 Sparse maximum eigenvalue

For a design matrix $X$ with $p$ columns, define ${\kappa }_m^2={\max }_{|M|=m}{\lambda }_{\max }(X_M^T{X}_M/n)$ for $1\le m\le p$. Equivalently this bounds $\Vert Xv{\Vert }_2^2/(n\Vert v{\Vert }_2^2)$ for vectors supported on at most $m$ coordinates. These quantities are nondecreasing: extend the support of a maximizing vector by zero coordinates and use the variational characterization of the largest eigenvalue. Values at $m>p$, including infinity, require a separate convention.