Intrinsic Dimension for Large-Scale Geometric Learning
This work addresses the challenge of efficiently determining intrinsic dimension for complex geometric data, which is incremental as it builds on prior axiomatic approaches to make them practical for real-world applications.
The authors tackled the problem of computing intrinsic dimension (ID) for large-scale geometric datasets by deriving a computationally feasible method based on an axiomatic foundation, and demonstrated its application to graph data with experiments on the Open Graph Benchmark.
The concept of dimension is essential to grasp the complexity of data. A naive approach to determine the dimension of a dataset is based on the number of attributes. More sophisticated methods derive a notion of intrinsic dimension (ID) that employs more complex feature functions, e.g., distances between data points. Yet, many of these approaches are based on empirical observations, cannot cope with the geometric character of contemporary datasets, and do lack an axiomatic foundation. A different approach was proposed by V. Pestov, who links the intrinsic dimension axiomatically to the mathematical concentration of measure phenomenon. First methods to compute this and related notions for ID were computationally intractable for large-scale real-world datasets. In the present work, we derive a computationally feasible method for determining said axiomatic ID functions. Moreover, we demonstrate how the geometric properties of complex data are accounted for in our modeling. In particular, we propose a principle way to incorporate neighborhood information, as in graph data, into the ID. This allows for new insights into common graph learning procedures, which we illustrate by experiments on the Open Graph Benchmark.