A Concentration Inequality for the Covariance Matrix of an Arbitrary Subset of Random Vectors
Measuring uncertainty when you choose which data points to analyze
When statisticians select which data to use based on what that data looks like, standard mathematical guarantees break down. This paper proves new rules for how reliable sample covariance matrices remain even after such data-dependent selection—and shows these new rules are much tighter and more practical than existing workarounds. The results extend to realistic scenarios with weakly dependent observations and apply directly to clustering problems.
Many real machine-learning algorithms pick or filter their data based on what they see, not randomly in advance. Without reliable guarantees for this setting, practitioners can't know whether their statistical conclusions are trustworthy. This work closes that gap, providing theoretical backing for algorithms that adaptively select subsets of data while maintaining provable recovery guarantees—particularly relevant for clustering tasks where picking the right groups is inherently data-dependent.