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.