  We consider learning Ising tree models when the observations from the nodes
are corrupted by independent but non-identically distributed noise with unknown
statistics. Katiyar et al. (2020) showed that although the exact tree structure
cannot be recovered, one can recover a partial tree structure; that is, a
structure belonging to the equivalence class containing the true tree. This
paper presents a systematic improvement of Katiyar et al. (2020). First, we
present a novel impossibility result by deriving a bound on the necessary
number of samples for partial recovery. Second, we derive a significantly
improved sample complexity result in which the dependence on the minimum
correlation $\rho_{\min}$ is $\rho_{\min}^{-8}$ instead of $\rho_{\min}^{-24}$.
Finally, we propose Symmetrized Geometric Averaging (SGA), a more statistically
robust algorithm for partial tree recovery. We provide error exponent analyses
and extensive numerical results on a variety of trees to show that the sample
complexity of SGA is significantly better than the algorithm of Katiyar et al.
(2020). SGA can be readily extended to Gaussian models and is shown via
numerical experiments to be similarly superior.
