Research map / Learning theory and algorithms / Optimal transport
Fréchet Regression on the Bures-Wasserstein Manifold
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Studies existence, landscape, and projection-free Riemannian (full-batch and pairwise stochastic) optimization of signed-weight conditional barycenters (Fréchet regression) on the Bures-Wasserstein SPD manifold, under a Spectral Dominance of Positive Weights condition.
Problem
Fréchet regression with extrapolation produces negative weights, making the conditional barycenter on the Bures-Wasserstein SPD manifold a non-convex signed barycenter problem whose minimizers may not exist and whose iterates may leave the SPD cone.
Approach
A sufficient eigenvalue-based condition (positive-weight spectral dominance) guaranteeing a minimizer in the open SPD cone, a no-local-maximum result, projection-free Riemannian GD, and a pairwise stochastic reformulation pairing positive and negative weight terms so that stochastic steps stay SPD.
Key results
- On the ant networks, the BW method achieves modularity 0.13 to 0.18 versus ground truth around 0.30, closer than Frobenius and Fisher-Rao baselines, and lowest degree-distribution Wasserstein distance at all time steps.
- Full-batch Riemannian GD drives the gradient norm to about 1e-10 on all 11 days, with interpolation converging in under 10 iterations.
- Pairwise R-SGD on 100,000 synthetic DTI tensors drops the objective from about 0.6 to below 0.2 within 20 iterations and stays SPD without projections.
Surprising findings
- Under the stated condition, the objective has no local maxima; the Example shows a signed objective with no critical point in the SPD cone when the condition fails.
Limitations
- Existence condition is only sufficient and can be restrictive; uniqueness requires stronger assumptions with all data in a small ball.
- Stochastic reformulation requires a slightly stronger min/max eigenvalue condition than the original.
- Experiments are small-scale on the network data and use synthetic DTI; no real large-scale DTI data.
Experimental setup
Datasets: Ant Social Organization network dataset (Network Repository) synthetic helix DTI tensors
Benchmarks: Ant Social Organization temporal networks synthetic DTI helix regression
Metrics: gradient norm Wasserstein distance between degree distributions modularity Bures-Wasserstein error objective value
Baselines: Frobenius (arithmetic mean) Fréchet regression Fisher-Rao (geometric mean) Fréchet regression
Builds on: Fréchet regression Wasserstein barycenter Riemannian stochastic gradient descent
Closest papers by meaning
- Computing Optimal Transport Maps and Wasserstein Barycenters Using Conditional Normalizing Flows (ICML 2025 · similarity 0.87)
- Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm (NeurIPS 2020 · similarity 0.87)
- Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimization (ICML 2025 · similarity 0.85)
- On the Convergence of Projected Bures-Wasserstein Gradient Descent under Euclidean Strong Convexity (ICML 2024 · similarity 0.85)
- Estimating Barycenters of Distributions with Neural Optimal Transport (ICML 2024 · similarity 0.85)
- Energy-Guided Continuous Entropic Barycenter Estimation for General Costs (NeurIPS 2024 · similarity 0.84)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.