[Paper Review] Uniform Interpolation Constrained Geodesic Learning on Data Manifold

Cong Geng, Jia Wang+4 more|arXiv (Cornell University)|Feb 12, 2020

Generative Adversarial Networks and Image Synthesis22 references4 citations

TL;DR

This paper proposes a geodesic learning method that generates uniform, high-quality interpolations between data samples by enforcing constant-speed and minimizing geodesic constraints on a data manifold. Using an autoencoder with geometric regularization and an interpolation network with Riemannian metric-based losses, the method ensures smooth, realistic transitions that remain within the data distribution, outperforming linear interpolation in path length and visual quality on MNIST and Fashion-MNIST.

ABSTRACT

In this paper, we propose a method to learn a minimizing geodesic within a data manifold. Along the learned geodesic, our method can generate high-quality interpolations between two given data samples. Specifically, we use an autoencoder network to map data samples into latent space and perform interpolation via an interpolation network. We add prior geometric information to regularize our autoencoder for the convexity of representations so that for any given interpolation approach, the generated interpolations remain within the distribution of the data manifold. Before the learning of a geodesic, a proper Riemannianmetric should be defined. Therefore, we induce a Riemannian metric by the canonical metric in the Euclidean space which the data manifold is isometrically immersed in. Based on this defined Riemannian metric, we introduce a constant speed loss and a minimizing geodesic loss to regularize the interpolation network to generate uniform interpolation along the learned geodesic on the manifold. We provide a theoretical analysis of our model and use image translation as an example to demonstrate the effectiveness of our method.

Motivation & Objective

• To learn a minimizing geodesic on a data manifold that ensures smooth, realistic interpolation between data samples.
• To address the limitation of non-convex latent representations in autoencoders when modeling curved manifolds like swiss-roll or S-curve.
• To enforce uniform interpolation by parameterizing the geodesic with arc-length, ensuring constant-speed traversal.
• To provide a mathematically rigorous geodesic definition consistent with Riemannian geometry, unlike prior graph-based or latent-space-only approaches.
• To improve interpolation quality by combining autoencoder-based manifold learning with Riemannian metric constraints on the interpolation network.

Proposed method

• An autoencoder is trained with prior geometric information to promote convex latent representations, ensuring interpolated points remain within the data manifold.
• The Riemannian metric on the data manifold is induced via isometric immersion in Euclidean space, enabling geometric constraints in the latent space.
• A constant-speed loss is introduced to parameterize the interpolation curve using arc-length, ensuring uniform progression along the geodesic.
• A minimizing geodesic loss enforces the interpolation path to follow the shortest path on the manifold, consistent with Riemannian geometry.
• The full model combines the autoencoder and interpolation network, jointly optimized with both losses to generate high-quality, uniform interpolations.
• The method uses variational and adversarial autoencoders (VAE, AAE) to embed high-dimensional image data into a latent space for interpolation.

Experimental results

Research questions

• RQ1 Can a deep learning framework learn a minimizing geodesic on a data manifold that is consistent with Riemannian geometry?
• RQ2 How can uniform interpolation be achieved along a geodesic path using a parameterized curve with constant speed?
• RQ3 Can geometric regularization of the autoencoder ensure convex latent representations even on curved manifolds?
• RQ4 Does the proposed method generate more realistic and smoother interpolations than linear interpolation in image data?
• RQ5 How does the geodesic path length compare to linear interpolation in terms of minimizing path length and variance?

Key findings

• The proposed method achieves a geodesic path length of 1.1028 on a 3D curved manifold, closest to the true geodesic length of 1.0776, demonstrating high accuracy in geodesic approximation.
• On MNIST and Fashion-MNIST, the geodesic interpolation path has a significantly shorter average length and lower variance than linear interpolation, confirming its minimizing property.
• Visual results show that geodesic interpolation produces more realistic and semantically meaningful transitions between images, avoiding artifacts like ghosting seen in linear interpolation.
• The velocity curve of the geodesic interpolation remains smooth and consistent, while linear interpolation exhibits erratic speed variations, confirming the effectiveness of the constant-speed loss.
• Ablation studies confirm that combining both constant-speed and minimizing geodesic losses yields the best performance, with the shortest path and most uniform interpolation.
• The method maintains interpolants within the data manifold distribution, avoiding out-of-distribution samples, even on complex, curved manifolds.