Let $\{Y_j : j = 1,\ldots,n\}$ be independent observations in
$\mathbb{R}^m, m \geq 1$ with common distribution $Q$. Suppose that $Y_j = X_j
+ \xi_j, j = 1,\ldots,n$, where $\{X_j, \xi_j, j = 1,\ldots,n\}$ are
independent, $X_ j, j = 1,\ldots,n$ have common distribution $P$ and $\xi_ j, j
= 1, \ldots,n$ have common distribution $\mu$, so that $Q = P * \mu$. The
problem is to recover hidden geometric structure of the support of $P$ based on
the independent observations $Y_j$. Assuming that the distribution of the
errors $\mu$ is known, deconvolution statistical estimates of the fractal
dimension and the hierarchical cluster tree of the support that converge with
exponential rates are suggested. Moreover, the exponential rates of convergence
hold for adaptive versions of the estimates even in the case of normal noise
$\xi_ j$ with unknown covariance. In the case of the dimension estimation,
though, the exponential rate holds only when the set of all possible values of
the dimension is finite (e.g., when the dimension is known to be integer). If
this set is infinite, the optimal convergence rate of the estimator becomes
very slow (typically, logarithmic), even when there is no noise in the
observations.