Linear Recovery and Its Topological Relaxation

This manuscript studies the minimum number $\mu_{\mathbb R}(d,r)$ of real linear measurements that are injective on real $d\times d$ matrices of rank at most $r$. Its main result is an exact solution of a topological relaxation: one measurement below the complex generic count $4dr-4r^2$, a continuous odd nonvanishing map on the low-rank unit link exists if and only if the complex determinantal degree is even, and the corresponding evaluation bundle admits the required number of independent sections exactly in the even-degree case. These statements explain a limit of bundle-only arguments. A direct rank-one index calculation gives $\mu_{\mathbb R}(d,1)\ge4d-4-2h+(h\bmod2)$ for even $d$, where $h$ is the number of binary ones of $d/2-1$; in dimension six this gives nineteen. The existence or impossibility of a seventeen-dimensional real space of $6\times6$ matrices whose nonzero members have rank at least three remains undetermined.