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Learning a distance measure from the information-estimation geometry of data

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 7.0Download Report PDF
Distance functionsperceptual metricsimage quality measuresproximity measuresinformation-estimation relationsi-mmseriemannian metricinformation geometrymetric 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 distance 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.

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Overview

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
26
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: Learning a distance metric from probability density geometry. The field encompasses a rich interplay between statistical theory, differential geometry, and machine learning, organized into several major branches. Metric Learning from Probabilistic Information focuses on extracting distance functions directly from data distributions, often leveraging density estimates or probabilistic models. Distance Measures Between Distributions includes classical divergences and statistical distances such as those surveyed in Probability Density Distance Survey[3] and Probability Distribution Distances[12]. Information-Geometric Metrics emphasizes the Riemannian structure of probability manifolds, drawing on Fisher information and related constructs as seen in works like Kahler Fisher Metric[8] and Pulling Information Geometry[15]. Geometric and Manifold-Based Metric Learning addresses how to learn or adapt metrics on curved spaces, with contributions such as Riemannian Metric Learning[13] and Log Euclidean Metric[14]. Theoretical Foundations of Probabilistic Metric Spaces and Applications branches cover axiomatic treatments and domain-specific uses, while Auxiliary Topics capture related methodological developments. A particularly active line of work explores score-based and denoising-derived metrics, which leverage the geometry of score functions or diffusion processes to define distances that respect the underlying density landscape. Information Estimation Geometry[0] sits squarely within this emerging cluster, proposing a metric derived from the geometry of probability densities via information-theoretic principles. This approach contrasts with classical divergence-based methods like those in Probability Density Distance Survey[3], which often rely on integral functionals, and with purely Riemannian frameworks such as Probabilistic Geometries Metrics[5], which emphasize Fisher-information geodesics. By grounding the metric in score or denoising structures, Information Estimation Geometry[0] offers a computationally tractable alternative that naturally integrates with modern generative modeling, bridging classical information geometry and contemporary machine learning practice.

Claimed Contributions

Information-Estimation Metric (IEM)

The authors propose a new distance function that is induced by the geometry of a probability density. The IEM compares score vector fields of a blurred density around two signals over a range of noise amplitudes, adapting both locally and globally to the distribution's geometry.

7 retrieved papers
Closed-form local Riemannian metric

The authors derive a second-order expansion of the IEM that yields a Riemannian metric. This local metric is most sensitive in regions of high log-density curvature and to perturbations that induce large changes in signal probability, behaving like a locally adaptive Mahalanobis distance.

10 retrieved papers
Generalized Information-Estimation Metric

The authors introduce a generalized version of the IEM that incorporates a scalar function f to measure deviations of the log-probability ratio process from zero. This generalization allows the distance to adapt to different types of data by selecting an appropriate function f.

9 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Information-Estimation Metric (IEM)

The authors propose a new distance function that is induced by the geometry of a probability density. The IEM compares score vector fields of a blurred density around two signals over a range of noise amplitudes, adapting both locally and globally to the distribution's geometry.

Contribution

Closed-form local Riemannian metric

The authors derive a second-order expansion of the IEM that yields a Riemannian metric. This local metric is most sensitive in regions of high log-density curvature and to perturbations that induce large changes in signal probability, behaving like a locally adaptive Mahalanobis distance.

Contribution

Generalized Information-Estimation Metric

The authors introduce a generalized version of the IEM that incorporates a scalar function f to measure deviations of the log-probability ratio process from zero. This generalization allows the distance to adapt to different types of data by selecting an appropriate function f.

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