  A line of work has looked at the problem of recovering an input from distance
queries. In this setting, there is an unknown sequence $s \in \{0,1\}^{\leq
n}$, and one chooses a set of queries $y \in \{0,1\}^{\mathcal{O}(n)}$ and
receives $d(s,y)$ for a distance function $d$. The goal is to make as few
queries as possible to recover $s$. Although this problem is well-studied for
decomposable distances, i.e., distances of the form $d(s,y) = \sum_{i=1}^n
f(s_i, y_i)$ for some function $f$, which includes the important cases of
Hamming distance, $\ell_p$-norms, and $M$-estimators, to the best of our
knowledge this problem has not been studied for non-decomposable distances, for
which there are important special cases such as edit distance, dynamic time
warping (DTW), Frechet distance, earth mover's distance, and so on. We initiate
the study and develop a general framework for such distances. Interestingly,
for some distances such as DTW or Frechet, exact recovery of the sequence $s$
is provably impossible, and so we show by allowing the characters in $y$ to be
drawn from a slightly larger alphabet this then becomes possible. In a number
of cases we obtain optimal or near-optimal query complexity. We also study the
role of adaptivity for a number of different distance functions. One motivation
for understanding non-adaptivity is that the query sequence can be fixed and
the distances of the input to the queries provide a non-linear embedding of the
input, which can be used in downstream applications involving, e.g., neural
networks for natural language processing.
