unbalanced ot alignment

This work addresses the performance degradation of conventional Coherent Point Drift (CPD) in the presence of substantial outliers and partial overlap, which stems from enforcing normalization of the target point set’s probability mass. To overcome this limitation, the authors propose an unbalanced entropy-regularized optimal transport framework that replaces marginal constraints with a dual Kullback–Leibler divergence, yielding a fully unbalanced registration model. The formulation is efficiently solved via generalized Sinkhorn iterations while preserving CPD’s closed-form Procrustes transformation and variance update mechanisms. The method automatically anneals to transition from fuzzy to sharp correspondences without manual parameter tuning. Extensive experiments demonstrate state-of-the-art accuracy on synthetic data, cross-category benchmarks, and scan-to-CAD registration tasks, significantly enhancing robustness against outliers and partial overlaps.

In entropy-regularized optimal transport under Gaussian distributions, the Sinkhorn algorithm struggles to solve nonlinear transformations exactly. Method: This paper proposes a finite-dimensional recursive Sinkhorn algorithm. It establishes, for the first time, an explicit recursive analytical form of Sinkhorn iterations in the Gaussian setting, deeply coupling iterative scaling with the Kalman filter and Riccati matrix difference equation frameworks. Contributions/Results: We derive closed-form solutions for both the entropic transport map and the Schrödinger bridge. Moreover, we provide the first complete convergence analysis, rigorously proving linear convergence. The method enables exact, efficient, and analytically tractable numerical computation for multivariate Gaussian settings—without approximation. By unifying optimal transport, Schrödinger bridges, and filtering theory, it offers a novel tool for probabilistic modeling and dynamic inference.

Conventional unbalanced optimal transport (UOT) in heterogeneous spaces suffers from fixed, pre-specified ground cost functions that fail to capture intrinsic geometric structures of data. Method: We propose a cost-regularized unbalanced Gromov–Wasserstein (NR-UGW) framework that jointly optimizes the transport plan and a learnable ground cost function. The cost is parameterized via linear transformations within an inner-product family and integrated with entropic regularization for computational efficiency. NR-UGW supports mass creation/destruction and structural alignment across Euclidean spaces. Contribution/Results: Unlike standard UOT, NR-UGW dynamically adapts to geometric discrepancies between heterogeneous measures. On single-cell multi-omics data, it significantly improves contour alignment accuracy under sample-missing conditions—particularly where cell-level correspondences are absent—enabling robust cross-modal integration without explicit matching.

To address the challenges of outlier sensitivity, poor robustness to class imbalance, and high computational cost in unbalanced entropic optimal transport (UEOT), this paper proposes a lightweight, theoretically rigorous solution framework. Methodologically, we formulate a non-minimax objective function amenable to analytical optimization, integrating entropy regularization with relaxed marginal constraints, and employ a lightweight parameterized network trained via gradient-based optimization. Theoretically, we establish, for the first time, general approximation guarantees and generalization error bounds for UEOT solutions. Empirically, our method efficiently solves UEOT problems within minutes on CPU hardware, consistently outperforming existing heuristic and multi-network approaches on both synthetic and real-world datasets. It achieves superior robustness to label noise and class imbalance, higher accuracy, and better scalability—demonstrating a favorable trade-off among statistical fidelity, computational efficiency, and practical applicability.

This work addresses the rigidity of classical optimal transport in scenarios where mass conservation does not hold and its poor sample complexity in high dimensions. The authors investigate the finite-sample theory of entropy-regularized unbalanced optimal transport, introducing a translation-invariant dual formulation and analyzing its geometric structure. They establish, for the first time, high-probability convergence bounds for empirical optimal couplings at the coupling level. Their analysis demonstrates that entropy regularization not only substantially mitigates the curse of dimensionality and reduces the required sample size but also enhances estimation stability, all while preserving compatibility with efficient solvers such as Sinkhorn algorithms.

This work addresses the limitations of traditional Bayesian inference, which relies on exact likelihoods and suffers when the likelihood is misspecified, intractable, or misaligned with the target discrepancy. The authors propose the first integration of Sinkhorn divergence as a generalized Bayesian loss within Hamiltonian Monte Carlo (HMC) and its adaptive variant NUTS, accommodating both balanced and unbalanced optimal transport settings. They further incorporate common random numbers to handle stochastic simulators and introduce a heuristic for hyperparameter selection that preserves gradient consistency with Hamiltonian dynamics. Empirical evaluations on Gaussian models, noisy spiral manifolds, pulse misalignment, and CIFAR-10 image patch alignment demonstrate the method’s effectiveness, robustness, and the nuanced differences between transport mechanisms.