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The second derivatives of the energy required for the Hessian, eq.~\ref{eq:hessian}, can be calculated either via finite differences, what is known as \texit{numerical} \textit{numerical}  derivatives, or using perturbation theory, usually called \textit{analytical} derivatives~\cite{Pulay1969,Jorgensen1983,Baroni2001,Kadantsev2005}. A property of the calculations of the energy derivatives is that the numerical cost does not depend on the direction they are calculated. This can be readily seen in the case of finite differences, as the cost of calculating \(E(\vec{R}^1,\ldots,\vec{R}^j+\Delta,\ldots,\vec{R}^N)\) is essentially the same as computing \(E(\vec{R}^1 + \Delta^1,\ldots,\vec{R}^j+\Delta,\ldots,\vec{R}^N + \Delta^N)\). This is an essential requirement of our method. If on the contrary, there is a preferred basis where the matrix is cheaper to compute, to compute it in a different basis might offset the reduction in cost offered by compressed sensing.