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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s \in \{0,1\}^{\leq n}$ </tex-math></inline-formula> , and one chooses a set of queries <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$y \in \{0,1\}^{ \mathcal {O}(n)}$ </tex-math></inline-formula> and receives <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d(s,y)$ </tex-math></inline-formula> for a distance function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> . The goal is to make as few queries as possible to recover <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula> . Although this problem is well-studied for decomposable distances, i.e., distances of the form <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d(s,y) = \sum _{i=1}^{n} f(s_{i}, y_{i})$ </tex-math></inline-formula> for some function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> , which includes the important cases of Hamming distance, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{p}$ </tex-math></inline-formula> -norms, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> -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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula> is provably impossible, and so we show by allowing the characters in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$y$ </tex-math></inline-formula> 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.