Lipschitz Continuity of Mahalanobis Distances and Bilinear Forms
This provides theoretical guarantees for machine learning applications relying on these metrics, though it is incremental as it builds on existing mathematical frameworks.
The paper tackled the problem of establishing Lipschitz continuity for Mahalanobis distances and bounded-space bilinear forms, deriving tight Lipschitz constants for these metrics, which had not been formally proven before.
Many theoretical results in the machine learning domain stand only for functions that are Lipschitz continuous. Lipschitz continuity is a strong form of continuity that linearly bounds the variations of a function. In this paper, we derive tight Lipschitz constants for two families of metrics: Mahalanobis distances and bounded-space bilinear forms. To our knowledge, this is the first time the Mahalanobis distance is formally proved to be Lipschitz continuous and that such tight Lipschitz constants are derived.