Math Goal

By the end of the summer I want to have read all of the following papers and have an outline of my own original contribution.

Foundational & General Approaches

1. A Riemannian Framework for Optimal Transport

   • Authors: M. Arjovsky, A. Doucet, et al.
   • Develops a geometry-aware framework for OT over manifolds, making PCA-like generalizations more natural in curved spaces.

2. Learning Generative Models with Sinkhorn Divergences

   • Authors: Genevay et al.
   • While not PCA-specific, this introduces latent representations learned with OT-based losses—relevant for embedding/latent space OT.

PCA-Like and Subspace Methods with OT

1. Wasserstein Principal Geodesic Analysis

   • Authors: Seguy, Cuturi
   • Generalizes PCA to the Wasserstein space using geodesic analysis, important for structured data like distributions.

2. Wasserstein Principal Component Analysis: Sparse Optimal Transport Based Dimensionality Reduction

   • Authors: Wang et al.
   • An explicit method for Wasserstein PCA, adapted for probability distributions rather than Euclidean vectors.

3. A New Formulation of Principal Geodesic Analysis in the Wasserstein Space

   • Authors: Bigot et al.
   • Focuses on computing principal components in Wasserstein space more efficiently, relevant for shape and distributional data.

Embedded Manifolds & Latent OT

1. Learning Optimal Transport Maps using Generative Adversarial Networks

   • Introduces learning OT in latent/embedded spaces, allowing for manifold-constrained transport.

2. Learning Optimal Transport for Domain Adaptation

   • Authors: Damodaran et al.
   • Uses PCA for latent dimensionality reduction, then applies OT—an instance of embedded OT.

3. Manifold Matching using OT

   • Proposes OT over non-Euclidean geometries (e.g., spheres), and matching distributions on these curved spaces.

Autoencoders, Latent Space + OT

1. Sliced-Wasserstein Autoencoders

   • Uses Sliced-Wasserstein distance in a latent representation learning setting.

2. Autoencoding Probabilistic PCA with Optimal Transport

   • Combines PCA, probabilistic models, and OT regularization.

Other Notable Contributions

1. Wasserstein Dictionary Learning

   • Extends PCA to the OT setting via sparse dictionary learning over probability distributions.

2. Sliced-Wasserstein Flow: Nonparametric Generative Modeling via Optimal Transport and Diffusions

   • Uses diffusion + OT to generate data in embedded spaces, related to low-dimensional transport learning.

Summary of Concepts

Concept	Relation to OT
-------------------------	-----------------------------------------
Wasserstein PCA	PCA in probability/measure space
Geodesic PCA (PGA)	PCA generalized to curved OT geometry
Latent OT / Embedded OT	OT after PCA or in learned subspaces
Sliced Wasserstein	Efficient approximation of OT for high-D
Manifold OT	Optimal transport on Riemannian manifolds
Autoencoding + OT	Embedding generation with OT-based loss