  Single-molecule experimental techniques track the real-time dynamics of
molecules by recording a small number of experimental observables. Following
these observables provides a coarse-grained, low-dimensional representation of
the conformational dynamics but does not furnish an atomistic representation of
the instantaneous molecular structure. Takens' Delay Embedding Theorem asserts
that, under quite general conditions, these low-dimensional time series can
contain sufficient information to reconstruct the full molecular configuration
of the system up to an a priori unknown transformation. By combining Takens'
Theorem with tools from statistical thermodynamics, manifold learning,
artificial neural networks, and rigid graph theory, we establish an approach
Single-molecule TAkens Reconstruction (STAR) to learn this transformation and
reconstruct molecular configurations from time series in
experimentally-measurable observables such as intramolecular distances
accessible to single molecule F\"orster resonance energy transfer. We
demonstrate the approach in applications to molecular dynamics simulations of a
C24H50 polymer chain and the artificial mini-protein Chignolin. The trained
models reconstruct molecular configurations from synthetic time series data in
the head-to-tail molecular distances with atomistic root mean squared deviation
accuracies better than 0.2 nm. This work demonstrates that it is possible to
accurately reconstruct protein structures from time series in
experimentally-measurable observables and establishes the theoretical and
algorithmic foundations to do so in applications to real experimental data.
