Empirical Error Estimates for Graph Sparsification
This work addresses reliability issues for users of graph-based learning algorithms, offering a practical solution to replace theoretical bounds, though it is incremental as it builds on existing sparsification techniques.
The paper tackles the problem of uncertainty in graph sparsification by proposing empirical error estimates to provide practical numerical guidance, demonstrating their versatility in four use cases and showing manageable computational cost.
Graph sparsification is a well-established technique for accelerating graph-based learning algorithms, which uses edge sampling to approximate dense graphs with sparse ones. Because the sparsification error is random and unknown, users must contend with uncertainty about the reliability of downstream computations. Although it is possible for users to obtain conceptual guidance from theoretical error bounds in the literature, such results are typically impractical at a numerical level. Taking an alternative approach, we propose to address these issues from a data-driven perspective by computing empirical error estimates. The proposed error estimates are highly versatile, and we demonstrate this in four use cases: Laplacian matrix approximation, graph cut queries, graph-structured regression, and spectral clustering. Moreover, we provide two theoretical guarantees for the error estimates, and explain why the cost of computing them is manageable in comparison to the overall cost of a typical graph sparsification workflow.