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Minimax Optimal Estimation of Transport-Growth Pairs in Unbalanced Optimal Transport
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Derives minimax-optimal rates for estimating the transport-growth pair (Monge map and growth factor) in unbalanced OT with quadratic cost and KL marginal penalties, via a plan-based 1NN estimator and a cosine-kernel plug-in estimator, using a value-based stability reduction.
Problem
Statistical estimation of Monge-type objects (transport map and growth map) in unbalanced optimal transport with KL marginal penalties, where existing theory is limited to semi-dual potentials or non-Monge quantities.
Approach
Treats the population target as a transport-growth pair (T, lambda) and reduces map/growth risk to perturbations of the UOT objective value via a stability bound and first-order expansion, using a cosine (Neumann) kernel to handle boundary issues on the hypercube.
Key results
- Kernel-based estimator attains rate Õ(N^{-2α/(2α-2+d)}) matching a minimax lower bound up to log factors on the hypercube.
- Plan-based 1NN estimator attains Õ(N^{-2/d}) rate for transport and growth estimation under unbalanced setting.
- In simulations, plug-in kernel and wavelet estimators achieve lower MSE and faster empirical rates than plan-based estimators and SSUOT.
Surprising findings
- Plan-based estimators and SSUOT show empirical learning rates faster than the theoretical upper bound in simulations.
Limitations
- Setup does not cover entropic regularization.
- Assumes known or separately estimated total masses; mass estimation error enters the rate.
- Lower bound and kernel rate restricted to smooth densities on hypercube [0,1]^d.
Experimental setup
Datasets: Completion3D
Benchmarks: Completion3D
Metrics: MSE empirical learning rate
Baselines: SSUOT PI-Wavelet
Builds on: unbalanced optimal transport (Gaussian-Hellinger / KL penalties) balanced OT minimax map estimation semi-dual UOT statistical rates
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