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Learning maps between data samples is fundamental. Applications range from representation learning, image translation and generative modeling, to the estimation of spatial deformations. Such maps relate feature vectors, or map between feature spaces. Well-behaved maps should be regular, which can be imposed explicitly or may emanate from the data itself. We explore what induces regularity for spatial transformations, e.g., when computing image registrations. Classical optimization-based models compute maps between pairs of samples and rely on an appropriate regularizer for well-posedness. Recent deep learning approaches have attempted to avoid using such regularizers altogether by relying on the sample population instead. We explore if it is possible to obtain spatial regularity using an inverse consistency loss only and elucidate what explains map regularity in such a context. We find that deep networks combined with an inverse consistency loss and randomized off-grid interpolation yield well behaved, approximately diffeomorphic, spatial transformations. Despite the simplicity of this approach, our experiments present compelling evidence, on both synthetic and real data, that regular maps can be obtained without carefully tuned explicit regularizers, while achieving competitive registration performance.
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This content will become publicly available on December 1, 2025
Correlational Lagrangian Schrödinger Bridge: Learning Dynamics with Population-Level Regularization
Modeling population dynamics is a fundamental problem with broad scientific applications. Motivated by real-world applications including biosystems with diverse populations, we consider a class of population dynamics modeling with two technical challenges: (i) dynamics to learn for individual particles are heterogeneous and (ii) available data to learn from are not time-series (i.e, each individual’s state trajectory over time) but cross-sectional (i.e, the whole population’s aggregated states without individuals matched over time). To address the challenges, we introduce a novel computational framework dubbed correlational Lagrangian Schrödinger bridge (CLSB) that builds on optimal transport to “bridge" cross-sectional data distributions. In contrast to prior methods regularizing all individuals’ transport “costs” and then applying them to the population homogeneously, CLSB directly regularizes population cost allowing for population heterogeneity and potentially improving model generalizability. Specifically our contributions include (1) a novel population perspective of the transport cost and a new class of population regularizers capturing the temporal variations in multivariate relations, with the tractable formulation derived, (2) three domain-informed instantiations of population regularizers on covariance, and (3) integration of population regularizers into data-driven generative models as constrained optimization and an approximate numerical solution, with further extension to conditional generative models. Empirically, we demonstrate the superiority of CLSB in single-cell sequencing data analyses (including cell differentiation and drug-conditioned cell responses) and opinion depolarization.
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- Award ID(s):
- 1943008
- PAR ID:
- 10596092
- Publisher / Repository:
- NeurIPS 2024 Workshop on AI for New Drug Modalities (AIDrugX)
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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