H(\Sigma) = \frac{n}{2}- \frac{1}{2}\mathrm{tr}(\Sigma_1^{-1}\Sigma) + \frac{1}{2}\log(\det(\Sigma_1^{-1}\Sigma))
H(A) = \frac{n}{2}- \frac{1}{2}\mathrm{tr}(\Sigma_1^{-1}A\Sigma_0A^\top) + \log(\det(A))
Final gradient flow
(\Sigma_0 = I)
\dot P = P^{-1} - \frac{1}{2}(\Sigma_1^{-1}P + P \Sigma_1^{-1})
Convergence
Lemma:
-\mathrm{Hess}(H|_{K})_P \geq \alpha \mathcal{G}_P \quad \alpha > 0
Corollary:
d^2(P(t),P_\infty) \leq \, \mathrm{e}^{-2\alpha t} d^2(P(0),P_\infty)
Example
P = \begin{bmatrix}
3 & -1 \\
-1 & 2
\end{bmatrix}
\Sigma_1 = \pi(P) = PP^\top
Example
-H(P(t))
d^2(P(t),P_{\infty})
Outlook: inf-dim version
H(\nabla\phi) = -\int_{\mathbb{R}^n}\frac{\rho_0\circ (\nabla\phi)^{-1}}{\det(\nabla^2\phi\circ(\nabla\phi)^{-1})}\log\left( \frac{\rho_0\circ(\nabla\phi)^{-1}}{\rho_1 \det(\nabla^2\phi\circ (\nabla\phi)^{-1})} \right)dx
\dot\phi = -\nabla\cdot\rho_0\nabla\cdot (\nabla^2\phi)^{-1} - \nabla\cdot (\nabla^2\phi)^{-1}\cdot\nabla\rho_0 + \nabla\cdot\rho_0 \frac{\nabla\rho_1\circ(\nabla\phi)^{-1})}{\rho_1\circ (\nabla\phi)^{-1})}
Observation and conjecture
Observation: if log-concave then
\rho_1
-\mathrm{Hess}(H)_{\nabla\phi} \geq \alpha \mathcal{G}_{\nabla\phi}
Conjecture: if log-concave then convergence
\rho_1