Outline

• Riemannian geometry
• Manifold of multivariate Gaussians
• Fisher-Rao metric
• Homogeneous space structure
• QR (and Cholesky)
• Convexity and convergence
• Numerical example

Riemannian manifold

v
g_x(v,v) = \langle v,v\rangle_x
\displaystyle L(\gamma) = \int_0^1 \sqrt{g_{\gamma(s)}\big(\gamma'(s),\gamma'(s)\big)}\mathrm{d}t
x
\displaystyle A(\gamma) = \int_0^1 g_{\gamma(s)}\big(\gamma'(s),\gamma'(s)\big)\mathrm{d}t

Information geometry

\displaystyle p(x;\mu,\Sigma) = \frac{1}{\sqrt{(2\pi)^n|\Sigma|}}\exp(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1}(x-\mu))

[Amari, ~80's]

Information geometry

\displaystyle p(x;\Sigma) = \frac{1}{\sqrt{(2\pi)^n|\Sigma|}}\exp(-\frac{1}{2}x^\top \Sigma^{-1}x)

[Amari, ~80's]

Manifold of inverse covariance matrices

P(n) = \{ W\in \mathbb{R}^{n\times n}\mid W=W^\top, W>0 \}
\displaystyle p(x;W^{-1}) = \sqrt{\frac{|W|}{(2\pi)^n}}\exp(-\frac{1}{2}x^\top W x)

Fisher-Rao metric on P(n)

T_W P(n) = \{ U\in \mathbb{R}^{n\times n}\mid U=U^\top \}
U
W
g_W(U,U) = \frac{1}{2}\mathrm{tr}(W^{-1}UW^{-1}U)

Geodesics on P(n)

\ddot W - \dot W W^{-1}\dot W = 0

Explicit distance function

\displaystyle d(W_0,W_1)^2 = \frac{1}{2}\mathrm{tr}\big(\log(W_1W_0^{-1})\log(W_1W_0^{-1}) \big)

Geodesic equation

W_0
W_1

Where is QR?!

Homogeneous space structure

I

fiber

\pi

fiber

I
W_1
P(n)
A
Q
\mathrm{GL}(n)
\mathrm{O}(n)\backslash \mathrm{GL}(n) = \{ [A] \mid A\in\mathrm{GL}(n), [A]=\mathrm{O}(n)\cdot A \}
\mathrm{O}(n)\backslash \mathrm{GL}(n) \simeq P(n) \quad\text{by}\quad \pi\colon A\mapsto A^\top A

Principal bundle

Fisher-Rao invariance

U
W
g_W(U,U) = \frac{1}{2}\mathrm{tr}(W^{-1}UW^{-1}U)
\mathrm{GL}(n)\times P(n) \ni (A,W) \mapsto A^\top W A \in P(n)

Right action of GL(n) on P(n)

g_{A^\top W A}(A^\top U A,A^\top U A) = g_W(U,U)

Compatible metric on GL(n)

\displaystyle \bar g_A(V,V) = \frac{1}{2}\mathrm{tr}\big(\ell(VA^{-1})^\top\ell(VA^{-1})+\sigma(VA^{-1})\sigma(VA^{-1}) \big)
V
A

horizontal slice

I
R

fiber

\pi

fiber

I
W_1
K
P(n)
A
Q

Horizontal distribution

\mathrm{Hor}_A = \{ V\in T_A\mathrm{GL}(n) \mid \ell(VA^{-1}) = 0 \}
K = \{ R\in \mathrm{GL}(n)\mid \ell(R)=0, R_{ii}>0 \} \Rightarrow T_R K = \mathrm{Hor}_R

horizontal slice

I
R

fiber

\pi

fiber

I
W_1
K
P(n)
A
Q

Voilà!

horizontal slice

I
R
I
W_1
A

K
\mathrm{P}(n)
1. Convex functional      on           with
2. Lifted functional on

3. Consider gradient flow on horizontal slice
F
\mathrm{P}(n)
\nabla_{g}F(W_1) = 0
\mathrm{GL}(n)
\bar F(A) = F(\pi(A)) = F(A^\top A)
\dot R = -\nabla_{\bar g} \bar F(R)

Use relative entropy

H(W) = \frac{n}{2}- \frac{1}{2}\mathrm{tr}(W_1W^{-1}) + \frac{1}{2}\log(\det(W_1W^{-1}))
\displaystyle \dot R = \nabla_{\bar g}\bar H(R), \qquad \bar H(R) = H(R^\top R)

\displaystyle \dot R = \frac{1}{2} R^{-\top}(W_1-R^\top R) + ZR, \qquad Z\in\mathfrak{o}(n)

Use your favorite Runge-Kutta method!

That is, use RK4

Convergence

Convexity lemma:

-\mathrm{Hess}(\bar H)_R = \bar g_R

Corollary:

\bar d^2(R(t),R_\infty) \leq \, \mathrm{e}^{-2 t} \bar d^2(R(0),R_\infty)
R = \begin{bmatrix} 3 & -1 \\ 0 & 2 \end{bmatrix}
W_1 = \pi(R) = R^\top R

Example

-\bar H(R(t))
\bar d(R(t),R_\infty)^2
\text{slope of }\exp(-2t)

By Klas Modin

Silly method for QR factorization

Presentation given as CAM seminar 2017-02-22.

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