Dataset distillation (DD) is an increasingly important technique that focuses on constructing a synthetic dataset capable of capturing the core information in training data to achieve comparable performance in models trained on the latter. While DD has a wide range of applications, the theory supporting it is less well evolved. New methods of DD are compared on a common set of benchmarks, rather than oriented towards any particular learning task. In this work, we present a formal model of DD, arguing that a precise characterization of the underlying optimization problem must specify the inference task associated with the application of interest. Without this task-specific focus, the DD problem is under-specified, and the selection of a DD algorithm for a particular task is merely heuristic. Our formalization reveals novel applications of DD across different modeling environments. We analyze existing DD methods through this broader lens, highlighting their strengths and limitations in terms of accuracy and faithfulness to optimal DD operation. Finally, we present numerical results for two case studies important in contemporary settings. Firstly, we address a critical challenge in medical data analysis: merging the knowledge from different datasets composed of intersecting, but not identical, sets of features, in order to construct a larger dataset in what is usually a small sample setting. Secondly, we consider out-of-distribution error across boundary conditions for physics-informed neural networks (PINNs), showing the potential for DD to provide more physically faithful data. By establishing this general formulation of DD, we aim to establish a new research paradigm by which DD can be understood and from which new DD techniques can arise.

本研究解决了数据集蒸馏（DD）理论支持不足的问题，提出了一种正式的DD模型，强调了在个别应用中针对特定推断任务的优化问题的必要性。通过系统分析现有DD方法的优缺点，我们展示了DD在医学数据分析与物理信息神经网络等领域的新应用，可能对新技术的发展产生深远影响。

基于第一原则的数据集蒸馏：整合核心信息提取和有目的的学习