Resilient Coresets and Clustering

Many machine learning problems are geometric at their core, relying on metric representations of data for tasks such as clustering, prototype selection, nearest-neighbor search, and graph-based learning. Furthermore, data is constantly evolving and it is routinely transformed through dimensionality reduction, random projections, feature embeddings, compression, or privacy-preserving mechanisms. These transformations are designed to preserve geometry approximately. As a result, they preserve objective values for many geometric optimization problems, but they fail to guarantee that algorithmic outcomes remain consistent. In this work, we study resilient data summaries for geometric optimization. Building on the notion of $\gamma$-resilient algorithms from Ahmadian, we introduce $\gamma$-resilient coresets. A $\gamma$-resilient $(k,\varepsilon)$-coreset is a compact, weighted summary that guarantees a $(1+\varepsilon)$ approximation to the objective and enforces stability at the level of assignments. We complement our positive result with a lower bound showing that to obtain a tight approximation for resilient clustering it is necessary to use a bi-criteria solution.