Erik Jansson
WASP and KAW postdoctoral fellow in the Cambridge Image Analysis Group and CRA at Wolfson College.
Sub-Riemannian landmark matching and ResNets
Joint work with Klas Modin
ResNets?
Fully-connected equal layers
with a fixed final node
Skip connection \(\implies \) Explicit Euler
\ldots
ResNets?
Infinite number of layers \(\implies\)
Time-continuous optimal control problem
Skip connection \(\implies \) Explicit Euler
\ldots
Diffeomorphic shape matching
f_0 \in C^\infty(M)
f_1 \in C^\infty(M)
Landmark matching
x_1,\dots, x_n \in M
c_1,\dots,c_n \in M
Landmark matching
\dot y_i = v(t,y_i), ~ y_i(0)=x_i, v(t,\cdot) \in \mathcal{X}(M)
Evolve points along a time-dependent vector field!
Which vector field?
Landmark matching
Penalize missing targets:
\sum_{i=1}^n d_M^2(y_i(1),c_i)
Penalize weird vector fields:
\int_0^1 \int_M Lv(t) \cdot v(t)\mathrm{d}x \mathrm{d}t
\,\, E(v) = \int_0^1 \int_M Lv(t) \cdot v(t)\mathrm{d}x \mathrm{d}t+ \sum_{i=1}^n d_M^2(y_i(1),c_i) \,\,
Key question: How does \(v\) evolve in time?Â
Landmark matching
 Trajectory of \(m_t\) does not care about matching!
Cannonball does not care about target
Landmark matching
Use the Lagrangian \({\int_0^T \int_M Lv(t) \cdot v(t)\mathrm{d}x\mathrm{d}t}_{}\)Â
Calculus of variations gives:
\,\dot m = \nabla^T_v m - \nabla_m v + m \operatorname{div}v\,\\
m = Lv
\dot m = \operatorname{ad}_v^T m
Modifying landmark matching
x_1,\dots, x_n \in M
c_1,\dots,c_n \in M
Modifying landmark matching
x_1,\dots, x_n \in M
c_1,\dots,c_n \in {\color{red} N}
h
Modifying landmark matching
In usual LM, all vector fields are available
Idea: Constrain set of vector fields
Sub-Riemannian landmark matching!
Modifying landmark matching
v \in \mathcal{S} \iff
\exists u \in \mathcal{U} \text{ s.t. } v(x) = F(u)(x)
\(\mathcal{S}\)
\(v = F(u)(x)\)
\(\mathfrak{X}(M)\)
\(F(u)\)
\(\mathcal{U}\)
\(u\)
Modifying landmark matching
v = F(u)
Lagrangian becomes \({\int_0^T \int_M LF(u(t)) \cdot F(u(t))\mathrm{d}x\mathrm{d}t}_{}\)Â
Points evolve by \(\dot y_i = F(u(t))(y_i)\)
Modifying landmark matching
A(u)\dot u = (D_u^T m \circ L^{-1}) \cdot \operatorname{ad}^T_v m
A(u) = D_u^T m \circ L^{-1} \circ D_u m
How to find dynamics of \(u\)?
Idea: Apply chain rule to \(\dot m = \operatorname{ad}_v^T m\)
A(u)\dot u = (D_u^T m \circ L^{-1}) \cdot \operatorname{ad}^T_v m
\dot m = \operatorname{ad}^T_v m +L\mathcal{M}(u)
Modifying landmark matching
\(\mathcal{S}\)
\(v = F(u)(x)\)
\(\mathfrak{X}(M)\)
\(F(u)\)
\(\mathcal{U}\)
\(u\)
Control-affine systems
\mathcal U= \mathbb{R} ^m
F(u){=}X^0 {+} {\sum_{i=1}^m} u_i X^i
[A(u)]_{ij}{=}\int_{M} LX^i \cdot X^j
Simplification of geometry!
- Set \(M=\mathbb{R}^2/(2\pi \mathbb{Z})^2 \)
- Let \(X^i\) be eigenfunctions of toroidal Laplacian
m=16, \\
\cos(2y)\partial y, \sin(3x)\partial x,\cos(y)\partial x,...
Computational example
Control-affine systems
Â
- Initial guessof inital control \(u^{[0]}_0\)
- Evolve \(u^{[0]}_0\) and landmarks
- Evaluate energy functional \(E\).
- Update control by \(u^{[1]}_0 \leftarrow u^{[0]}_0-\epsilon \nabla_{u^{[1]}_0}E\)
A(u)\dot u = (D_u^T m \circ L^{-1}) \cdot \operatorname{ad}^T_v m
\dot y_i = F(u)(y_i), ~y_i(0)=x_i
Computational example
"Adjust initial conditions until target is hit"
Control-affine systems
Goal: Move landmarks from one fish to the other
Control-affine systems
Control-affine systems
Goal: Move landmarks on torus to \([0,1]\)
[{{\color{maroon}0},\color{blue}1}]
Projection
Control-affine systems
Badly behaved transformation!
Increase regularization strength?
Control-affine systems
Control-affine systems
Why study sR-LM?
When matching landmarks, points are moved by a vector field parametrized by some control variable.
A neural networks moves points along a vector field determined by weights and biases.
- Landmarks
- Image
- Shooting
- Warping new landmarks
- Control parameters
- Input data, e.g. images
- "Meta-image"
- Training network
- Testing
- Weights and biases
 View networks with shape analysis glasses!
Why study sR-LM?
Why study sR-LM?
Available vector fields:
Iterated Lie brackets
If distribution is integrable, we
move only along blue line
To have reachability, we must destroy integrability
\(\implies\) Nonlinearity!
Move from
to
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We can parametrize landmark matching and retain geometric frameworkÂ
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There is a connection to neural networks
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Future work: leverage connection for concrete AI applications
MAGIC 2023
By Erik Jansson
