Displacement-Sparse Neural Optimal Transport

• URL: http://arxiv.org/abs/2502.01889v1
• Date: Mon, 03 Feb 2025 23:44:17 GMT
• Title: Displacement-Sparse Neural Optimal Transport
• Authors: Peter Chen, Yue Xie, Qingpeng Zhang,
• Abstract summary: Optimal Transport (OT) theory seeks to determine the map $T:X to Y$ that transports a source measure $P to a target measure $Q$.<n>We introduce a sparsity penalty into the minimax Wasserstein formulation, promote sparsity in displacement vectors $Delta(mathbfx) := T(mathbfx) := T(mathbfx) := T(mathbfx) := T(mathbfx) := T(mathbf
• Score: 6.968698312185846
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Optimal Transport (OT) theory seeks to determine the map $T:X \to Y$ that transports a source measure $P$ to a target measure $Q$, minimizing the cost $c(\mathbf{x}, T(\mathbf{x}))$ between $\mathbf{x}$ and its image $T(\mathbf{x})$. Building upon the Input Convex Neural Network OT solver and incorporating the concept of displacement-sparse maps, we introduce a sparsity penalty into the minimax Wasserstein formulation, promote sparsity in displacement vectors $\Delta(\mathbf{x}) := T(\mathbf{x}) - \mathbf{x}$, and enhance the interpretability of the resulting map. However, increasing sparsity often reduces feasibility, causing $T_{\#}(P)$ to deviate more significantly from the target measure. In low-dimensional settings, we propose a heuristic framework to balance the trade-off between sparsity and feasibility by dynamically adjusting the sparsity intensity parameter during training. For high-dimensional settings, we directly constrain the dimensionality of displacement vectors by enforcing $\dim(\Delta(\mathbf{x})) \leq l$, where $l < d$ for $X \subseteq \mathbb{R}^d$. Among maps satisfying this constraint, we aim to identify the most feasible one. This goal can be effectively achieved by adapting our low-dimensional heuristic framework without resorting to dimensionality reduction. We validate our method on both synthesized sc-RNA and real 4i cell perturbation datasets, demonstrating improvements over existing methods.

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