# Geometric Active Inference
## Overview
Geometric Active Inference provides a differential geometric framework for understanding active inference and the free energy principle. This approach reveals deep connections between information geometry, symplectic geometry, and optimal control theory.
## Geometric Structures
### 1. Statistical Manifolds
#### Definition
```math
(\mathcal{M}, g, \nabla^{(\alpha)}, \nabla^{(-\alpha)})
```
where:
- $\mathcal{M}$ is manifold of probability distributions
- $g$ is Fisher-Rao metric
- $\nabla^{(\alpha)}$ are dual connections
#### Fisher-Rao Metric
```math
g_{ij}(\theta) = \int p_\theta(x) \frac{\partial \log p_\theta(x)}{\partial \theta^i} \frac{\partial \log p_\theta(x)}{\partial \theta^j} dx
```
### 2. Belief Space Geometry
#### Tangent Space
```math
T_p\mathcal{M} = \text{span}\left\{\frac{\partial}{\partial \theta^i}\right\}_{i=1}^n
```
#### Cotangent Space
```math
T^*_p\mathcal{M} = \text{span}\{d\theta^i\}_{i=1}^n
```
### 3. Symplectic Structure
#### Canonical Form
```math
\omega = \sum_i dp^i \wedge dq^i
```
where:
- $p^i$ are momenta
- $q^i$ are coordinates
#### Hamiltonian Flow
```math
\dot{q}^i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q^i}
```
## Geometric Free Energy
### 1. Free Energy as Action
#### Action Functional
```math
S[q] = \int_0^T \left(g_{ij}\dot{\theta}^i\dot{\theta}^j + F(q_\theta)\right)dt
```
where:
- $F(q_\theta)$ is variational free energy
- $g_{ij}$ is Fisher metric
#### Euler-Lagrange Equations
```math
\frac{d}{dt}\frac{\partial L}{\partial \dot{\theta}^i} - \frac{\partial L}{\partial \theta^i} = 0
```
### 2. Natural Gradient Flow
#### Gradient Flow
```math
\dot{\theta}^i = -g^{ij}\frac{\partial F}{\partial \theta^j}
```
#### Parallel Transport
```math
\nabla_{\dot{\gamma}}\dot{\gamma} = 0
```
## Geometric Policy Selection
### 1. Policy Manifold
#### Structure
```math
\mathcal{P} = \{P_\pi : \pi \in \Pi\}
```
where:
- $P_\pi$ is policy distribution
- $\Pi$ is policy space
#### Metric
```math
h_{ij}(\pi) = \mathbb{E}_{P_\pi}\left[\frac{\partial \log P_\pi}{\partial \pi^i}\frac{\partial \log P_\pi}{\partial \pi^j}\right]
```
### 2. Expected Free Energy
#### Geometric Form
```math
G(\pi) = \int_{\mathcal{M}} g_{ij}(\theta)\dot{\theta}^i\dot{\theta}^j d\mu(\theta)
```
#### Policy Update
```math
\dot{\pi}^i = -h^{ij}\frac{\partial G}{\partial \pi^j}
```
## Implementation
### 1. Geometric Integration
```python
class GeometricIntegrator:
def __init__(self,
manifold: RiemannianManifold,
hamiltonian: Callable):
"""Initialize geometric integrator.
Args:
manifold: Riemannian manifold
hamiltonian: Hamiltonian function
"""
self.M = manifold
self.H = hamiltonian
def symplectic_euler(self,
q: np.ndarray,
p: np.ndarray,
dt: float) -> Tuple[np.ndarray, np.ndarray]:
"""Perform symplectic Euler step.
Args:
q: Position coordinates
p: Momentum coordinates
dt: Time step
Returns:
q_next,p_next: Updated coordinates
"""
# Update momentum
grad_H = self.compute_gradient(self.H, q)
p_next = p - dt * grad_H
# Update position
q_next = q + dt * p_next
return q_next, p_next
def parallel_transport(self,
v: np.ndarray,
gamma: Geodesic,
t: float) -> np.ndarray:
"""Parallel transport vector along geodesic.
Args:
v: Tangent vector
gamma: Geodesic curve
t: Parameter value
Returns:
v_t: Transported vector
"""
# Compute connection coefficients
Gamma = self.M.christoffel_symbols(gamma(t))
# Solve parallel transport equation
v_t = self.solve_transport_equation(v, gamma, Gamma, t)
return v_t
```
### 2. Natural Gradient Methods
```python
class NaturalGradientOptimizer:
def __init__(self,
manifold: StatisticalManifold,
learning_rate: float = 0.1):
"""Initialize natural gradient optimizer.
Args:
manifold: Statistical manifold
learning_rate: Learning rate
"""
self.M = manifold
self.lr = learning_rate
def compute_natural_gradient(self,
theta: np.ndarray,
grad_F: np.ndarray) -> np.ndarray:
"""Compute natural gradient.
Args:
theta: Parameters
grad_F: Euclidean gradient
Returns:
nat_grad: Natural gradient
"""
# Compute Fisher information
G = self.M.fisher_metric(theta)
# Solve metric equation
nat_grad = np.linalg.solve(G, grad_F)
return nat_grad
def update_parameters(self,
theta: np.ndarray,
grad_F: np.ndarray) -> np.ndarray:
"""Update parameters using natural gradient.
Args:
theta: Current parameters
grad_F: Euclidean gradient
Returns:
theta_next: Updated parameters
"""
# Compute natural gradient
nat_grad = self.compute_natural_gradient(theta, grad_F)
# Update parameters
theta_next = self.M.exp_map(
theta,
-self.lr * nat_grad
)
return theta_next
```
### 3. Geometric Policy Optimization
```python
class GeometricPolicyOptimizer:
def __init__(self,
policy_manifold: RiemannianManifold,
efe_function: Callable):
"""Initialize geometric policy optimizer.
Args:
policy_manifold: Policy manifold
efe_function: Expected free energy
"""
self.P = policy_manifold
self.G = efe_function
def optimize_policy(self,
pi_init: np.ndarray,
n_steps: int = 100,
learning_rate: float = 0.1) -> np.ndarray:
"""Optimize policy using geometric methods.
Args:
pi_init: Initial policy
n_steps: Number of steps
learning_rate: Learning rate
Returns:
pi_opt: Optimized policy
"""
pi = pi_init.copy()
for _ in range(n_steps):
# Compute EFE gradient
grad_G = self.compute_efe_gradient(pi)
# Compute policy metric
h = self.P.metric_tensor(pi)
# Update policy
nat_grad = np.linalg.solve(h, grad_G)
pi = self.P.exp_map(pi, -learning_rate * nat_grad)
return pi
```
## Applications
### 1. Geometric Control
- Optimal transport
- Path planning
- Trajectory optimization
- Feedback control
### 2. Information Processing
- Belief propagation
- Message passing
- Information geometry
- Statistical inference
### 3. Learning Theory
- Natural gradient descent
- Information bottleneck
- Geometric deep learning
- Manifold learning
## Best Practices
### 1. Geometric Methods
1. Preserve invariants
2. Use natural coordinates
3. Implement symplectic integrators
4. Handle parallel transport
### 2. Numerical Stability
1. Monitor geodesic distance
2. Check metric positivity
3. Regularize curvature
4. Control step size
### 3. Implementation
1. Efficient tensor operations
2. Adaptive discretization
3. Geometric integration
4. Parallel computation
## Common Issues
### 1. Technical Challenges
1. Coordinate singularities
2. Metric degeneracy
3. Geodesic completeness
4. Computational complexity
### 2. Solutions
1. Multiple charts
2. Regularization
3. Adaptive methods
4. Efficient algorithms
## Related Topics
- [[differential_geometry]]
- [[information_geometry]]
- [[symplectic_geometry]]
- [[optimal_control]]
- [[path_integral_free_energy]]
- [[variational_methods]]