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Geodesic Calculus on Implicitly Defined Latent Manifolds

This paper proposes a robust framework for performing discrete Riemannian calculus on implicitly defined latent manifolds of autoencoders by learning an approximate projection via a denoising objective, thereby enabling the computation of geodesic paths and exponential maps independent of the underlying autoencoder architecture.

Original authors: Florine Hartwig, Josua Sassen, Juliane Braunsmann, Martin Rumpf, Benedikt Wirth

Published 2026-06-17

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CC BY 4.0

Original authors: Florine Hartwig, Josua Sassen, Juliane Braunsmann, Martin Rumpf, Benedikt Wirth

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, messy pile of data—thousands of photos of faces, thousands of 3D models of human poses, or thousands of images of a rotating toy cow. In the world of machine learning, we often use a tool called an autoencoder to squash all this high-dimensional chaos down into a tiny, neat "latent space" (a small, compressed map).

Think of this latent space like a flat sheet of paper. But here's the catch: the actual data doesn't fill the whole sheet. Instead, it lives on a specific, crumpled, twisted shape inside that sheet. Maybe it's a flat circle, maybe it's a twisted torus (like a donut), or maybe it's a complex, invisible surface.

The problem is that standard machine learning tools often treat this latent space as if it were perfectly flat and empty everywhere. If you try to draw a straight line between two points on this crumpled shape, that line might cut right through "empty air" (data that doesn't exist), leading to weird, broken results when you try to turn the data back into an image or a 3D model.

This paper proposes a new way to navigate this crumpled shape. Here is the breakdown of their approach using simple analogies:

1. The Problem: The "Straight Line" Trap

Imagine you are walking on the surface of the Earth. If you want to get from New York to London, a "straight line" drawn through the center of the Earth (tunneling through the core) is mathematically straight, but it's useless for a traveler. You need to follow the curve of the Earth.

In machine learning, if you just draw a straight line between two data points in the latent space, you are tunneling through the "core" of the data manifold. The result? When you decode that line back into an image, you get gibberish or distorted shapes (like a person with a shoulder inside their head).

2. The Solution: The "Invisible Trampoline" (Implicit Representation)

The authors realized that instead of trying to map out every single point on this crumpled shape (which is hard and often impossible), they should treat the shape as an invisible boundary.

They use a special neural network to learn a "projection." Imagine a trampoline with a specific shape. If you drop a ball anywhere near it, the projection function tells you exactly where on the trampoline surface that ball would land.

The Innovation: They don't need to know the exact formula for the trampoline's shape. They just train a network to act like a magnet that pulls any point in the latent space back onto the "data surface."
The "Denoising" Trick: To teach this network, they take a bunch of valid data points, add a little bit of "noise" (shake them up), and train the network to push them back to their original, clean position. This teaches the network what the "true" surface looks like, even if the data is a bit messy.

3. The Tool: "Rubber Band" Geodesics

Once they have this "invisible trampoline" (the implicit representation), they need a way to walk along it. They use a method they call Discrete Geodesic Calculus.

Think of a geodesic as the shortest path between two points staying on the surface.

The Analogy: Imagine you have a chain of beads (a discrete path) connecting Point A to Point B. You want to pull them tight so they form the shortest path, but you have a rule: Every single bead must stay glued to the surface of the trampoline.
The Physics: They treat the chain like a series of rubber bands. They pull the chain tight (minimizing energy) while constantly checking that no bead falls off the trampoline. If a bead tries to float into the "empty air," a "penalty" pushes it back down.
The Result: This creates a smooth, curved path that hugs the data perfectly. When they decode this path back into images or 3D models, the transition is natural and realistic (e.g., a person smoothly turning their head, rather than their head suddenly melting).

4. Why This Matters (According to the Paper)

The authors tested this on three different types of data:

3D Shapes: They could morph one human pose into another without the limbs passing through each other (a common problem with straight lines).
Motion Capture: They could animate a skeleton moving naturally, respecting the complex geometry of how joints actually move.
Images: They could rotate a 3D object in an image smoothly, keeping the object looking real.

The Bottom Line

The paper doesn't claim to invent a new type of AI or a new medical scanner. Instead, it offers a better navigation system for the "hidden maps" that AI already creates.

By treating the hidden data space as a specific, curved surface (defined by a "projection" function) and using a "rubber band" method to walk along that surface, they can create smooth, realistic transitions between data points. It's like giving the AI a pair of shoes that grip the terrain, so it doesn't slip off the edge of reality when it tries to move from one idea to another.

Technical Summary: Geodesic Calculus on Implicitly Defined Latent Manifolds

Problem Statement

In machine learning, autoencoders provide low-dimensional latent representations of high-dimensional data (e.g., images, shapes) based on the manifold hypothesis. However, these latent spaces typically lack explicit geometric structure, making it difficult to perform rigorous geometric operations such as interpolation, extrapolation, or geodesic computation. While shape analysis extensively uses Riemannian manifolds to model collections of shapes (e.g., via LDDMM), applying these tools to autoencoder latent spaces is challenging because:

Lack of Explicit Parametrization: Latent manifolds often have non-zero codimension in the ambient latent space, making global parametrizations difficult or impossible due to topological obstructions.
Imperfect Representations: Practical implicit representations (learned from data) are often inaccurate or "imperfect," causing standard numerical methods to fail or produce paths that deviate from the true data support.
Metric Dependency: Existing methods often restrict geodesic computation to specific autoencoder architectures or specific metric choices (e.g., Euclidean), failing to incorporate application-dependent metrics (e.g., pullback metrics from the data manifold).

The core challenge is to equip latent manifolds with a usable geometric structure that allows for the computation of geodesic paths (interpolation and extrapolation) that adhere to the latent sample distribution, even when the manifold representation is imperfect.

Methodology

The proposed framework addresses these challenges through three main components: an implicit representation of the latent manifold, a discrete variational geodesic calculus, and a learning strategy for the projection operator.

1. Implicit Representation via Learned Projection

Instead of learning a global parametrization, the authors describe the latent manifold $Z \subset \mathbb{R}^l$ as an implicit submanifold defined by the zero level set of a function $\zeta$ .

Definition: $\zeta(z) = z - \Pi(z)$ , where $\Pi: \mathbb{R}^l \to \mathbb{R}^l$ is a projection mapping any point in the latent space to the nearest point on the manifold $Z$ .
Learning Strategy: Since the true projection $\Pi$ $Π$ is unknown, the authors propose learning an approximate projection $\Pi_\sigma$ $Π_{σ}$ by minimizing a denoising objective. This objective, inspired by denoising autoencoders (Alain & Bengio, 2014), treats the latent samples as a point cloud and learns $\Pi_\sigma$ $Π_{σ}$ to minimize the expected squared distance between a noisy sample and its projection.
- The loss functional is $Q(\Pi) = \int \int |z - \Pi(y)|^2 f_\sigma(z-y) \, dz \, dy$ , where $f_\sigma$ is a Gaussian kernel.
- This approach is independent of the underlying autoencoder architecture and supports arbitrary codimensions.

2. Discrete Variational Geodesic Calculus

To compute geodesics on these potentially imperfect implicit manifolds, the authors employ a time-discrete variational approach (based on Rumpf & Wirth, 2015).

Discrete Path Energy: A continuous path $z(t)$ is approximated by a discrete sequence $z = (z_0, \dots, z_K)$ . The path energy is approximated by a sum of local distance approximations $W(z_{k-1}, z_k)$ :
$E_K(z_0, \dots, z_K) = \sum_{k=1}^K W(z_{k-1}, z_k)$
Metric Flexibility: The framework supports different Riemannian metrics via the choice of $W$ $W$ :
- Euclidean: $W_E(z, \tilde{z}) = |\tilde{z} - z|^2$ (inherited from ambient space).
- Pullback Metric: $W_{PB}(z, \tilde{z}) = |\psi(z) - \psi(\tilde{z})|^2$ (distance in data space).
- General Riemannian: $W_M(z, \tilde{z}) = \text{dist}_M^2(\psi(z), \psi(\tilde{z}))$ (using the intrinsic metric of the data manifold).
Optimization: The geodesic is found by minimizing $E_K$ subject to the constraint $\zeta(z_k) = 0$ for all $k$ . The authors use an Augmented Lagrangian method to solve this constrained optimization. This method is robust to "imperfect" constraints (where $\zeta_\sigma$ does not vanish exactly) by penalizing deviations from the manifold while iteratively updating Lagrange multipliers.

3. Discrete Exponential Map

For geodesic extrapolation (shooting), the authors define a discrete exponential map $\text{Exp}_K$ . Given an initial point $z_0$ and a discrete velocity (defined by $z_1$ ), the method iteratively minimizes a functional $F$ that enforces the saddle point conditions of the discrete geodesic equation and the manifold constraint. This allows for the generation of geodesic paths extending beyond given endpoints.

Key Contributions

Time-Discrete Geodesic Calculus for Imperfect Manifolds: The paper introduces a numerical framework for computing geodesics on implicit latent manifolds that are not perfectly represented. The method is robust to inaccuracies in the implicit function $\zeta_\sigma$ .
Denoising-Based Projection Learning: A novel approach to learning an approximate projection onto a latent manifold of unknown codimension by minimizing a denoising objective, independent of the specific autoencoder used.
Flexible Metric Integration: The framework allows for the incorporation of application-dependent Riemannian metrics (e.g., pullback metrics) directly into the geodesic computation without requiring explicit parametrization of the manifold.
Empirical Validation: The approach is evaluated on various autoencoders (isometric, spherical VAE, low-bending) trained on synthetic and real data (discrete shells, motion capture, image rotations), demonstrating realistic interpolation and extrapolation compared to linear interpolation.

Results

The authors evaluate the method across four distinct scenarios:

Discrete Shells (Isometric Autoencoder): On the SCAPE dataset of human poses, geodesic interpolation on the learned latent manifold successfully avoids self-intersections that occur with linear interpolation in the latent space.
Motion Capture (Spherical VAE): Using a spherical VAE on CMU motion data, the method computes geodesics using a pullback spherical metric. The resulting decoded poses are realistic, whereas linear interpolation produces unnatural joint contractions and self-intersecting limbs.
Image Data (Low Bending/Distortion): On a dataset of rotating 3D objects, the method performs geodesic interpolation and extrapolation in a 16-dimensional latent space, producing smooth transitions in the decoded images.
Pose-NDF (Distance Function): The method is extended to work with neural distance fields (Pose-NDF) as the implicit representation, successfully computing geodesics on the manifold of plausible human poses where linear interpolation fails.

In all cases, the computed geodesics adhere to the data manifold's geometry, avoiding the "empty space" traversed by linear interpolation. The discrete exponential map successfully generates extrapolated paths that follow the manifold's curvature.

Significance and Claims

The paper claims to provide a practical and robust tool for performing geometric operations on latent manifolds, bridging the gap between the theoretical richness of Riemannian geometry and the practical limitations of learned latent representations.

Robustness: The primary significance lies in the method's ability to function with "imperfect" implicit representations, a common reality in practical machine learning applications where exact manifold descriptions are unavailable.
Generality: By decoupling the geometric calculus from the specific autoencoder architecture and allowing for arbitrary metrics, the framework enables new reduced-order methods for shape spaces and generative modeling.
Future Directions: The authors modestly suggest that future work could connect projection-based representations with distance-based ones, extend the calculus to latent distributions (e.g., in VAEs), and utilize discrete parallel transport for detail transfer. They acknowledge limitations regarding scalability to very high dimensions and the lack of a full mathematical error analysis combining learning errors with discretization errors.

The work aims to open new ways of utilizing latent representations in machine learning by enabling geodesic interpolation and extrapolation that respect the underlying data geometry.