  We reformulate unsupervised dimension reduction problem (UDR) in the language
of tempered distributions, i.e. as a problem of approximating an empirical
probability density function by another tempered distribution, supported in a
$k$-dimensional subspace. We show that this task is connected with another
classical problem of data science -- the sufficient dimension reduction problem
(SDR). In fact, an algorithm for the first problem induces an algorithm for the
second and vice versa.
  In order to reduce an optimization problem over distributions to an
optimization problem over ordinary functions we introduce a nonnegative penalty
function that ``forces'' the support of the model distribution to be
$k$-dimensional. Then we present an algorithm for the minimization of the
penalized objective, based on the infinite-dimensional low-rank optimization,
which we call the alternating scheme. Also, we design an efficient approximate
algorithm for a special case of the problem, where the distance between the
empirical distribution and the model distribution is measured by Maximum Mean
Discrepancy defined by a Mercer kernel of a certain type. We test our methods
on four examples (three UDR and one SDR) using synthetic data and standard
datasets.
