Title: Learning a distance measure from the information-estimation geometry of data
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
Keywords: Distance functions, perceptual metrics, image quality measures, proximity measures, information-estimation relations, i-mmse, riemannian metric, information geometry, metric learning

Abstract: We introduce the Information-Estimation Metric (IEM), a novel form of distance function derived from an underlying continuous probability density over a domain of signals.
The IEM is rooted in a fundamental relationship between information theory and estimation theory, which links the log-probability of a signal with the errors of an optimal denoiser, applied to noisy observations of the signal.
In particular, the IEM between a pair of signals is obtained by comparing their denoising error vectors over a range of noise amplitudes.
Geometrically, this amounts to comparing the score vector fields of the *blurred* density around the signals over a range of blur levels.
We prove that the IEM is a valid global metric and derive a closed-form expression for its local second-order approximation, which yields a Riemannian metric.
For Gaussian-distributed signals, the IEM coincides with the Mahalanobis distance.
But for more complex distributions, it adapts, both locally and globally, to the geometry of the distribution.
In practice, the IEM can be computed using a learned denoiser (analogous to generative diffusion models) and solving a one-dimensional integral.
To demonstrate the value of our framework, we learn an IEM on the ImageNet database.
Experiments show that this IEM is competitive with or outperforms state-of-the-art supervised image quality metrics in predicting human perceptual judgments.

TL;DR: We introduce the Information-Estimation Metric (IEM), a theoretically grounded unsupervised distance measure that adapts to data geometry. It shows competitive performance with supervised perceptual metrics.