Many modern learning tasks require models that can take inputs of varying sizes. Consequently, dimension-independent architectures have been proposed for domains where the inputs are graphs, sets, and point clouds. Recent work on graph neural networks has explored whether a model trained on low-dimensional data can transfer its performance to higher-dimensional inputs. We extend this body of work by introducing a general framework for transferability across dimensions. We show that transferability corresponds precisely to continuity in a limit space formed by identifying small problem instances with equivalent large ones. This identification is driven by the data and the learning task. We instantiate our framework on existing architectures, and implement the necessary changes to ensure their transferability. Finally, we provide design principles for designing new transferable models. Numerical experiments support our findings.

翻译：许多现代学习任务要求模型能够处理可变尺寸的输入。因此，在输入为图、集合和点云等领域，人们提出了维度无关的架构。图神经网络的最新研究探讨了在低维数据上训练的模型能否将其性能迁移到更高维的输入上。我们通过引入一个跨维度可迁移性的通用框架来扩展这一研究体系。我们证明，可迁移性精确对应于一个极限空间中的连续性，该空间通过将小规模问题实例与等价的大规模实例进行识别而形成。这种识别由数据和学习任务驱动。我们在现有架构上实例化了我们的框架，并实施了必要的修改以确保其可迁移性。最后，我们为设计新的可迁移模型提供了设计原则。数值实验支持了我们的发现。