Recovery From Non-Decomposable Distance Oracles

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 instances including edit distance, dynamic time warping (DTW), Fréchet distance, earth mover’s distance, and others. We initiate the study and develop a general framework for such distances. Interestingly, for some distances such as DTW or Fréchet, 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. One motivation for understanding non-adaptivity is that the query sequence can be fixed and provide a non-linear embedding of the input, which can be used in downstream applications involving, e.g., neural networks for natural language processing.