  We study the Pareto frontier for two competing norms $\|\cdot\|_X$ and
$\|\cdot\|_Y$ on a vector space. For a given vector $c$, the pareto frontier
describes the possible values of $(\|a\|_X,\|b\|_Y)$ for a decomposition
$c=a+b$. The singular value decomposition of a matrix is closely related to the
Pareto frontier for the spectral and nuclear norm. We will develop a general
theory that extends the notion of singular values of a matrix to arbitrary
finite dimensional euclidean vector spaces equipped with dual norms. This also
generalizes the diagonal singular value decompositions for tensors introduced
by the author in previous work. We can apply the results to denoising, where
$c$ is a noisy signal, $a$ is a sparse signal and $b$ is noise. Applications
include 1D total variation denoising, 2D total variation Rudin-Osher-Fatemi
image denoising, LASSO, basis pursuit denoising and tensor decompositions.
