  We provide high probability finite sample complexity guarantees for hidden
non-parametric structure learning of tree-shaped graphical models, whose hidden
and observable nodes are discrete random variables with either finite or
countable alphabets. We study a fundamental quantity called the (noisy)
information threshold, which arises naturally from the error analysis of the
Chow-Liu algorithm and, as we discuss, provides explicit necessary and
sufficient conditions on sample complexity, by effectively summarizing the
difficulty of the tree-structure learning problem. Specifically, we show that
the finite sample complexity of the Chow-Liu algorithm for ensuring exact
structure recovery from noisy data is inversely proportional to the information
threshold squared (provided it is positive), and scales almost logarithmically
relative to the number of nodes over a given probability of failure.
Conversely, we show that, if the number of samples is less than an absolute
constant times the inverse of information threshold squared, then no algorithm
can recover the hidden tree structure with probability greater than one half.
As a consequence, our upper and lower bounds match with respect to the
information threshold, indicating that it is a fundamental quantity for the
problem of learning hidden tree-structured models. Further, the Chow-Liu
algorithm with noisy data as input achieves the optimal rate with respect to
the information threshold. Lastly, as a byproduct of our analysis, we resolve
the problem of tree structure learning in the presence of non-identically
distributed observation noise, providing conditions for convergence of the
Chow-Liu algorithm under this setting, as well.
