High dimensional statistics - session 11

This lecture is part of a course on high-dimensional statistics. The instructor begins by reviewing the previous session, which introduced the Null Space Property (NSP) as a necessary and sufficient condition for sparse recovery via l1 minimization. The main topic of this session is the Restricted Isometry Property (RIP). The instructor defines RIP, which requires that for any sparse vector with support size at most s, the norm of the matrix-vector product is approximately preserved. To make the definition clear, the instructor reviews the operator norm, showing that it is the maximum singular value for symmetric matrices. They then prove the equivalence between the operator norm definition and a quadratic form definition. The instructor also provides a geometric interpretation of the operator norm as the maximum stretching of unit vectors. After establishing the definition, the instructor shows that RIP implies the Null Space Property, which is crucial for sparse recovery. The lecture concludes with a discussion of how RIP can be achieved with high probability for random matrices, setting the stage for future sessions on recovery guarantees.