  We show that in finite-dimensional nonlinear approximations, the best
$r$-term approximant of a function $f$ almost always exists over $\mathbb{C}$
but that the same is not true over $\mathbb{R}$, i.e., the infimum
$\inf_{f_1,\dots,f_r \in Y} \lVert f - f_1 - \dots - f_r \rVert$ is almost
always attainable by complex-valued functions $f_1,\dots, f_r$ in $Y$, a set of
functions that have some desired structures. Our result extends to functions
that possess special properties like symmetry or skew-symmetry under
permutations of arguments. For the case where $Y$ is the set of separable
functions, the problem becomes that of best rank-$r$ tensor approximations. We
show that over $\mathbb{C}$, any tensor almost always has a unique best
rank-$r$ approximation. This extends to other notions of tensor ranks such as
symmetric rank and alternating rank, to best $r$-block-terms approximations,
and to best approximations by tensor networks. When applied to
sparse-plus-low-rank approximations, we obtain that for any given $r$ and $k$,
a general tensor has a unique best approximation by a sum of a rank-$r$ tensor
and a $k$-sparse tensor with a fixed sparsity pattern; this arises in, for
example, estimation of covariance matrices of a Gaussian hidden variable model
with $k$ observed variables conditionally independent given $r$ hidden
variables. The existential (but not the uniqueness) part of our result also
applies to best approximations by a sum of a rank-$r$ tensor and a $k$-sparse
tensor with no fixed sparsity pattern, as well as to tensor completion
problems.

Numerical Analysis;Algebraic Geometry