Geodesic Calculus on Implicitly Defined Latent Manifolds

Latent manifolds of autoencoders provide low-dimensional representations of data, which can be studied from a geometric perspective. We propose to describe these latent manifolds as implicit submanifolds of some ambient latent space. Based on this, we develop tools for a discrete Riemannian calculus approximating classical geometric operators. These tools are robust against inaccuracies of the implicit representation often occurring in practical examples. To obtain a suitable implicit representation, we propose to learn an approximate projection onto the latent manifold by minimizing a denoising objective. This approach is independent of the underlying autoencoder and supports the use of different Riemannian geometries on the latent manifolds. The framework in particular enables the computation of geodesic paths connecting given end points and shooting geodesics via the Riemannian exponential maps on latent manifolds. We evaluate our approach on various autoencoders trained on synthetic and real data.

💡 Research Summary

This paper addresses the problem of equipping the latent space of autoencoders with a usable Riemannian geometry, enabling operations such as geodesic interpolation, extrapolation, and the computation of exponential maps directly on the learned latent manifold. Rather than requiring an explicit parametrization of the manifold, the authors treat the latent manifold Z as an implicit submanifold of the ambient latent space ℝ^ℓ, defined by a smooth function ζ(z)=z−Π(z) where Π is a projection onto Z. The projection Π is learned by minimizing a denoising objective: noisy latent vectors are passed through the encoder‑decoder pipeline and the network is trained to reconstruct the original clean latent code. This yields an approximate implicit representation ζ that points roughly in the normal direction of the true manifold, even when the codimension is unknown.

With an implicit representation in hand, the authors develop a time‑discrete variational calculus for geodesics. The continuous path energy E(z)=∫0^1 g_z(ẋ,ẋ)dt is approximated by a discrete sum E_K=∑{k=1}^K W(z_{k-1},z_k) where W is a local approximation of the squared Riemannian distance. Three concrete choices of W are discussed: (i) Euclidean metric W_E(z,ẑ)=‖ẑ−z‖², (ii) pull‑back metric through the decoder W_{PB}(z,ẑ)=‖ψ(ẑ)−ψ(z)‖², and (iii) a metric induced by the original data space M, W_M(z,ẑ)=dist_M²(ψ(z),ψ(ẑ)). The discrete geodesic is defined as the minimizer of E_K subject to the constraint ζ(z_k)=0 for all nodes.

Because the learned ζ is only approximate, the constrained optimization is solved using an augmented Lagrangian method. The Lagrangian L_a(z,Λ,μ)=E_K(z)−Λ:ζ(z)+½μ‖ζ(z)‖² introduces Lagrange multipliers Λ and a penalty parameter μ. Alternating updates of the primal variables z and the multipliers Λ, together with a geometric increase of μ, enforce the constraints while tolerating small violations of ζ. The authors show that as long as ζ and its Jacobian approximate the true normal direction sufficiently well, the method converges and remains stable even when the implicit function is imperfect.

For geodesic extrapolation, a discrete exponential map Exp_K is constructed. Given the first two points (z₀,z₁), an initial discrete velocity v₀=K(z₁−z₀) is defined. The next point z_{k+1} and an auxiliary multiplier λ_k are obtained by minimizing the functional F(z_{k+1},λ_k)=μ²‖ζ(z_{k+1})‖²+‖∂{z_k}L‖², where ∂{z_k}L represents the discrete Euler‑Lagrange condition. A BFGS optimizer solves this sub‑problem iteratively, and the final node z_K is taken as Exp_K(z₀,v₀). Theoretical results from Rumpf & Wirth (2015) guarantee that Exp_K converges to the continuous exponential map as K→∞.

The experimental evaluation covers synthetic manifolds (a torus embedded in ℝ³) and real image datasets where a standard autoencoder learns a 2‑dimensional latent representation of rotated and scaled letters. The learned projection Π produces a smooth implicit field ζ_σ, and the discrete geodesic calculus yields interpolation paths that stay on the data manifold, outperforming naïve linear interpolation both visually and quantitatively. The method works with various autoencoder architectures (deterministic AEs, VAEs) and with any of the three metric choices, demonstrating flexibility. Comparisons with prior latent‑space geodesic methods show that the implicit‑representation approach avoids the need for explicit manifold coordinates or pre‑specified codimension, and it handles imperfect representations gracefully thanks to the augmented Lagrangian scheme.

In summary, the paper makes four key contributions: (1) a denoising‑based learning scheme for an approximate projection onto an unknown‑codimension latent manifold, (2) a robust time‑discrete variational framework for geodesic computation on implicit manifolds, (3) a practical algorithm for discrete exponential maps enabling geodesic extrapolation, and (4) extensive empirical validation confirming that the approach yields accurate, distribution‑aware paths across a variety of latent spaces and metrics. By bridging implicit neural representations with discrete Riemannian calculus, the work opens a new avenue for geometric reasoning in deep generative models.