arXiv:2505.07124v3 Announce Type: replace Abstract: We study inverse problems where an unknown potential is observed only through samples from the measure it induces by a convex variational principle. Such problems arise in learning costs, energies, and dynamics from distributional data, but the associated forward solution map is typically nonlinear and implicit. We show that its optimality gap nevertheless yields convex empirical objectives for finite-dimensional potential classes, and we introduce sharpened Fenchel--Young losses that add a data-dependent discrepancy inside the forward proble
This paper represents a methodological advancement in machine learning, specifically in inverse problems, showing how to learn potentials from sampled distributional data, building on ongoing research in AI foundations.
Improved methods for learning from distributional data can lead to more robust and powerful AI systems, particularly in areas like reinforcement learning, generative models, and scientific discovery where underlying potentials are critical.
The proposed 'sharpened Fenchel-Young losses' offer a novel approach to handle typically nonlinear and implicit solution maps in inverse problems, potentially making such learning tasks more tractable and efficient.
- · AI researchers
- · Machine learning startups
- · Computational scientists
- · Generative AI developers
- · Traditional statistical modeling approaches
More accurate and efficient learning of underlying principles from complex high-dimensional data will be possible.
This could accelerate the development of autonomous AI systems that learn from observation, leading to more sophisticated AI agents.
Advances in understanding inverse problems could enable better control and design of complex dynamic systems, impacting fields from robotics to materials science.
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Read at arXiv cs.LG