  Sensor placement for linear inverse problems is the selection of locations to
assign sensors so that the entire physical signal can be well recovered from
partial observations. In this paper, we propose a fast sampling algorithm to
place sensors. Specifically, assuming that the field signal $\mathbf{f}$ is
represented by a linear model $\mathbf{f}=\pmb{\phi}\mathbf{g}$, it can be
estimated from partial noisy samples via an unbiased least-squares (LS) method,
whose expected mean square error (MSE) depends on chosen samples. First, we
formulate an approximate MSE problem, and then prove it is equivalent to a
problem related to a principle submatrix of $\pmb{\phi}\pmb{\phi}^\top$ indexed
by sample set. To solve the formulated problem, we devise a fast greedy
algorithm with simple matrix-vector multiplications, leveraging a matrix
inverse formula. To further reduce complexity, we reuse results in the previous
greedy step for warm start, so that candidates can be evaluated via lightweight
vector-vector multiplications. Extensive experiments show that our proposed
sensor placement method achieved the lowest sensor sampling time and the best
performance compared to state-of-the-art schemes.
