6.4Spectrum approximation

Let $\vec{A}\in\R^{n\times n}$ be a symmetric matrix with eigendecomposition $\vec{A} = \sum_{i=1}^{n} \lambda_i \vec{u}_i \vec{u}_i^\T$, where $\lambda_i$ are the eigenvalues and $\vec{u}_i$ are the orthonormal eigenvectors of $\vec{A}$.

We are interested in a coarse approximation to the spectral density of $\vec{A}$; e.g. in Wasserstein distance.

Relation to spectral sums¶

Spectrum approximation is closely related to the task of approximating spectral sums. Observe that

Likewise,

Hence approximating the spectral density and approximating spectral sums are equivalent; i.e. being able to approximate spectral sums gives us a way to approximate spectral densities and vise-versa.