Hyperbolic Latent Space

A hyperbolic latent space is a representation space with negative curvature. It is useful for branching or hierarchical structures because the available space grows rapidly as points move away from the centre.

Poincaré ball

The Poincaré ball is one coordinate model of hyperbolic geometry. It draws the space inside a ball using ordinary coordinates, but it uses a hyperbolic distance rule.

• all valid points stay inside the ball
• points near the boundary can be very far apart under hyperbolic distance
• geodesics are the shortest paths under this curved distance rule

Exponential map

In GeoWorld, the visual encoder first produces an ordinary Euclidean latent state. An exponential map at the origin maps this vector into the Poincaré ball.

Euclidean encoder output → exponential map → hyperbolic latent state

The exp symbol means an exponential map from a tangent space to a curved manifold. It does not mean the ordinary scalar function $e^x$.

Why GeoWorld uses it

From one state, several actions can produce several future states. Repeating this process creates a branching tree of possible futures. Hyperbolic space can represent this hierarchy with less distortion than a flat Euclidean space.

GeoWorld uses hyperbolic geodesic distance as the energy between predicted and target states. This creates the Energy-Based Model used by its planner.

Careful interpretation

• The mapping changes the geometry used for distance and planning, rather than adding an energy value as one more feature in the vector
• A geodesic is shortest in latent space, but it is not automatically the shortest robot motion or the lowest physical energy path
• Hyperbolic geometry and Geometric Reinforcement Learning both contribute to GeoWorld, so the full improvement cannot be assigned to geometry alone

My way of understanding the mapping

I initially thought the exp() operation was an exponential function, or $e$ to the power of the latent state. At least now I know it is an exponential map, not an exponential function.

My way of visualizing the Poincaré ball is that we draw a ball using Euclidean coordinates, but the ball represents a hyperbolic space because it uses a different distance rule. We keep every point inside this ball.

The current observation and goal image first become Euclidean latent representations. The exponential map then maps both representations into hyperbolic space. This does not add an extra energy feature to each vector. It changes how distance is measured between the representations.

I think of a geodesic path as the shortest path along a curved manifold, such as the hyperbolic ball. My initial assumption was that this must also be the most energy-efficient physical path. The correction is that it is shortest under the learned latent-space distance, so the physical robot still needs separate validation.