Birdsong in state space
Zebra finch song is normally described as acoustics: a spectrogram and an inventory of syllables. The premise here was that this throws away what matters most, because the song is produced by a physical system with dynamics, and the recording is one measurement of that system.
The approach
Takens’ theorem says you can recover the geometry of a dynamical system’s attractor from a single measured channel, by treating delayed copies of that channel as independent coordinates. So: take the waveform, choose a delay, and plot each moment against itself one and two delays ago. The delay is not free — too short and the coordinates are redundant, too long and they are unrelated — and it was chosen from the first local minimum of the mutual information, the standard criterion. The embedding dimension and the Lyapunov exponents, which say whether trajectories nearby now stay nearby, were estimated alongside it.
Because a reconstruction like this will happily produce a picture from noise, the procedure was run on systems whose answers are already known — Lorenz and Rössler — and kept next to the song reconstructions as controls. The corpus was ten zebra finch songs, segmented so that the geometry could be compared syllable by syllable rather than over a whole recording.
Where it got to
It was presented at the APS March Meeting in 2019 and did not go further; the work stopped when I moved to Berkeley. The published claim is the abstract, and I would rather link it than restate it from memory seven years on.
The through-line is easy to see in hindsight. The question — is there structure in this signal that the conventional description throws away, and how would you know — is the same one the later work asks about word embeddings and about movement at home. The strange attractor turning on the front page is a Lorenz system, and it is there for this reason.
This page describes the method from the project’s own 2018–2019 files — the embedding code, the mutual-information criterion, the Lorenz and Rössler reconstructions kept as controls — and reports no results beyond that. The abstract is the only published claim. The work appeared under Xiao Zeng.