Abstract: While it is known that gradient descent recovers the optimal LQR gain without spurious local minima, it is hard to argue against just solving the Riccati equation. We revisit this question in the dual (filtering) setting and give a concrete answer: for large-scale sparse systems, gradient-based computation of the Kalman gain can reduce per-iteration cost to linear in the state dimension. Our starting point is a new formula expressing the gradient of the innovations loss as a product of two interpretable factors: the observability Gramian of the error dynamics and the cross-covariance between the estimation error and the innovation, a quantity that measures how far the current filter is from satisfying the Kalman orthogonality principle. This decomposition reveals that spurious stationary points arise if and only if the system loses observability, and it identifies a non-standard observability under which gradient descent converges geometrically to the Kalman gain. The convergence rate itself decomposes cleanly into a worst-case observability measure and a term capturing how steeply orthogonality violation penalizes the cost. Finally, we show that under natural sparsity assumptions, the gradient can be approximated using only sparse matrix-vector products, and we demonstrate order-of-magnitude speedups over standard Riccati solvers on systems with several thousand state variables.