Learning a distance measure from the information-estimation geometry of data
Authors: Guy Ohayon
Pierre-Etienne H Fiquet
Florentin Guth
Jona Ballé
Eero P Simoncelli
Venue: ICLR 2026 Poster (2026)
Keywords: information–estimation metric
metric learning
information geometry
score-based generative models
denoising diffusion models
Riemannian metrics
unsupervised representation learning
full-reference image quality assessment
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We introduce the Information-Estimation Metric (IEM), a theoretically grounded unsupervised distance measure that adapts to the geometry of the data. It shows competitive performance with supervised perceptual metrics.

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.

Published as a conference paper at ICLR 2026
LEARNING A DISTANCE MEASURE FROM THE INFORMATION-ESTIMATTON GEOMETRY OF DATA
Guy Ohayon
Pierre-Etienne H. Fiquet
Flatiron Institute
Flatiron Institute
@ohayyasfflatirroohssiittttttt
priquettflaatiroonnsiittt......
Florentin Guth
Jona Ballé New York University jona. balle@nyu. edu
Eero P. Simoncelli
Flatiron Institute
Flatiron Institute
New York University
New York University eero. wimoncelli@nyu edu
fl Florentin.c Plorentin.gutheeeeee gww.iiiu..ttuu....mm edu
ABSTRACT
We introduce the Information- Enstmmttnnnntiiaiiionnnn Metric (IEM), novel form of distance function derived from an underlying continuous probability density over a domain of signals. The IEM is rooted in fundamental relationship between information theory and estimation theory, which links the log-probability of a signal with the errors of an optimal denoiser, applied noisy observations of the signal. In partic- ular, the IEM between pair of signals is obtained by comparing their denoising error vectors over range of noise amplitudes. Geometrically, this amounts comparing the score vector fields of the blurred density around the signals over a range of blur levels. We prove that the IEM is valid global distance metric and derive closed-form expression for its local second-orde approximation, which yields a Riemannian metric. For Gaussian-distribtted 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 frame- work, learn IEM the ImageNet database. Experiments show that this IEM is competitive with or outperforms state-of-the-ard supervised image quality metries predicting human perceptual judgments.
INTRODUCTION
Distance functions central to many scientific and engineering enterprises, enabling systematic comparison, organization, and interpretation of data. In some cases, a meaningful notion of dis- tance arises from the structure distribution of the data (e.g., geodesics in Riemannian manifolds, z-scores for Gaussian distributions), or from the requirements of the task (e.g.. Hamming distance in crror-correcting codes, edit distance text processing). However, this often not the case. For instance, algorithms that process natural signals (e.g., compression engines) should ideally be evaluated in terms of human perception, for which no precise mathematical definition is available. Numerous algorithms aiming to mimic human perception have been proposed (Wang et al., 2004; Heusel al., 2017; Zhang et al., 2018: Ding al., 2022; Chen al., 2024), with the most successful approaches to date being those trained (supervised) on databases of human perceptual judgments. Nevertheless, this reliance on human-labeled data is problematic, as data annotation is a highly costly and noisy procedure. More importantly, supervised approaches ematically, making it harder to explain the principles that underlie our perceptual judgments of similarity between natural signals (Barlow, 1989). Deriving a perceptual metric solely based unlabeled data remains a fundamental open problem of both scientific and practical importance.
difficult to interpret math-
A natural opportunity for developing such metric arises from the concept of coding efficiency. Biological sensory systems are believed to decompose incoming signals in a manner that maxi- mizes the transmission of information about those signals, subject biological constraints (e.g..

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noise, metabolic cost) (Attneave, 1954; Barlow, 1961; Laughlin, 1981; van Hateren, 1992; Atick & Redlich, 1992; Olshausen & Field, 1996; Barlow, 2001; Simoncelli & Olshausen, 2001). Put differently, sensory pathways function communication chamnels optimized for natural signals, implying that our ability to discriminate between natural signals depends on their statistical prop- erties. Indeed, for one-dimensional sensory attributes, previous work has shown that perceptual sensitivity to small signal perturbations increases with the probability of the signal (Laughlin, 1981; Ganguli & Simoncelli, 2014; Wei & Stocker, 2017). However, in the multivariate setting, such as color discrimination (i.e. detecting changes in hue or saturation), humans exhibit complex patterns of sensitivity that vary with the direction of the signal's perturbation (MacAdam, 1942). This leaves us with a conundrum: how can a probability density, which is scalar function, induce Riemannian metric, let alone a global distance function between any pair of signals in the domain?
In an attempt to construct distance function from probability density, it is natural to resort to principles from information theory. Unfortunately, poormatioonthhorettii quantities are agnostic to the geometry of the probability distribution. For example, the mutual information between random variables is invariant to bijective (even discontinuous) transformations of the variables. In con- trast, estimation quantities such denoising emror are explicitly tied to the geometry of the density through an assumed observation model (e.g., additive Gaussian noise) and loss function (e.g., square error). Despite this salient difference, line of work (Guo al., 2005; 2013) rooted in information theory (Stam, 1959) and empirical Bayesian methods (Robbins, 1956) has revealed an extensive cor- respondence between these scemingly unrelated quantities. It takes the form of a set of relationships that express information quantities in terms of estimation quantities, thereby linking probability (in- formation) with geometry (the shape of the "data support"). In particular, scalar probability values can be decomposed into denoising error vectors, which provide a natural way to characterize the geometry of the signal density. Indeed, denoising errors are proportional to the score (gradient of the log) of the signal density blurred through convolution with a Gaussian density. This relationship between denoising errors and scores, known as the Prreddee-Myysawa formula (Robbins, 1956; Miyasawa, 1961), is the foundation of generative diffusion models (Sohl-Dicksterr al., 2015; Ho et al., 2020; Song et al., 2020).
Building on these tion which is derived from the geometry of a given probability density. Our distance, coined the Information -sstimationntniiiinnnn Metrie (IEM), compares the score vector fields of the blurred density in the vicinity of two given signals. More specifically, it is defined as the mean square error (MSE) between these score vector fields, integrated over range of blur levels (i.e., Gaussian noise magnitudes). We prove that the IEM is a valid distance metric (in the mathematical sense), and show that it coin- cides with the Mahalanobis distance (Mahalanobis, 1936) when the prior density is Gaussian. For more complex priors, however, the IEM reflects the structure of the "data addaptinn to the global geometry of the density. Furthermore, we analyze the local behavior of the IEM by deriving the second-order expansion of the distance between signal and its perturbed version, which yields a Riemannian metric. We show that this Riemannian metric is most sensitive (1) in regions where the curvature of the log-density is highest, and (2) to perturbations that induce the largest changes in the signal's 's probability. This implies that the IEM behaves like a locally adaptive Mahalanobis distance conforming to the local geometry of the density. Importantly, the IEM can be efficiently learned from samples by training a denoiser (i.e., a diffusion model). We train such a denoiser on ImageNet (Deng et al., 2009) and use compute the IEM. Although the IEM is learned unsuper- vised from unlabeled image data, we find that it is competitive with supervised perceptual distance measures in terms of predicting human judgments of image similarity.
relationships, we introduce a novel form of distance func-
2 THE COTRMMATION-ESTIMAAIE METRIC
We aim to construct distance function that is induced by the geometry of a given probability density. In information theory, it is natural to compare two signals Φ and x2 using their log- probability ratio, which may be turned into "distance" by taking its square value. However, this is a poor choice, as it depends solely on the (scalar) values of the density at the two points. Instead, we would like distance measure that is associated with the geometry of the density (e.g., the curvature of the density around the two signals). To this end, we build upon a fundamental equation that decomposes the oooorbbbyyyyity of signal in terms of the geometry of the probability density in the
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vicinity of the signal. We then apply this decomposition to the log oog-pobbbiliity ratio of two signals, yielding a distance metric that adapts to the density's 's geometry.
Observation channel. Let px denote the probability density function of random vector taking values in Ra (ie., the signal). To decompose Px. we introduce an observation process y- such that Py, gradually "zooms" into Px as the eignal-to-noie ratio (SNR) y is increased, analogously to how diffusion models generate samples. Specifically, we define yy as a Gaussian channel
Y7 7x + Wy:
(1)
where w.y ~ (0, I)) is standard Wiener process which is independent of x. Since the noise w is statistically independent of x, the distribution py. is obtained by blurring Px through convolution with the Gaussian density Pw.. Viewing log Py, (y-) stochastic process that evolves with y, we can decompose Po Px (x) in terms of the increments of this process. By combining two fundamental relations from previous work, we show next that these increments characterize the geometry of log Px in the vicinity of x.
Pointwise I-MMSE. Venkat & Weissman (2012) proved that log Pyr (yr) for any fixed I can be expressed in terms of the denoising error vectors of the minimum mean square error (MMSE) estimator of x from Y7. E[x ynl, integrated across all SNR levels 7 E [0, rj. Formally,
1 - llog Pyr (yr) L.¹' (x- E(x|x>) dw,, + 2 ||x ( log Pwr (WT), (2)
where this equality holds with probability one (almost surely), i.e., it holds pointwise for almost every realization of the signal x x and of the Wiener process trajectory {w,, w., ) Equa- tion (2), which we refer to as the pointwise I-MMSE formula, is a generalization of the 1-MMSE formula (Guo et al., 2005), whose roots date back de Bruijn's ' identity from the 1950s (Stam, 1959, see App. A.2 for more detailed background). When r - co, Eq. (2) expresses the log-density of the original signal, logpx (x), in terms of the denoising errors at all 7 E [0, o0).
Geometric interpretation. Denoising errors are related to the gradients of log Py, ie., the scores of the blurred density py,, via the Tweedie -Weedie-Miyysaw formula (Robbins, 1956; Miyasawa, 1961):
x ExExxy
W Poggy, (y-).
where the gradient on the right-hand side is taken w.r.t. yy- Substituting this formula into Eq. (2), we now see that log (x) (a scalar) can be decomposed in terms of the local geometry of log Py, (y-)), particularly the gradients V llgg Py, (y=). at all SNR levels y. We refer to this decomposition as the information-estinen esttimatio geometry of the density px-
Definition of the Inpermation-Essimatin Metric (LEM). The relationships above suggest natural way to compare two arbitrary points Ø1 and æ2, by tracking the increments of their log-probability ratio under the blurred density, log (Py, (y + w-)/Py, (yæz + w,)). Doing so amounts to comparing the local geometry of log Px around Iι and Ø2, as illustrated in Fig. 1. Specifically, by combining Eqs. (2) and (3), obtain
Pyr (Tx1 + WT) = [ (V Py, (1x1 + -W,) Log + +w,) dw.,
log
Pyr (T22 + wT))
2 L." (IVlogpy, (yx1 + ww, w. / V logpy, (1322 + w.)) + wr//>/l/)dr, (4)
1
Since Eq. (4) is an Itô process, it is natural to quantify the sum of its squared increments by taking the expected quadratic variation of the process, which is simply the second moment of the diffusion coefficient integrated over the range E [0, []. This leads to our proposed distance function.
Definition 1. The Information-Estiiattoon Metric (IEM) induced by the density px is defined as
= (0)=(())
(J E V log Py, (y21 + w.,) Vlogyy, (1 W. w
1/2
(M1 I)
where the expectations are taken over Pw., for each 7-
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logg Px, (190) Wc)
(taPg Wc)
log
Y (SNR)
Figure 1: The wawermmaatoo-ssimmttii geometry around two points. We show Gaussian mixture log-density and its gradient vector fields around the points y2z and 722 for three different SNR levels y. The space is rescaled by y and the distribution collapses the density (small 7), the two modes merge, and the gradients around 7 point toward either of the modes. When the two modes are far enough apart (large 7), most gradient vectors point toward their closest mode. Thus, the local gradients around a given point can capture different geometrical features of the distribution, depending Def. 1) between the two points æ1 and 22 is the square error between the local gradient fields around them, weighted by Gaussian window (illustrated by the opacity of the gradients' arrows) and integrated over all levels of SNR [).
point at 7 O. When blurring
the SNR 7. The information-Estimaaii
Metric (IEM,
For ease of notation, we write IEM x22) = IEM 22, oo). Although our construction does make it obvious, the IEM is proper distance metric (see proof in App. C.1):
Theorem 1. For every 1 > 0, the IEM is proper distance metric: it is symmetric, non-negative, equal to zero if and only if x = x2, and it satisfies the triangle inequality.
In App. B, we discuss an intriguing relationship between the IEM and the Kullback-Leibler (KL) divergence between distributions. Specifically, we interpret the IEM as local decomposition of the KI. divergence between two translated copies of Px, centered at æ and æ2- We also define mismatched IEM, generalizing the IEM to the case where 1 and D2 are assumed to come from different distributions.
2.1 LOCAL GEOMETRY
To gain insight into the properties of the IEM, we study its local behavior, namely the dis- tance between given signal and its perturbation a + for small c. As for any distance, ==0 global minimum, and we can express the quadratic expansion of the IEM in as IEM c. = G(=, + ( The positive-defintte matrix Gix,, I') then acts as a local metric (in the Riemannian sense), but note that its relationship with the IEM is one-way: the IEM is not equivalent to the geodesic distance that corresponds to G(æ, D). The local metric G( I) is characterized in the following theorem (see proof in App. C.2):
Theorem 2. The local Riemannian metric derived from the second-order Taylor expansion of the squared IEM is given by
G(x,I)= S' 7²B (V2 loggy. Py, (yx w.)) dr
(5)
1/ (1-~ qx + w,,1)² d7,
(6)
where the expectations are taken over Pw., for each 7, Moreover, for ao we have
E[G(x)) = B 2 1o (x (x)] E log log (x)
where we denote G(x) G(x,, oc) and the expectations are taken over Px.
Theorem 2 gives two equivalent expressions for the local metrie induced by the IEM. In particu- lar, Eq. (5) shows how this local metric, G(æ, I),, is tied to the local curvature of log around the point Indeed, the Hessian 2 log Py, (yar wy). which is (nonlinear) smoothing of

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2 log (xt)), describes the local curvature at blur level y. In fact, the relationship between G( I) and V log Px (xx becomes clearer when taking I - 0c and averaging over Px. as expressed by Eq. (7). Qualitatively, this demonstrates that, locally around the signal æ, the IEM is more sensitive to perturbations e that change the log- probability of æ the most. Note that it is not true in gen- eral that G(x)) 2 log Psx (ar) pointwise, as the Hessian of the log-density may not be negative semi-definite, whereas G(x) = O by construction. The local metric G( thus acts as positive semi-definite smoothing of V log Px(x))
Furthermore, Eq. (6) relates the local metric G(æ, I) to the covariance of x The + w., which compared to 1/n--thh the covariance of rescaled observation noise: x + wa/2 Nxx I/7). Equa- tion (6) therefore provides additional intuition about the behavior of G I). First, when the noisy observations y + W-, can effectively denoised across many SNR levels 7, the posterior covari- ance for such values of y is substantially smaller than that of the noise, which results in (relatively) high sensitivity to small perturbations of æ. A simple practical example of this scenario is when is a "smooth" signal (e.g., an image of clear blue sky). Second, perturbations e that can be effectively denoised also lead to large local distance values. For instance, the density Px is supported on a oow-dimensional manifold, then the local metric G(x,I) I) is more sensitive around points æ that are near the manifold, and in directions e that are orthogonal to the local tangent subspace.
2.2 ILLUSTRATIVE EXAMPLES
Gaussian prior. The IEM depends on the distribution of the data Px. When this distribution Gaussian, Px (H, Σ). and r === oo, the IEM coincides with the well-known Mahalanobis distance (see App. C.3 for proof):
IEM(ar = (a) xx2) - -x) 222).
(8)
In other words, the IEM is the Euclidean distance after whitening the data: Σー Displacements in directions of small variance of the data are thus amplified and contribute more to the final distance, as visualized in the center column of Fig. 2.
μt).
This closed-form expression of the Gaussian IEM comes from the linearity of the corresponding optimal denoisers. While more complicated distributions Px have non-linear optimal denoisers, they are often locally linear (Milanfar, 2013; Mohan et al., 2020), so that the corresponding IEM behaves like a Mahalanobis distance locally, adapting to the "local covariance" of the data. This in agreement with our observations above about the local behavior of the IEM for general priors. Together, they paint a picture of how the IEM adapts to the geometry of the data distribution.
Since the IEM coincides with the Mahalanobis distance when Px is Gaussian, it is important to examine the behavior of the IEM when Px is no longer Gaussian.
Gaussian mixture prior. First, consider a two-dimensional, two-mode Gaussian mixture model. To compute the IEM and the local metric G(æ, I),, we numerically solve the integrals in Def. and Eq. (5), using closed-form expressions for log Py, and related quantities (see details in App. E.4). To illustrate the global behavior of the IEM, we choose a reference point Tyet, and evaluate its distance from each of a uniform grid of points x. We then plot the resulting equidistant contours (Fig. 2, top row). and compare with unimodal Gaussian density to illustrate how the IEM adapts to the prior. The IEM clearly adapts to the density's 's global geometry: the equidistant contours resemble the shape of the log-density contours. Interestingly, the regions delimited by equidistant contours can be disconnected: points belonging to one mode are closer to points belonging to the other mode than they are to points lying in between the modes. This is because the local curvature of the log- density can be similar in the vicinity of two points, even if their Euclidean distance is large.
Furthermore, we eigendecompose G(x,, -)- 2 at each point on the grid, and draw an ellipse cen- tered at æ, whose axes and radii are the resulting eigenvectors and the corresponding eigenvalues, respectively. These ellipses represent the discrimination thresholds of the metric across space, which are inversely proportional to its local sensitivities. In other words, the ellipses illustrate the direc- tions that require larger perturbations to induce the same change in distance. Figure 2 shows that the discrimination thresholds align with the direction of the local covariance, i.e., the metric G(x, I') be- haves like locally adaptive Mahalanobis metric. We also note that G( T) is more sensitive in the
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Px Laplace
Px
Gaussian
Px Gaussian mixture
High
Equicisttan
contours
Heavier tnils
Extra made
log Px
Discrimination thresholds
Low
Figure 2: Illustrating the global and local geometry of the InformationnEssimationn Metric (IEM) on three different prior densities. Top row: Equidistant IEM contours relative to an ex- ample reference point (white star). When Px is Gaussian (middle column), the IEM coincides with the well-known Mahalanobis distance. For a separable Laplacian prior (left column), the equidistant contours cluster and curve around the axes, following the high-probability ridges. For Gaussian mixture prior (right column), the contours reflect the shapes of the modes. These examples illustrate how the IEM adapts to the global geometry of the given prior density. Bottom row: Ellipses repre- senting the local discrimination thresholds of the local Riemannian metric G( I') (Eq. (5)). Larger ellipse radii correspond to higher discrimination thresholds, i.e., lower sensitivity local perturba- tions. For the Gaussian prior, the local metric is constant actoss the entire domain (identical to the Mahalanobis metric). For the Laplace (heavy-tailed) prior, the discrimination thresholds are smaller in high-probabilityy theories. Moreover, the orientations of the ellipses align with the equiprobable log-density contours, implying that G') is more sensitive to perturbations that yield larger change in the probabil- ity of æ. For the Gaussian mixture density, the discrimination thresholds are smaller between the modes, and the major axes of the ellipses align with the direction of larger local variance. Overall, these examples illustrate that Gix:, I) is more sensitive in regions of higher log-density curvature and to perturbations that induce larger local changes in probability.
with human perception and predictions of efficient coding
local minima of probability in between the modes, as illustrated by the ellipses with smaller radii (smaller discrimination thresholds). This is consistent with Thm. 2, as the signals lying between modes incur large denoising errors due to the uncertainty about the mode they belong to.
Laplace prior. To further illustrate the influence of the density's curvature on the IEM, we now consider the case where Px is two-dimensional Laplace distribution, formed by taking the prod- uet of one-dimensional Laplace densities. As shown in Fig. 2, this prior density induces dis- crimination thresholds that increase as they move away from the high-probability ridges that lie along the axes. From the point of view of Thm. 2, this reflects the fact that the Hessian matrices 2 log Py. (yx + w,) (specifically, their negative eigenvalues) decrease in magnitude away from the axes. For this sparse and heavy-tailed distribution, curvature is correlated with probability, so that discrimination thresholds are larger in low- pow-babbyyylity regions, consistent with predictions from prior work on efficient coding (Ganguli & Simoncelli, 2014). From the global behavior of the dis- tance, we also see that the equidistant contour lines tend to cluster around the axes: under a sparse prior such the Laplace density, flipping the sign of one several coordinates of x (landing on the other side of the higb-probabilityy ridge) incurs a large cost as measured by the IEM.
Additional illustrative examples on one-dimensional prior densities are provided in App. E.4.
2.3
COOEMAATION-ESTIMAATION GENERALIZED INFORMATION ESTIMATION METRIC
The IEM is defined as the expected quadratic variation of the Itô process
Py. (yar + w.))
z, (x1, x2) = log
Py, (n02 + w.,)
Note that the quadratic variation of z, is unaffected by additive shifts of the process (the drift co- efficient is ignored). Namely, 2., may have small large average values while yielding the same

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quadratic variation. It is therefore interesting to take such shifts into account by quantifying the de- viation of z, from zero, A natural way to achieve this is to measure the quadratic variation some scalar function ?(z that increases with l.,l. thereby generalizing the IEM. When f is twice differentiable, Itô's lemma shows that the diffusion coefficient of the process ((, 7) equals that of 2ny multiplied by (" r)- derivative of f(:7)) w.r.t. the first argument. We thus define:
Definition 2. For any twice differentiable scalar function f. the generalized IEM is defined
y 11, (21, I2, := (%. E[
log (y01 + w.y) Vlog Pog (102 + w,)ll (
where the expectations are taken over Pw., for each 7--
For 7() Zny , recovers the IEM from Def. 1. Moreover, when (z, satisfies (2(,,, = if and only if 2, O, we have (EM,(x1,22) definiteness). Unlike the IEM, however, the IEM, generally not a proper metric, the symmetry or the triangle inequality axioms. This may or may not be considered a limitation, depending the intended application of the distance. Definition 2 can be extended any non- anticipative functional f((z., =0 7) whose input is the entire history of the process Zy up to SNR y. this case, f' is Dupire derivative (Dupire, 2009; Cont & Fournie, 2010).
if and only if æ1 = 22 (positive
may violate
In App. C.4, we establish two important properties of the process Z,, which are inherited by the family of IEMs. Specifically, we show that 2, is invariant under Euclidean isometries, i.e., it invariant to the choice of orthonormal coordinate system. Moreover, 2, is invariant to sufficient statistics of yy, a property that the IEMs share with the Fisher information metric (Chentsov, 1981).
3 EXPERIMENTS
We assess how well our proposed distances predict human judgments of similarity between pho- tographic images. Specifically, we evaluate the IEMs on pairs of images taken from databases of psychophysical experiments, and compare the predicted distances with human similarity ratings. Computing the IEMs requires access the score function V Vlgg or equivalently to an MMSE estimator E[x| Yyl, at each SNR level 7. We approximate this estimator with learned neural denoiser Do(y.: which is trained to predict x from (y-.7) by minimizing MSE (similarly to unconditional diffusion models). To evaluate our distance functions, we plug the trained denoiser into Defs. 1 and and solve the integral numerically (see App. E. for more details).
3.1 IMPLEMENTATION
Neural denoiser architecture. We use the Hourglass Diffusion Transformer (HDiT) (Crowson et al., 2024) as our denoiser model because it can be trained efficiently and scales linearly with image resolution. We train a denoiser model from scratch on the ImageNet-Ik (Deng et al., 2009) dataset, cropping the images to size 256 x 256. We follow most of the implementation choices of Crowson et al. (2024), but use significantly smaller models and a log-unifome schedule for the noise level. Additional training details and hyperparameters are disclosed in App. E.2.
Choosing f. The generalized IEM, (Def. 2) depends on the choice of the scalar function f. so we examine three options: (1) Identity function: Setting f(z,, 7) = 2, corresponds to our first IEM distance (Def. 1). (2) Quadratic function: We take f(z,, 7) zł, and denote by IEMg,, the resulting distance. We find this simple choice sufficient to demonstrate that IEM, can adapt to different types of human data by selecting an appropriate function f. without supervision. (3) Learned function: We consider learning a parameterized function f from labeled data. The purpose of this choice is to assess whether our proposed family of distances can match human perception across several kinds of psychophysical experiments simultaneously. This is challenging problem, since the distance must adapt to both "local" distortions near the visual sensitivity threshold (e.g., small additive noise) and "global" distortions (e.g., images containing similar-lood looking textures). Moreover, this choice provides a fairer comparison with competing methods, all of which are supervised algorithms. We implement f" as simple causal (non-anticipative) fully-connected network, where the output
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SSIM
VIP
FSIM
PicAPP
MAD
LPIPS
DISTS
TEM Other
Reference
IEM < Other
Figure 3: Illustrating the disagreement between different types of perceptual distance mea- sures. We ranked the distorted images associated with each reference image in the LIVE and CSIQ databases (middle row), according to the IEM and several other metrics. Each column displays the distorted images with the largest positive (bottom row) or negative (top row) rank differences be- tween the IEM and the compared metric (denoted in the title of the column).
TID2013
LIVE
CSIQ
TQD
0.89
SRCC
0.75
0.92
0.90
0.68
0.60
0.87
0.85
0.45
0.46
0.83
0.79
0.23
0.31
0.78
0.74
0.00
GMSD
TOPIQ TEM
IEM..
PSNR
MAD
NLPD
PieAPP
DISTS
IEM
Figure 4: Spearman's rank correlation coefficient (SRCC) results on full-reference image sim- ilarity benchmarks. On TID2013, LIVE, and CSIQ, the IEM performs competitively with previ- ous state-of-the-art supervised methods, but struggles on TQD (texture similarity data), as do most methods. In contrast, the unsupervised IEMa, performs surprisingly well on TQD. Our supervised variant, which only learns fw. achieves strong results on both types of databases simultaneously.
SNR y depends on all previous samples f1z In all experiments, f.. is trained on data disjoint from the evaluation data. See App. E.3 for more details about learning this function.
3.2 PREDICTING MEAN OPINION SCORES
We evaluate our solutions using several full-reference image quality assessment databases containing mean opinion scores (MOS). In App. D.1 we report additional experiments BAPPS (Zhang al., 2018) different type of database consisting of two-alternative forced choice (2AFC) rankings.
Common benchmarks. We consider several standard full-reference image quality assessment benchmarks, including TID2013 (Ponomarenko et al., 2015), CSIQ (Larson Chandler, 2010), and LIVE (Sheikh et al.. 2006). Since the learned denoiser model is suited for images of size 256 x 256, we adjust the resolution of the images in the considered databases by first center-cropping each image to the length of its shorter edge, and then resizing it to 256 x 256. We compare against PSNR, SSIM (Wang et al., 2004), VIF (Sheikh & Bovik, 2006), MAD (Larson Chandler, 2010), FSIM (Zhang et al., 2011), GMSD (Xue et al., 2014), NLPD (Laparra al., 2016), PicAPP (Prash- nani et al., 2018), LPIPS (Zhang et al., 2018), DISTS (Ding et al., 2022), and TOPIQ (Chen et al., 2024). We find that the IEM with r 1/4 yields surprisingly strong results, even though it is computed solely based on denoising errors and is not exposed to human labels. Indeed, as shown in Fig. 4, this same choice of produces strikingly high Spearman's rank correlation coefficient
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(SRCC) with the human MOS across all of the aforementioned datasets. Additional performance measures demonstrate similar trends, so we report them in Figs. 7 and 8 in App. D.2.
To illustrate the differences between the IEM and the compared distance measures, we present in Fig. 3 several example images for which the IEM rankings differ the most from those of the compared methods. For each reference image in the dataset, we rank all of its distorted counterparts according to each distance measure. We then compute the difference between the ranks assigned by the IEM and those assigned by each compared method. From these differences, we take the maxi- mum and minimum values, and sum their absolute magnitudes to quantify the degree of disagree- ment. Finally, we display the reference and distorted images that achieve the largest disagreement. This systematic procedure for comparing image similarity models on a given dataset is analogous to the maximum differentiation competition (Wang & Simoncelli, 2008). The results show that VIF, FSIM, PieAPP, and LPIPS can assign smaller distances to image pairs that are perceptually distin- guishable, whereas the IEM correctly detects that the images are different. and DISTS tend to disagree with the IEM in cases involving perceptually noticeable distortions. For example, DISTS appears to favor noise over blur, whereas the IEM shows the opposite preference.
comparison, MAD
Texture images. We further evaluate our distance measures on the TQD textures dataset (Ding al., 2022). Unlike the previously considered benchmarks, which contain general natural images, this dataset consists of texture images (e.g., leaves or brick walls), paired both with visually similar textures and with distorted versions of the same texture (e.g.. Gaussian blur, JPEG compression). In this setting. human observers are expected to judge two images of the same texture as more similar each other than a clean and distorted pair. Thus, the TQD benchmark assesses whether a perceptual distance measure produces scores consistent with human perception even when the compared images are substantially different in terms of their Euclidean distance. As the previous experiments, we use the 256 x 256 denoiser model with the same preprocessing to resize the images.
As shown in Fig. 4, we find that our distance IEM.g. with r 10 outperforms all other methods, except for DISTS, which was explicitly designed to handle texture images. However, IEM.q. does not perform well on the TID2013, LIVE, and CSIQ datasets. This highlights the flexibility of the IEM, to accommodate very different types of distortions by choosing f. An important question, then, is whether a single mapping f can realize both types of functions. The answer is positive: our learned distance IEM,, fo achieves strong performance across all datasets simultancously. indicating the significance of f and the flexibility of our distances in practice.
3.3 MAXIMUM DIFFERENTIATIE COMPETITION AGAINST THE PSNR MEASURE
To further demonstrate the behavior of the IEM, we conduct a maximum differentiation competition against PSNR. Specifically, we minimize or maximize each perceptual distance (IEM, DISTS, etc.) via projected gradient descent or ascent, respectively, where the projection step during optimization constrains the PSNR of the distorted image to a fixed, pre-determined level. Figure 5 illustrates this experiment and compares the optimized images produced by the IEM and DISTS, using PSNR constraint of 10dB. Comparisons with additional perceptual distance measures on varying levels of PSNR constraints are shown in Figs. 9 to 12. Implementation details disclosed in App. D.3.
Interestingly, we find that minimizing the IEM consistently yields high perceptual quality images that preserve the overall geometric structure of the reference image, even under very low PSNR constraints (see Figs. 5 and 9 to 12). In contrast, minimizing other perceptual metrics produces images with unnatural artifacts, even for relatively high PSNR constraints (e.g. 25dB). These re- sults suggest that, unlike previous perceptual metrics, the IEM may serve a stand-alone robust optimization objective (e.g.. for solving inverse problems), and we encourage future work to explore this potential. Furthermore, maximizing the IEM reveals that it is most sensitive to unstructured noise perturbations that push an image off the "data support" (decreasing the image's probability), consistent with our theoretical analysis in Sec. 2.
4 DISCUSSION
We have introduced the Information-Estimation Metric (IEM), a novel form of distance measure induced by the geometry of an underlying probability density, and provided means of learning this

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Hypersphere of dissortted images with PSNR 10dB
Roderence image
Best PSNR = 10dB
Best DISTS for PSNR 10dB
Minimize IEM
Maximize IEM
Add noise
Minimize DISTS
Worst IEM for PSNR = 10dB
Maximize DISTS
Worst PSNR = 10dB
Initall disterted image
Figure 5: Visual illustration of the maximum differentiation competition. We corrupt a given reference image with random noise produce distorted image with PSNR 10dB. This distorted image lies on the surface of a hypersphere in IR centered at the reference image, with radius equal the Euclidean norm of the added noise, as illustrated above. Starting from the distorted image, we then minimize or maximize the perceptual distance to the reference image while constraining the PSNR to 10dB. Even under such restrictive constraint, minimizing the IEM yields artifact- free images that are perceptually similar to the reference image, whereas minimizing state-of-the-an supervised perceptual metrics such as DISTS yields unrealistic images with noticeable artifacts. Furthermore, the results obtained by maximizing the IEM support our theoretical analysis in Sec. 2, demonstrating that the IEM is most sensitive to unstructured distortions (e.g., additive noise) that perturb the reference image outside the "data support."
(unsupervised) metric from samples. The definition of the IEM relies on fundamental principles that link the probability density with its local geometry, namely the pointwise I-MMSE and Tweedie- Miyasawa formulas. This relationship between probability and geometry arises from the choice of an estimation problem, in our case Gaussian denoising. Different choices of the estimation problem may yield different types of IEMs. For instance, it is possible wise I-MMSE relation for Poisson channels (Jiao et al., 2013), and the empirical Bayes relation for Poisson denoising (Raphan & Simoncelli, 2011). We leave these as opportunities for future work. Furthermore, the IEM does not assume that the density is supported on low-dimensional manifold, in contrast to manifold learning approaches (see App. A. for further discussion). fact, the IEM is well-defined for any valid probability density. We proved that the IEM is a valid distance metric and analyzed its local and global properties through both thcoretical results and illustrative examples. To demonstrate the value of our proposed framework, we trained an IEM on the ImageNet database and found that it aligns surprisingly well with human judgments of image similarity.
define an IEM using the point-
The proposed IEMs (Defs. and 2) require choosing scalar hyperparameter I. This hyperparam- eter sets the maximum SNR over which the integral is computed, effectively controlling the finest resolution which the metric is adapted to the density. Such hyperparameter should presumably be chosen based on the fine-scale geometry of the density, or for a learned density, the complexity and size of the training set. Moreover, the generalized IEM (Def. 2) depends on the choice of the function f. which qualitatively controls the relative importance of log- oog-robbabylity ratio values com- pared to score differences. A systematic principle for determining both I' and f remains open problem. Perhaps the most important limitation of the IEM is its computational cost: Numerical estimation of the integral is more computationally demanding than evaluation of existing supervised perceptual metrics (e.g., LPIPS or DISTS). This is acceptable for applications to collections of im- ages (as in our comparison to human perceptual data), but would limit its use as an optimization objective (e.g., for solving inverse problems, optimizing compression systems). We believe that the IEM may be evaluated in a single forward pass, e.g., using a strategy similar to Guth et al. (2025).
There are many potential future applications for the IEM framework. For example, it offers new opportunities for unsupervised data clustering (as illustrated in App. D.4), information retrieval, evaluating (or optimizing) image restoration and compression engines, and discriminating between generative models (using the mismatched IEM proposed in App. B). It is also natural to consider applying the principles presented in this paper to other forms of continuous signals, such audio.
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REPRODUCIBILITY STATEMENT
Section 3 and Apps. D and E provide all the details necessary to reproduce our results, includ- ing the training hyperparameters of the denoiser model used in the computation of the IEM, the implementation details of the learned function fw. the data preprocessing procedures, and the maximum differentiation competition experiments. Our code is available online https: //github. com/ ohayonguy informat on-oonotttteet wet.eestitoonteett..
ACKNOWLEDGMENTS
We thank our colleagues from the Center for Computational Neuroscience at the Flatiron Institute for their helpful comments and suggestions. G.O. gratefully acknowledges the Viterbi Fellowship from the Faculty of Electrical and Computer Engineering at the Technion.
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Appendix
Related work and background
16
A.1 Related work
16
A.2 Origins the pointwise I-MMSE formula
18
Mismatched IEM and local decomposition of Kullback-Leibler divergence
18
Proofs
19
C.1 Proof of Thm.
19
C.2 Proof of Thm.
C.3 Proof that the IEM coincides with the Mahalanobis distance for Gaussian priors
23
C,4 Additional properties of the process 2,
24
Additional experimental results
27
D.I Predicting two-alternative forced choice judgements
27
D.2 Predicting mean opinion scores: additional metrics
28
D.3 Maximum differentiation competition against PSNR: additional details and results.
28
D.4 A toy clustering example
35
Training and implementation details
35
E.1
Numerical integration
35
E.2
Denoiser training
36
E.3 Learning f%
36
E,4 Implementation of the illustrative examples
36
LLMs usage
37
A RELATED WORK AND BACKGROUND
A.1 RELATED WORK
Metric learning. Metric learning methods aim learn a distance or similarity function from data that is suited to a particular clustering task (e.g., face verification or recognition, image or text re- trieval), such that points from the same cluster are considered closer each other than points from different clusters (Xing et al., 2002; Kulis, 2013). Thus, in metric learning, the metric is, by de- sign, informed either by a specific downstream task or by some other specification of the desired clustering structure within the signal domain. In contrast, the IEM we introduce in this paper is not informed by any downstream task, but is rather derived directly from the probability distribution of the (unlabeled) data. Several approaches in the literature rely on self-supervised learned representa- tions, such as the latent space of generative models, in order to induce a Riemannian metric in the input space (Arvanitidis et al., 2017). In contrast, approach does rely explicit latent space and does not require computing geodesic distances.
Another path to metric learning through dimensionality reduction, where the metric is derived by measuring distances (e.g., Euclidean) in a low-dimensional embedding space. For example, diffusion maps (Coifman & Lafon, 2006) aim to reveal the manifold structure of data. Given
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similarity matrix and its corresponding graph, this method characterizes the geometry of the un- derlying manifold by simulating random walks. Computing the eigenvectors of the corresponding diffusion operator provides coordinates for embedding data into a lower-dimensional space, where Euclidean distances correspond to distances along the manifold and are referred to as "diffusion distances." However, constructing the similarity matrix requires the choice of a kernel in data space, and the diffusion "time" needs to be carefully calibrated (Shan & Daubechies, 2022). In contrast, the IEM does not make manifold assumption and instead relies on denoising, which corresponds to reversing a diffusion process in the full signal space.
Defining a Riemannian metric through the score of the blurred density. Recent studies pro- posed different ways define a Riemannian metric using the score of the blurred density (i.e. the density of roise-corrupted signals) (Saito & Matsubara, 2025; Azeglio & Bernardo, 2025). These Riemannian metries are then used to compute geodesics, which can, e.g., be applied to interpo- late between a given pair of images. Such approaches differ from our proposed IEM framework in several important ways. First, the IEM is, by definition, a global distance function from which we derive a local Riemannian metric, rather than the other way around. In fact, the IEM is not the geodesic distance corresponding to the associated Riemannian metric we derive (Eq. (5)). Second, the Riemannian metrics proposed in (Saito & Matsubara, 2025; Azeglio & Bernardo, 2025) are de- fined based on the score at a single blur level ", whereas the IEM (and its associated Riemannian metric) integrates over range of blur levels y E [), Third, the IEM is constructed from first principles and satisfies several important properties, e.g., it reduces to the well-known Mahalanobis distance when the prior density is Gaussian.
Information geometry. Information geometry (Amari & Nagaoka, 2000) uses tools from differ- ential geometry to analyze statistical models. In particular, it considers probability distributions as points lying on a Riemannian manifold whose metric is derived from the KL divergence. In the case of a family of conditional distributions (-y]x(-|z> x E R²), this metric is given by the Fisher information matrix (Fisher, 1922), which quantifies the amount of information that y carries about (here, x is considered as an unknown "parameter"). Specifically, the Fisher information matrix is defined as
I(x) = E [
V log x (x(x))())]))))))))))))))))))(( "].
(10)
where the expectation is taken over Py (- in the domain of px (although different x's 's can also be compared directly with the KL divergence between the associated conditional distributions). Note that the Fisher information metric is derived solely from the given observation model, namely the representation, Pylx. which can be completely unrelated to the prior Px (or otherwise, one has to specify explicitly how Pylx depends on px). Our approach departs from this classical framework in two ways (although there are qualitative analogies, see paragraph on invariance to sufficient statistics in App. C.4). First, the IEM depends directly on the prior Px. rather than through a potentially prior-dependent observation model Pylx- Second, the IEM is global distance metrie from which we derive the local Riemannian metric G (Eqs. (5) and (6)), but this is a one-way relationship: the IEM is not a geodesic distance.
(r). This metric can be used to define geodesic distance
Deriving a metric from a prior using Jeffreys rule. Solving a Bayesian inference problem re- quires both likelihood function Pylx and a prior Px. However, in some cases only the likelihood is available. In such situations, it is natural to choose a non-informative prior using Jeffreys rule (Jef- freys. 1946), which states that the prior should be proportional to the square root of the determinant of the Fisher information matrix. More relevant to our case, this relationship has also been applied in the reverse direction, where the prior is known but the likelihood not. In particular, it has been shown that a likelihood for which the Jeffreys prior matches the data distribution satisfies the principles of efficient coding (Ganguli & Simoncelli, 2014). The reverse use of Jeffreys prior has also been explored in machine learning. For example, Lebanon (2002) considered a Riemannian metric under which the data is uniformly distributed. Since the prior density is a scalar function, they assumed an isotropic metric of the form M (x) x I(x) where X(x) p(æ))/d and æ E I In both of these cases, the result depends only on a scalar quantity and therefore cannot account for the varying magnitudes of discrimination thresholds in different perturbation directions (Berardino et al., 2017). In contrast, the IEM builds on the local geometry of the data distribution to define a distance that captures its anisotropic structure.
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A.2 ORIGINS OF THE POINTWISE I-MMSE FORMULA
I-MMSE. The I-MMSE relation (Guo et al., 2005), which is closely related to de Bruijn's 's identity from the 1950s (Stam, 1959), is a fundamental connection between information theory and esti- mation theory for Gaussian noise channels. Specifically, the I-MMSE formula relates the mutual information between x and y, to the integrated MMSE achievable when estimating x from the noisy channel. Formally, letting I yx) denote the mutual information between x and Ny, the I-MMSE formula (Guo al., 2005) states that
yr) = E log
-
(11)
(yr)
2
where the expectation on the left-hand side is taken over the joint distribution Px.yr. while on the right-hand side it is taken over Px.y. for each y. The result above holds for any Gaussian channel with SNR y, and not only for the channel defined in Eq. (1).
Pointwise I-MMSE. By interchanging the order of expectation and integration the right-hand side of Eq. (11), we obtain
Pyr Pyrix(Vr/x) Pyr (yr)
)]
1 2 2
E log
E
||x
F
11
(12)
This reformulation highlights that the two sides the I-MMSE formula correspond to random vari- ables that are equal in expectation. Venkat & Weissman (2012) showed that these random variables satisfy the pointwise 1-MMSE formula
PynIe(()T)x)) Pyr
f (x- dw, + /
/" ||x [[]xxxy
log
(13)
where this equality holds with probability one (i.e., almost surely). As noted by Venkat & Weissman (2012), taking expectations Eq. (2) immediately recovers the original I-MMSE formula. Indeed, the stochastic integral on the right-hand side of Eq. (2) is martingale with zero mean, while the left- hand side corresponds to the pointwise mutual information between x and yy, whose expectation yields the mutual information between these two random vectors. Using the fact that
(yrl))) = Pwr (wr),
(14)
it is straightforward to see that the pointwise I-MMSE equation in Eq. (13) is equivalent to Eq. (2).
B MISMATCHED IEM AND LOCAL DECOMPOSITION OF KULLBACK-LEIBIA DIVERGENCE
Here, show that the IEM compares the local behavior of the density in way that resembles "local KL divergence." This is formalized through a direct relationship between the average squared IEM between point x and its additive perturbation x x s in fixed direction s, and the KL divergence between the distributions Px and px- To motivate this relationship. it is helpful to first introduce generalization of the IEM.
Intommation-Estiaatii Metric between samples from different distributions. While the IEM from Sec. 2 is distance function associated with single distribution Px. can be straightforwardly generalized to the case where æt and De are assumed to come from two different distributions Px, and Px,. respectively. We refer to this generalization as the mismatched IEM, analogously to the term "mismatched estimation" that appears in the information-estimant
relations literature (Guo
et al., 2013).
Our construction mirrors that of Sec. 2. We define the mismatched IEM as the expected quadratic variation of the log probaabibyy ratio log (Pyl, (yæ: + w,)/Pys. (yæ2 + w,)), where Py.., is the distribution of y,,- 7x, Wy, This log probability ratio involves taking the difference between two denoisers corresponding to the two priors Px, and Pxz- Similarly to Sec. 2, these denoising emrrr can be expressed in terms of the gradients of log Py.,. Specifically,
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Definition 3. The mismatched IEM induced by the densities Pxi and Pxa is defined as
1/2
IEMp px; :Peg (I1, L2, D) = ( V (121 + Wy) - V log Pya, (702 + (w,) ws))l(aar))
where the expectations are taken over Pw., for each 7--
Intuitively, the mismatched IEM compares the local geometry of logpx; in the vicinity of with the local geometry of Px, Pxe the vicinity æ2. When Px, = Pxg. we trivially recover the IEM given in Def. 1.
Relation to Kulbback-Leibler divergence. The mismatched IEM distance is directly related to the KL divergence between Px: and Рx.- Indeed, Venkat & Weissman (2012) showed that this KL divergence can be expressed as the expected quadratic variation of the log-probability ratio when the two distributions are evaluated the same point, log (pg(P,,, (7x1 + w,)/Pya., (7x1 + wx)). yield- ing
1 (xx Pxa 2
/
///00//
IE V PyL, (1x1
Pog
w,)) log (7x1 + w,,)ll (15)
where the average is taken over Px,Pw. This result also appeared in Verdá (2010). immediately follows that the KI. divergence can be expressed as the average mismatched IEM with I oo, as follows:
DKL (Px: Px.) _ IEM Pxj Pxa
x11.
(16)
Generalizing the above for I 30 is trivial yielding KL divergence between the blurred ver- sions Px, and Px,- Equation (16) allows us to interpret the mismatched IEM between the same points samples coming from two different distributions, KL divergence between these two distributions. It formalizes the intuition that the (mismatched) IEM compares the local behavior of two (potentially different) densities around the two points, since the average over Px, yields global comparison (given by the KL divergence). Note that log(px, (x))/xx (x1)) also qualifies local decomposition KL divergence, but unlike the mismatched IEM, it is not a valid distance (for instance, it can take negative values). The mismatched IEM can thus be thought of a positive-definite decomposition of the pog-probability ratio.
a local decomposition of the
the sense that its average yields the
We now reinterpret Eq. (16) in the case of the IEM given by Def. Consider the scenario where the two distributions are px and Ps, where X = px (x(@) (x + 8), and thus IEMpx. (x, Φ) IEM(æ, ă + 8). In this setting, Eq. (16) becomes
for some additive (fixed) shift s. We then have
DKL. (px ll px)) = EE [IEM² (x, x s)]
(17)
By taking s = I1. this equation allows us to interpret IEM (11, x2) as the term corresponding x Φ1 in the local decomposition introduced above of the KL divergence between Px and its translation by 21. Again, it formalizes the intuition that the IEM compares the local behavior of the density around the two points xı and Ø2.
C PROOFS
C. PROOF OF TIM.
Theorem 1. For every I >> 0, the IEM is proper distance metric: it is symmetric, non-negative, equal zero if and only if x = 22, and satisfies the triangle inequality.
Proof. We verify the four metric axioms.
Symmetry. Swapping æn and &e leaves the squared norm unchanged, so
IEM (z), 02 D) [EM T)-
(18)
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Non-negativity. By definition, the integrand is squared norm and is thus nonnegative. The inte- gral and square root preserve nonnegativity, hence IEM (1, 22, r) 2 O.
Positive definiteness. that
Suppose IEM 11, 22, I) 0. Then for Lebesgue-a.e. y E [0, we have
EE
V loggy, (ro)) + ww) log Py, (y +
w/)]l²] +w,)+++,,110
=
(19)
which implies
V log Py, (1 + w,) = V log Py, (1 + wy) a.s. in Wy.
(20)
Because W., has a strictly positive density on R2 it follows that
V log Py, (y) = V loggy, (y + ( -x2)) for Lebesgue-a.e. y.
(21)
Thus the function is constant a.c., say 9- (y) = G. Exponentiating, we obtain
9- (y) log Py, (y + (x1 x22) log Py, (3)
(22)
Py, (y + 7x1 - x2)) = c" py, (y))
(23)
Integrating both sides over TR gives
1= / Py. (y + -
= Es / (y)dy -
(24)
so = 0. Hence
Py, (y + 7 - x2)) py, (v) for Lebesgue-a.e. y.
(25)
Fix some 7 0 so that the above holds. This means that Py, is invariant under translations by the vector 7(0) - 22). However, there no probability density on RR that is invariant under nonzero translation. To show that this is true, if œ + 22, consider the sets
B
y E
Rd
k 5
y
7||21
22
x2
2
< k +1
(26)
for k E Z. B3 forms a partition of IRd, so Σ kEz. SP. Py, (y)dy= 1. But by translation invariance, the terms in the sum do not depend on k, which is a contradiction. Therefore the translation vector must be zero, i.e., (xx 222) 0, which implies 21 22.
EkEZ
Triangle inequality. For each y, define
ly, Py, "y, + wx) V log (700 + ws))ll?) 1/2
F
|| log
(27)
My trivially satisfies the triangle inequality:
M, 0a) S M, (2)) + M, x:3)-
(28)
Integrating over and applying the Minkowski inequality, we get
IEM (2), I) = /-' M2, 2x,,))1)))d 2dy
< 1''0 (Mr x2) + Mn xxa)) 2d7 dr
I. My (2) 22y + V. My xz) M1((22, 2dy
2
(29)
Taking the square root on both sides gives
(EM L3, T) < IEM(0), 22, + IEM
(30)
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C.2 PROOF OF THM.
Theorem 2. The local Riemannian metric derived from the second-order Taylor expansion of the squared IEM is given by
G( = S/¹ B²B (22 logPy, Py, (rx +w,) dy
(5)
L B -1-
y + w. d7,
(6)
where the expectations are taken over pw., for each 7.. Moreover, for 0o we have
E[G(x)] B[ V Px
log log Px())) J,
where we denote ( == G (z) oo) and the expectations are taken over px.
Proof. We begin by taking the Taylor expansion of the IEM to derive first expression for the local metric G( D) in terms of the Hessian matrix of log Py, (App. C.2.1). We then derive an equivalent expression in terms of the covariance of the posterior Pxly,, (App. C.2.2). Finally, we relate the average of the metric G(x) the average of the Hessian of log (App. C.2.3).
C.2.1 TAYLOR EXPANSION OF THE IEM
We Taylor-expand the IEM distance between æ and + in
IEM x + e, D) =
E V log (yx + + w.y) - V Vloggy, (rx + w
(31)
E
log Py, (72 w, + + dy
(32)
T
22 V loggy, (1x + e o(le)]))
(33)
We thus have
G(a, = (-² ,² (Vology
loggy, (r0 + wz))
(34)
C.2.2 FROM THE HESSIAN OF THE NOISY CHANNEL LOG-DENSITY TO THE POSTERIOR COVARIANCE
We now show that the Hessian of log Py, can be expressed in terms of the posterior covariance of x conditioned on y-;
1 I. 7
2 logpy, (y) Cov[x|y, y)
(35)
This relationship has already appeared in the literature in several contexts, and is often referred to as the "second-order Tweedie identity." To the best of our knowledge, was first derived by Hatsell & Nolte (1971, Proposition 3). For completeness and notational consistency, we include derivation here.
We have
logPy, (v) log ( (xla))y,,
(36)
Differentiating w.r.t. y gives
1 py, (v) / px(x) uPy, be (x))(
V loggy, (A)
(37)
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Differentiating again w.tt. y gives
V 1yg (3) / Vpp xx (g)x))d
Py, (2)2 (ʃ (AaLeNWaRr.MMl ( (IasleNoRss
(38)
V 1x(xxx)))
yz
1x (3/x))
[[V, log Py, P (x)()) Iy, = y) E. [ log bog
(39)
Here, 8- 1~~ (2xx, ). A direct calculation gives
V v log (x = = y, and
(40)
V (x
1 y
( x
y/ 9
I.
(41)
(
?
Substituting into Eq. (39) and rearranging then yields
1 I.
V logPy, (y) Cov(x yy y)
(42)
Finally, injecting Eq. (42) into Eq. (34) gives the second expression for the local metric:
G(x, = / B B Covlx = y + wr,) d7.
(43)
C.2.3 AVERAGE LOCAL METRIC
We begin by decomposing the Hessian of the log-density log Px. Specifically, we use Eq. (4) to express the log probability ratio between point æ and a perturbed version of + e. Taking T - oo and averaging over Pw. gives
px(x=) { )
1 /
B[le,(0 + e, w,,, - (a, wr))lpjar
log
(44)
(x
Next, we take the Taylor expansion of the tracking error. From Tweedie-Miyasun
e,(2, = x -[xxyy, = 70 + w,] W.y V Vloggy, + wr);
(45)
we have
(,, W + loggy, (720 + w,)) V log Py, (7,0 Wyye
1 3 log Py, (1æ (,, + o(|le|)))
+
(46)
2
where we write A(x,y) = (2, Aiju:,3RR); for the partial contraction of symmetric third-order
tensor A against the vectors y. By inserting Eq. (46) into the expression of the log-probability ratio in Eq. (44) and expanding the square, we obtain
(x(x)
log
Px(x + E)
1
W. + V loggy, (7++ w.), Wo),,, (1op),,( wy)e + 22 v 2
E
2
V'''logyy (7æ (x)
2
+| log (10 + w.,) dx o((|E|13)
(47)
1
42 1og bloggyy (1 + w W,
v Vlog (700 + w.)) yp,
3 log + wx) (,w,,, + Vlog (yx + +w)) ((.)))(pa)))
v log Py, (1x + w²))²
2
(48)
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By identification, it follows that
- -v² Px
/ E8 V2
log Py, (r + w,)) + v
Py,
Vo log Py (ro + w.) ( + loggy, + w.)) +w.)))
)
(49)
Taking the expectation over Px. we obtain
V log Py, (yr))² + vo Py, (yr) { Ww,, V log Py, (yn) 72d7,
E| log [xggx(x)]
E
Vlog
(50)
where we used the fact that y7 7x Wy. The second term in the expectation then vanishes due to Stein's lemma (Stein, 1981),
V Py, (y+) ? W )
(,,r,)
E
E AE|
log Py, Po, (y+))
(51)
while applying integration by parts yields
E V log (y,) V logpy, (y+))] -E A (y:)]
(52)
Finally, we obtain
log = / (v log (x-)) (x))) -EG()))
(53)
which is the desired relationship.
The last equality in the theorem is generic and classical. By expanding the derivatives, we have
-f Pxx2z (xxx) log Px
(x dx (ogx
- (2px(x))
(54)
The second term in the integral on the right- side vanishes through integration by parts.
C.3 PROOF THAT THE IEM COINCIDES WITH THE MAHALANOBIS DISTANCE FOR GAUSSIAN PRIORS
Here, we prove the following proposition:
Proposition 1. Letx be a Gaussian random vector with mean şt and covariance Σ > 0. For 2 0, define
y, X + Wy: W,, NOO,,I), W., Lx
Then the MMSE estimator of given y, given by
E[x y-1 = p + K, (y- - 714);
where K, = ;=>=(=E) 71) Further, for any 21, 22 with A 20 322 if Σ >
IEM (2), = f. ( |e,, (2), w.) -
e (222, )1 = A
where (x, w.) = x - E(xly~ = 7x + w.l. If > 0 is singular, the integral equals A Σ A provided A E range(E) ()) (where Σt is the pseudoinverse of E) and diverges to too otherwise.
Proof. If x is Gaussian, then (x, is jointly Gaussian, with mean (p47444 and joint covariance
Σ
Σ
(55)
ΣΣ
2
By elementary properties of Gaussian distributions, follows that conditioned on y,, also has Gaussian distribution, with mean given by
E x yz p K. (y, 714,,
(56)
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where K, = >>>><>>>>>> + .. The denoising error vector is then, for any æ.
e, (2, w.) p (7 - (4) W,).
(57)
Hence,
ey (2 W.) - e, (22, (K-
(58)
> -
Since Kg === = +
we have
1-1
(+ I)-
(59)
Therefore
(2 w2) w,,) A I)
(60)
Assume first E 0 and diagonalize E UAUT with A Aa). 2 > 0. Writing = A
af
A (+E
(61)
(1 (1+)1)²
Integrating termwise,
dy
(62)
(1+ 72,)2
X
1+ 1
Ai
Thus,
d Σ A
|
A (+
-
(63)
If Σ 0 is singular, decompose with A; 0 and note that the integral of a3/(1 + TX,) diverges when A; = 0 and a; ff O, while it equals 0 when a; = 0. Thus finiteness requires A range(>)) in which case the same computation the nonzero eigenvalues yields A T A.
C.4 ADDITIONAL PROPERTIES OF THE PROCESS Z-
Invariance to Euclidean isometries. In Sec. 2.3 we show that one may derive different kinds of distance functions from the process Z-y. A natural question, then, is whether such distances preserved under Euclidean isometries. In other words, do these distance functions depend on the coordinate system in which is represented? The following proposition establishes that the process z, is invariant to such isometries, in the sense that changing the coordinate system of x yields the same stochastic process 2, up to a reparameterization distances introduced in the previous sections are invariant to Euclidean isometries as well.
the Wiener process. Consequently, the
Proposition 2. Let (bx))= Ax + b with A Bddd orthogonal (A A I) and b IR., Define
Y, 7X + Way y = T0(x) + Wyy
where W. ~N) is standard Wiener process statistically independent of x. Then for all F1 22 E Rd and all have
(Pyg(() ())((((X2)) +x,)) Pyg (70(x1) + w.,)
log Py, (y02 + w'r) Py, (yx1 + w())
log
where w/, AT Aww, W,y also standard Wiener process (w², d w.)-
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Proof. Let w/ == A wy. Aww,. Since A is orthogonal, w'y ~ ~N(00, and remains standard Wiener process independent of x. Define y' = 7x + Wy. Then y', Ny and
y 7Ax + +7b + wy
A(7x A Alw,) w.)) + +
AAy +
(64)
Now, since |det A| 1, the change of variables formula gives, for every y,
pyy yy) = PAy; +2b())
Py; (A (y -b))
Py, (A (A (y -b)).
(65)
Thus, for any fixed æ; it holds that
Py (yp(ro(2;)) + w.) = Py, (A T (1)((A)r) + 7b + W., - -bb)
= Py, (y TTw w.))
(66)
Hence
Py (70(x2) + w.,)
Py, (,,22 + AA (w.)
Py, (7,32 + W'y)
log
llog
= llog
(67)
py. (70 + w.y)
Py, (1 A (ww,))
Py, Py, (y00 + w()
Invariance to sufficient statistics. Traditionally, perceptual distance functions are obtained by first defining a stochastic representation y of x through likelihood function Pylx, and then "pulling back" the information geometry of this conditional density to the signal space, as given by the Fisher information of the representation. One can make a qualitative analogy to the IEM distance, interpreting the Gaussian channel y- the "representation" of x. The process Z, (x1, 222) then compares the signals æj and D2 by estimating each from its respective representation, and measuring the discrepancy in the resulting estimation errors back in the original signal space. Although the "pull back" is now done through estimation quantities rather than information geometry, one may wonder whether they share similar properties. In particular, an important property of the Fisher information metric is its invariance under sufficient statistics of the representation. We show below that z,, and thus the IEM, is also invariant under sufficient statisties of yn.
Proposition 3. Let y T(y.,") be sufficient statistic of y, with respect to for every % namely yn ++ y + is a Markov chain. Then
Pyt (T(122 wy
Py, (1x2 + (w,) Py, (yar + w,,)
log
Pyt (T(12) Wy. loog
7))
Proof.
From the law of total expectation, we have
E|x = E[[[x (x.x yr]
(68)
Since y, + y. <> x is Markov chain, then PxlyT
Px y..
So we have
E yn] E y
(69)
Substituting Eq. (69) into Eq. (68), we get
[lx|y_]= E y yr]]
(70)
Finally, since y is function of y,, then E [x y? is function of y, well. We therefore pull [[x >² out the expectation, and get
B x y,
= E x y.]
(71)
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From here, it is easy to see that the processes
Py, (7002 + wx)
2, log
Py, (701 + wx)
and
(72)
Pyg (7 (7x222 + w3))
z log
(73)
Pyg (T + w>))
are exactly the same, as they only depend on E[x yy) and Elxx y respectively.
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CNN
Traditional
Super-res reperrres..
Frame int.
Deblur.
Color.
84.0
79.3
71.2
63.6
61.3
66.0
2AFC Score
81.3
71.8
68.4
61.0
09.6
62.1
78.6
64.3
65.7
58.4
58.0
58.1
73.9
56.8
63.0
33.9
56.4
54.2
60.3
53.3
54.7
50.2
SSIM PSNR
FSIM
VIP
GMSD
LPIPS DISTS
TOPIQ IEM
IEMaa.
MAD
NLPD
PleAPP
IEM feu
Figure 6: Two-alternative forced choice (2AFC) performance comparison on the different dis- tortion categories in the BAPPS dataset. The unsupervised IEM achieves competitive perfor- mance in most types of distortion. Our supervised variant, JEM,., further improves the results.
TID2013
LIVE
CSIQ
TQD
0.70
0.83
0.80
0.71
KRCC
0.57
0.77
0.74
0.53
0.45
0.71
0.68
0.35
0.21
0.58
0.56
0.00
S5IM PSNR
FSIM MAD
GMSD PieAPP
LPIPS DISTS
TOPIQ 1EM
IEMaq.
NLPD
IEM
Figure 7: Kendall correlation coefficient (KRCC) results on several full-reference image simi- larity benchmarks.
TID2013
LIVE
CSIQ
TQD
0.91
0.96
0.95
0.89
PLCC
0.77
0.91
0.91
0.68
0.62
0.86
0.86
0.48
0.48
0.81
0.82
0.28
0.33
0.76
0.77
0.07
GMSD
TOPIQ IEM
IEM
PSNR
MAD
NLPD
PleAPP
DISTS
IEM p.
Figure 8: Pearson linear correlation coefficient (PLCC) results on several full-reference image similarity benchmarks. Following common practice, map the similarity scores to the MOS scores by fitting four-parameter logistic function.
D ADDITIONAL EXPERIMENTAL RESULTS
D.1 PREDICTING TWO-ALTERNATIVE FORCED CHOICE JUDGEMENTS
Here, we evaluate our solutions on the BAPPS dataset (Zhang et al., 2018), which consists of image triplets: reference image Tee and two distorted versions, 21 and 22 along with probability- of preffeeeenece score for each triplet. These probabilities are derived from human evaluations of similarity between the reference image and each distorted version. The probability score is defined as the fraction of evaluators who preferred (Kref, 21) over (Rief, 222). Specifically. if evaluators preferred (ærer, 21) and n2 preferred (æref; 22),, then the triplet (Krer, æ1 .22));))))))))) is assigned the label p n1/(m4 n2). The two-alternative forced choice (2AFC) score for given similarity measure is then computed by averaging p- 1,, +11 1e, +0.5 (0.5-1.,=1 1,, across the entire dataset (Zhang al., 2018).
To compute the IEM, we train an additional neural denoiser model on ImageNet- kk with images of size 64 64, which is the native resolution of images in the BAPPS dataset. We use the same type of denoiser architecture as Sec. 3 (see more details in App. E). Finally, we compute the IEM, using this trained denoiser, similarly to Sec. 3. The results reported in Fig. 6.
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D.2 PREDICTING MEAN OPINION SCORES: ADDITIONAL METRICS
We extend the experiments from Sec. 3.2 by reporting additional metrics on MOS predictions for the same methods and datasets. In Figs. and 8, we present Kendall's rank correlation coefficient (KRCC) and Pearson's s linear correlation coefficient (PLCC), respectively. These correlation scores exhibit trends similar to those observed with Spearman's rank correlation coefficient (SRCC, see Fig. 4). Note that to compute the PLCC, we follow common practice and first fit four- Kour-parameter logistic function.
D.3 MAXIMUM DIFFERENTIATION COMPETITION AGAINST PSNR: ADDITIONAL. DETAILS AND RESULTS
This section provides additional details and results for the maximum differentiation competition experiment described in Sec. 3.3.
Overview of the maximum differentiation competition optimization procedure. To conduct a maximum differentiation competition between given perceptual distance measure and PSNR, we begin with a reference image æ. We then sample a white Gaussian noise vector and normalize it such that the MSE between æ and x + equals a predetermined constant C, derived from target PSNR level. Formally, this normalization is defined as
C E |||
(74)
where C 2vdd 10 PSNR/20 and the image pixels value range is [-1, 1]. Given a total of N optimization steps, we iteratively update by optimizing (minimizing or maximizing) the perceptual distance between æ and t using projected gradient descent, where the radial projection in Eq. (74) is applied after each step to maintain a fixed PSNR throughout the optimization.
Optimization of the IEM. Using the IEM as optimization objective is challenging (e.g., when aiming to minimize the distance between a pair of images). In particular, since the IEM involves computing a one-dimensional integral of denoising errors over wide range of noise levels, the corresponding denoiser gradients must be computed and aggregated across this range. This results in large deviations in both the scale and variance of the loss gradients across noise levels, as the scale and variance of the denoising errors at low-SNR regions are substantially larger than those at high- SNR regions, causing the low-SNR regions to dominate the optimization. Reweighting the integral through a change of variables does not fully resolve this problem, it does not address the variance issue. To alleviate this issue, we propose an annealing strategy that enables stable optimization of the IEM in the context of the maximum differentiation competition against PSNR.
For an integration range (T, I in Def. 1 (where Iy is a small value that replaces the lower bound of the integral), optimize the IEM by progressively optimizing the integrand in Def. 1, starting at the lowest SNR To and gradually increasing the SÑR until it reaches I. Formally, we construct sequence (n, 022,. , ON uniformly spaced between log To and log and define 7 exp(a;). Then, at each optimization step i = 1,2,.... N, we update e by minimizing or maximizing the objective
k V Pog,y., (7:2: + w..) V Vlggyy, (7.(2: + +6) 13
(75)
where Wy, NN I) is sampled randomly and independently at each step i. Note that the optimization objective in Eq. (75) corresponds to the integrand in the definition of the IEM (Def. 1), where the expectation is approximated with a single noise sample (similarly to standard stochastic optimization), and the change of variables a = log applied (which introduces a multiplicative factor 7, in Eq. (75)).
The distortion e optimized using the Adam optimizer (Kingma, 2014), whose momentum term aggregates gradients across SNR values over the N optimization steps. Thus, the update at iteration i implicitly incorporates information from all previous steps. After each update, we re-normalize e according to Eq. (74) to ensure that the MSE between and æ + remains equal to C' throughout the optimization. We fix To- = 10 (lower bound of the integral) and r = 1 (upper bound of the integral) and use learning rate of 5 10-2 with N 1000 optimization steps. The Adam running
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averages coefficients are 81 0.9 and B2 0.999. These hyperparameters are kept fixed across all image examples.
Optimization of other perceptual distance measures. To illustrate the difference between the IEM and other perceptual distance measures (DISTS, LPIPS, VIF, SSIM, TOPIQ, PieAPP, NLPD, and GMSD), we use each measure to minimize or maximize the perceptual distance between a and + c, while maintaining a fixed PSNR level (we apply the same projection procedure after each optimization step, as defined Eq. (74)). All optimizations are performed using the Adam optimizer with a learning rate of 5 x 10-2, running averages coefficients of 81 0.9 and A8 = 0.999, and N 1000 optimization steps (consistent with the optimization procedure used for the IEM). We find that all evaluated perceptual distance measures, including the IEM, are effectively optimized using these hyperparameter settings, as shown in Fig. 13. We employ the implementations these metrics provided by the IQA-PyTorch package (Chen & Mo, 2022).
We note that TOPIQ, VIF, and SSIM are perceptual similarity measures, for which higher values indicate smaller perceptual distances. Thus, for these measures, minimizing (or maximizing) the perceptual distance corresponds to maximizing (or minimizing) the similarity measure.
Results. We conduct the maximum differentiation competition described above on several im- ages selected from the DIV2K database (Agustsson & Timofte, 2017), which contains high high-quality general-content images. Similarly Sec. 3.2, we preprocess each image by first center-cropping to the length of its shorter edge, and then resizing it to 256 x 256. For each method, the maximum differentiation competition performed five times under different PSNR constraints of 30dB, 25dB, 20dB, 15dB, and 10dB. The final results are shown in Figs. to 12.
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Reference image
Minimizo perceptual distance
Maximise percuptual distance
SSIM
VIF
LPIPS
DISTS
IEM
IRM
DISTS
LPIPS
VIF
SSIM
Figure 9: Visual results of the maximum differentiation competition. A reference image (top) is corrupted with increasing levels of white Gaussian noise (30dB, 25dB, 20dB, 15dB, and 10dB), producing sequence of distorted images (middle column) with varying MSE (PSNR) values rel- ative to the reference. Starting from each noise-distorted image, we minimize or maximize each of the perceptual distance measures shown (IEM, DISTS, LPIPS, VIF, and SSIM), while keeping the MSE (equivalently, PSNR) fixed. Interestingly, minimizing the IEM consistently yields high perceptual quality images that preserve the overall geometric structure of the reference image, even under large MSE (low PSNR) constraints. In contrast, all other methods introduce noticeable and unnatural artifacts during optimization (zoom in for best view). Maximizing the IEM reveals that most sensitive to unstructured noise perturbations that push an image off the "data support." This consistent with our theoretical analysis Sec. 2.
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Refevnce image
Avil modiee
peroeptual distance
Maximise peccrptual
SSDM
VIF
LPIPS
DISTS
IEM
IBM
DISTS
LPIPS
VIF
SSIM
MSE
Figure 10: Visual results of the maximum differentiation competition. reference image (top) is corrupted with increasing levels of white Gaussian noise (30dB, 25dB, 20dB, 15dB, and 10dB), producing sequence of distorted images (middle column) with varying MSE (PSNR) values rel- ative to the reference. Starting from each noise-distorted image. we minimize or maximize each of the perceptual distance measures shown (IEM, DISTS, LPIPS, VIE, and SSIM), while keeping the MSE (equivalently, PSNR) fixed. Interestingly, minimizing the IEM consistently yields high perceptual quality images that preserve the overall geometric structure of the reference image, even under large MSE (low PSNR) constraints. In contrast, all other methods introduce noticeable and unnatural artifacts during optimization (zoom in for best view). Maximizing the IEM reveals that most sensitive to unstructured noise perturbations that push an image off the "data support." This consistent with our theoretical analysis in Sec. 2.
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Refeveuce image
Minimize perceptual distance
Maxiinize perceptual distance
GMSD
NLPD
PleAPP
TOPIQ
IRM
TOPQQ
PieAPE
NLPD
GMSD
MSE
Figure 11: Visual results of the maximum differentiation competition. reference image (top) is corrupted with increasing levels of white Gaussian noise (30dB, 25dB, 20dB, 15dB, and 10dB), producing a sequence of distorted images (middle column) with varying MSE (PSNR) values rela- tive to the reference. Starting from each noise-distorted image, we minimize or maximize each of the perceptual distance measures shown (IEM, TOPIQ, PieAPP, NLPD, and GMSD), while keeping the MSE (equivalently, PSNR) fixed. Interestingly, minimizing the IEM consistently yields high perceptual quality images that preserve the overall geometric structure of the reference image, even under large MSE (low PSNR) constraints. In contrast, all other methods introduce noticeable and unnatural artifacts during optimization (zoom in for best view). Maximizing the IEM reveals that most sensitive unstructured noise perturbations that push an image off the "data support." This is consistent with our theoretical analysis in Sec. 2.
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Refeveuce
Add miii
Minimize perceptual distance
Maximize pezceptual distauce
GMSD
NLPD
PieAPP
TOPIQ
IEM
IEM
TOPIQ
PieAPP
NLPD
GMLSD
Figure 12: Visual results of the maximum differentiation competition. reference image (top) corrupted with increasing levels of white Gaussian noise (30dB, 25dB, 20dB, 15dB, and 10dB), producing a sequence of distorted images (middle column) with varying MSE (PSNR) values rela- tive to the reference. Starting from each the perceptual distance measures shown (IEM, TOPIQ, PieAPP, NLPD, and GMSD), while keeping the MSE (equivalently, PSNR) fixed. Interestingly, minimizing the IEM consistently yields high perceptual quality images that preserve the overall geometric structure of the reference image, even under large MSE (low PSNR) constraints. In contrast, all other methods introduce noticeable and unnatural artifacts during optimization (zoom in for best view). Maximizing the IEM reveals that it is most sensitive to unstructured noise perturbations that push an image off the "data support." This is consistent with our theoretical analysis in Sec. 2.
image, we minimize or maximize each of
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0.8
3
200
0.6
-EM
DISTS
L
2
0.4
100
0.2
0.0
25
15
15
PSNR constraint (dB)
PSNR constraint (dB)
PSNR constraint
(dB)
200
3
0.4
PPiiAPP
GMMS
2
100
0.2
0
0.0
30
25
20
15
10
30
25
20
15
10
30
25
20
15
10
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(dB)
PSNR constraint
(dB)
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1.00
1.00
1.00
0.75
TOPIQ
0.75
SSIM->
0.75
VIF-
0.50
0.50
0.50
0.25
0.25
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0.00
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0.00
25
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PSNR constraint (dB)
PSNR constraint (dB)
Best
Initial
Worst
Figure 13: Best, worst, and initial values of different perceptual metrics in the maximum differ- entiation competition against PSNR. To verify that the evaluated perceptual metrics are effectively optimized the maximum differentiation competition against PSNR, we present the average best, worst, and initial values of each metric (shown as bar plots), together with their standard deviations (shown as error bars), computed over 30 images randomly selected from the DIV2K dataset. The best and worst values correspond to the final perceptual distances obtained through minimization or maximization of the distance under a fixed PSNR constraint, respectively, while the initial value rep- resents the perceptual distance between the initial distorted (noisy) image and the reference image. While global optima are not guaranteed, these results show that all perceptual distance measures are consistently optimized. For example, the best SSIM values remain close to their maximum possible value of across all PSNR constraints, whereas the worst SSIM values approach 0 as we decrease the PSNR constraint. This trend is consistent across all evaluated perceptual metrics. Thus, for each perceptual metric, the images on the left-hand side of Figs. 9 to 12 (in the column corresponding to reference image according to metrie, whereas the im-
that metric) are perceived as similar ages the right-hand side are perceived perceptually distant from the reference image. Finally, we note that TOPIQ, VIF, and SSIM are perceptual similarity measures, for which higher values indicate smaller perceptual distances. Thus, for these measures, minimizing (or maximizing) the perceptual distance corresponds to maximizing (or minimizing) the measure. For this reason, the "best" (blue) bar plots for these metrics are high, while the "worst" (orange) bar plots are low.
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Information-Estimation Metric
Euclidean distance
Cluster
Cluster
8
Medoid
122
-3
-3 T1
2
3
33
-2
0
2
T1
Figure 14: Utilizing the IEM to solve toy clustering Gaussian mixture problem. We apply the K-medoids algorithm twice: once with the IEM (left panel) and once with the Euclidean distance (right panel). While K-medoids with the Euclidean distance fails to recover the correct cluster separation, using the IEM yields accurate solution that aligns with the underlying modes.
D.4 A TOY CLUSTERING EXAMPLE
To illustrate potential future application of the IEM, we it to solve simple two-dimensional clustering problem in which the K-medoids algorithm (Kaufman & Rousseeuw, 2008), when cou- pled with the Euclidean distance, fails provide satisfactory result. Specifically, we consider Gaussian mixture density consisting of two modes,
00
0.95) ( + 0.5N 0;; (4.) 0.95
0.95
(x(x)) 0.5.V a;; 1 1..5
(76)
0.95
We apply the K-medoids algorithm on 500 samples drawn independently from this density, using either the IEM the Euclidean distance the distance measure. To compute the IEM, train simple unsupervised neural denoiser (a 5-layer fully-connected network with GELU activations) on the range log y [2-10 20 and compute the integral in Def. on the same range after changing the integration variable to log Importantly, the denoiser only depends on y- and so it is "unaware" of the clusters. We 200 discretization steps to numerically solve the integral, and approximate the expectation of the integrand by averaging over 50 random Brownian motion paths. Remarkably, the IEM resolves the clustering problem effectively: it selects medoids located at the means of each mode and assigns samples to their corresponding modes with high accuracy, as illustrated in Fig. 14.
E TRAINING AND IMPLEMENTATION DETAILS
E. NUMERICAL INTEGRATION
Computing any of our distance functions requires numerically solving an SDE. We use the Euler- Maruyama discretization for this purpose, applying change of variables so that the integral evaluated over log-SNR values instead of SNR values. This common technique improves numer- ical stability and distributes integration steps more effectively across the SNR domain. To reduce computational demands, we compute the integrals along single Brownian motion path in all our experiments. Compared with averaging over multiple paths, we observe no significant difference in the resulting approximated distances. An independent Brownian motion path is generated randomly each time we compute the distance. Throughout our experiments, we use 512 discretization steps, which is relatively large number. This choice ensures that our results reflect the intended distance
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measures more accurately. Empirically, find that using fewer steps (e.g., 128) does not alter the results.
E.2 DENOISER TRAINING
The HDIT denoiser model training nyper-parameters are given in Tab. 1. To resize the ImageNet- Ik training images to 256 x 256 resolution, we center-crop each image to its shorter edge dimension and then Lanczos resampling. For ImageNet-Ik 64 x 64, we the official resized training set provided on the ImageNet website.
We note that trained denoiser model uses the Variance Exploding (VE) formulation (Song al., 2020) (specifically, EDM (Karras et al., 2022)), where the SNR y and the noise level a are related via 7 1/0².
E.3 LEARNING f"
We describe the architecture of our learned f' in Tab. 2. This function contains about 3M parameters all our experiments.
In Sec. 3.2, we train f using the KADID- 10k (Lin et al., 2019) and DTD (Cimpoi et al., 2014) datasets, similarly to (Ding et al., 2022). We cap the integral in Def. 2 I 102. For KADID- 10k, we employ simple pairwise ranking loss to encourage agreement between the rankings of the predicted distances and the ground-truth MOS scores. Specifically, given a mini-batch triplets
(ar dist ()))
(
(+)
)=
(77)
(i) where ae is reference image, dia. its distorted version, and ((0) is the MOS score for this pair, we compute the distances
(i)
d IEM,, (steet Ffist
(78)
The pairwise logistic ranking loss is then defined as
1 N2 log exp ( ( (00))((00))00)0))))))))))))) -
L
(79)
where 7 is temperature parameter learned jointly with w. For DTD, we randomly crop subimages of size 256 x 256 from each texture image, and apply smooth L1 loss with _ = 10.0 to the distances obtained for these patches, encouraging IEM,, to small for textures the same kind.
In App. D.I, we train f using the training split of BAPPS. We cap the integral in Def. 2 at 10 Similarly to Zhang et al. (2018), the loss function is a standard cross-entropy loss applied to the output of small fully connected network with nonlinear activations. This network maps the raw distances IEM,, to logits, which are then passed to the cross-entropy loss. The additional fully connected network is trained jointly with l-
In all experiments, we use the Adam optimizer (Kingma, 2014) with a learning rate of 10-3 a batch size of 512, and train for 100 epochs. We apply an exponential learning rate decay with a factor 0.95 after each epoch.
E.4 IMPLEMENTATION OF THE ILLUSTRATIVE EXAMPLES
We provide additional illustrative examples in Fig. 15, where we consider three different one- dimensional prior densities.
Prior densities used in Fig. 2. In the middle column of Fig. 2, we use a Gaussian density
px(x) N x: ( ( 0.1 )))
(80)
In the right-most column of Fig. 2, Gaussian mixture density
(
0 ) 0.7N ax; (-1)
(15
0.5
(x(x) 0.3.V a;:
(81)
0.5 0.4
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A
E
h
G(x)
log Px(x)
C
a
Figure 15: Local sensitivity of the Information-Estimattio Metric (IEM) for several dimensional prior densities. In each subplot, the local metric G(x) (Eq. (5), using very large 1) is plotted and compared the log-density log Px. The integral in Eq. (5) computed over log E [2-5, 2 after applying change of variables. (A) For a Gaussian (light-tailed) density, G(æ) coincides with the Mahalanobis metric, which is constant everywhere. (B) For a Gaussian mixture density, ((z) decreases near the local maxima and increases near the local minima; it converges to a constant value only at the tails. (C) For a Laplace (heavy-tailed) density, the sensitivity the local metric increases with the probability of the signal æ. (D) For Laplace mixture density, the sensitiv- ity grows near both the local maxima and minima, where the absolute curvature of the log-density is relatively large. (E) For mixture of Laplace and Gaussian densities (left and right modes, respec- tively), the sensitivity is relatively constant only the region where the Gaussian density dominates the Laplace density. All of these plots illustrate that the sensitivity of G(x) is governed by the local curvature of log Px around æ.
In the left-most column of Fig. 2, we use two-dimensional Laplace density obtained by taking the product of two one-dimensional densities,
Px (xx Lap(#1; 0, b = 0.3) x Lap(x2; = 1, 0.1),
(82)
where x (x1,
Numerical computation of the IEM and the associated local metric G(æ, I). For each of the prior densities above, we write the function log in closed form in Py PyTocc and compute V log Py., and 2 log using torch. aut ourad. The IEM in Def. 1 (global distance) is com- puted using 200 uniformly spaced discretization steps over log [2 [2-10 2n0 for all prior densi- ties, with 50 random Brownian-motion paths to estimate expectations. The local metric G( I') paths,
in Eq. (5) is computed using the same number of discretization steps and while the integral is taken over log E [2 2,24.
F LLMS USAGE
We used Large Language Models (LLMs) for minor text polishing and assistance in generating figures.
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Table 1: HDiT architecture details and training hyperparameters for two configurations.
Hyper-parameter
ImageNet-64*
ImageNet-256
Parameters
11.6M
22. 1M
Training steps
400k
400k
Batch size
256
256
Image size
64x64
256x256
Precision
b6loat16
bíloat16
Training hardware
H100 80GB
H100 80GB
Training time
1 day
3 days
Patch size
4
Levels
1+1
1 1
(local global attention)
Depth
[2, 11]
[2, 11]
Widths
[64, 128)
(128, 256]
Attention heads
[1, 2]
[2, 4]
(width / head dim.)
Attention head dim.
64
64
Neighborhood
7
kernel size
Mapping depth
Mapping width
768
768
Data sigma
0.5
0.5
Sigma sampling density
log-uniform
log-uniform
Sigma range
[10 (10-3,10')
[10 [10-3,10*
Optimizer
AdamW
AdamW
Learning rate
5 10-1
5 10-4
Learning rate
Constant (no warmup)
Constant (no warmup)
scheduler
AdamW betas
[0.9, 0.95]
[0.9, 0.95]
AdamW eps.
10
10
Weight decay
10
10 10-2 2
EMA decay
0.9999
0.9999
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Table 2: Architecture and hyperparameters of our fa network.
Layer
Details
Output Shape
Input
{2, 1==1 R212
512
Absolute value
2y, 1,y,1
512
Scale
Multiply by learnable a E R552
512
Mas 512)
1-5
+ GELU
512
+ Dropout(O.1)
Output
Maskedd 512)
512
Final transform
log(1 x²²)
512
MaskedLineen iinear: fully connected linear layer with lower- triangular binary mask applied to its weight matrix. This enforces a causal (autoregressive) structure by ensuring that the i-th output depends only on the first inputs.
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Reviews:
[{"review_id": "4mrW5XZRWp", "reviewer": "Reviewer_hJYw", "summary": "The paper introduces a well-defined (proper) distance metric called the Information-Estimation Metric (IEM) induced by a probability distribution. Specifically, the metric is computed by comparing the denoising errors between the input signals when subject to varying levels of noise. ​It is built on the pointwise ​I-MMSE and the Tweedie-Miyasawa formulas. Inspiration is also drawn from the generative process in a diffusion model. The metric is shown to be the Mahalanobis metric when the underlying distribution is Gaussian. As an application, it is shown to correlate well with human judgment when applied to image pairs. Other interesting applications are identified for future investigation.\n\nThe paper makes a solid contribution with several potential follow-up avenues. The experiments are both illustrative and convincing, and the writing is top-notch. ​", "questions": "Could you please provide some intuition on how iso-IEM contours look like at larger scales (or for larger epsilon)?  \n\nAs a follow-up question, can you please comment on the IEM landscape in the context of image quality assessment? Specifically, how do images that lie on an iso-IEM contour or a ball with the reference at the center look like? How do they compare with, say, iso-VIF or iso-LPIPS images?\n\nCan you please point to the prior distribution assumed for the IQA experiments? I may have missed this detail. \n\nWhy does the IEM correlate well with human judgment?\n\nCan you please comment on the motivation for exploring the connection with the KL divergence? From what I understand, the IEM is defined for points drawn from the same underlying prior data distribution. The KL divergence, on the other hand, compares distributions. The translation of p_x does answer this to some extent, but some more insights could help the readers.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "​The formulation is novel and has many potential applications.\n\n​The exposition and the illustrations are excellent.", "weaknesses": "A better connection between IEM and the image quality experiment could be made.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ktxwrpbFoe", "reviewer": "Reviewer_o82F", "summary": "The authors introduce a novel distance metric called the Information-Estimation Metric (IEM). The IEM is defined based on fundamental principles connecting probability density with local geometry (i.e., connecting the estimation error to the gradient of the log of the marginal). Authors provide a rigorous theoretical framework, as well as experiments to further explain the usefulness of the proposed metric. Hence, I recommend this paper to be accepted.", "questions": "How do authors expect to solve the computational complexity issue?\n\nHave authors conducted experiments on other types of data?\n\nThe authors show that for Gaussian distributions, the IEM coincides with the Mahalanobis distance. Are there other known probability distributions where the IEM has a closed-form expression?\n\nHow does the IEM relate to other geometrically-informed distance measures like optimal transport distances (e.g., Wasserstein distance)?\n\nThe paper focuses on supervised versus unsupervised approaches. Could a semi-supervised approach yield better results with less labeled data?\n\nHow stable is the IEM to small perturbations in the underlying data distribution? Is there a way to quantify its robustness?\n\nHow might the IEM perform in domains where human perceptual ground truth is unavailable or difficult to obtain?\n\nCould the IEM be applied to other perceptual tasks like anomaly detection or out-of-distribution detection?\n\nHow does the local metric G(x) derived from the IEM compare to other local metrics derived from different principles in manifold learning?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well-written, the motivation is clear and sound. Theoretical framework is thorough, experimental section is provided. While the computational complexity can be an issue, authors still provided experimental results, which is appreciated.\n\nThe paper introduces a new distance metric that relies on the geometry of the distribution. The idea is interesting and useful. One strength is that the method is not informed by any downstream task, and is rather derived directly from the probability distribution of the data. Second, the IEM does not rely on the manifold hypothesis, and is well-defined for any valid probability density.\n\nThe IEM depends directly on the prior $p_{\\mathbf{x}}$, rather than through an observation model, i.e., the representation $p_{\\mathbf{y} | \\mathbf{x}}$, that may or may not be dependent on the prior. And, the IEM is a proper metric, from which the authors derive a local metric $\\mathbf{G}$ such that it is a one-way relationship. I.e., the IEM is not a geodesic distance.\n\nLastly, the experiments show interesting results -- the distance metric can predict human perceptual judgement.\n\nOverall, the paper presents a significant theoretical advancement in connecting information theory, estimation theory, and geometry to derive perceptually meaningful distance metric.", "weaknesses": "An obvious weakness, as the authors acknowledge in the Discussion section, is the computational complexity of the approach. Numerical estimation of an integral is expensive, and there exist other works that provide a metric but have a smaller cost. It would be great if authors could come up with a way to make this practical and usable for the community. Also, the method needs training a diffusion model, which is also expensive.\n\nThe paper provides a metric for continuous distributions, which is often not available or well-defined for all types of data.\n\nThe experiments are focused on natural images. One of the strengths that the authors mention is that the IEM does not rely on the manifold hypothesis. While I understand that the theory of IEM does not rely on the manifold hypothesis, it strikes me that natural images \\textit{do} lie on a manifold. It would make the paper stronger to see experiments on data that do not necessarily lie on a manifold (e.g., text).\n\nDespite these limitations, the paper presents a significant theoretical advancement in connecting information theory, estimation theory, and geometry to derive perceptually meaningful distance metrics.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ysv6bbKdWy", "reviewer": "Reviewer_CAxo", "summary": "This paper proposes the IEM and primarily validates it on image quality assessment tasks. The core idea involves measuring distribution shifts, and the method is not inherently limited to visual data. The experimental results demonstrate its performance on specific datasets like TQD.", "questions": "1.  **Generalizability and Scope:** The paper's motivation seems broader than IQA. Have you considered, or could you perform, validation on non-visual modalities (e.g., audio, text) to truly demonstrate IEM's general applicability for measuring distribution shifts?\n2.  **Theoretical Interpretation and Measurement:** A key motivation relates to sensitivity varying with the *direction* of perturbation. In high-dimensional spaces (e.g., latent spaces), how can the \"direction\" of distributional change be effectively measured? Furthermore, how might the similarity or difference between these \"directions\" of change impact IEM's performance? Is IEM inherently sensitive to this directional information?\n3.  **Application to Generative Models:** Could IEM be applied to quantify the difference between generated and real images, thus serving as a metric for evaluating the performance of generative models?\n4.  **Comparative Advantage:** How does IEM's performance compare directly to other information-theoretic measures like KLD, JSD, and EMD on your tasks? This comparison is vital for establishing IEM's value.\n5.  **Explanation of Results and Definitions:** What explains the contradictory performance of `IEM_sq` (good on TQD, poor on photo quality assessment)? Can you provide a theoretical hypothesis or detailed experimental analysis?\n6.  **Robustness to Parameters:** The performance is highly sensitive to `Γ`. What is the underlying reason for this sensitivity? Are there strategies to make the method more robust to the choice of `Γ`, or to determine an optimal `Γ` in a more principled, task-agnostic way?\n7.  **Sensitivity to Denoiser Choice:** Is the performance of the proposed method sensitive to the selection of the specific denoiser used?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "*   **General Formulation:** The proposed IEM is not fundamentally tied to visual data, suggesting potential applicability to other media types like audio or natural language, as it makes no specific assumptions about the modality.\n*   **Interesting Motivation:** The motivation based on \"humans exhibit complex patterns of sensitivity that vary with the direction of the signal’s perturbation\" is compelling and aligns with nuanced aspects of perception.", "weaknesses": "*   **Insufficient Experimental Validation:**\n    *   The paper's core motivation and problem formulation do not appear exclusively focused on Image Quality Assessment (IQA), yet validation is conducted *only* within IQA tasks. This is insufficient. If the goal is a general measure of distribution shift, experiments on other modalities (e.g., sound, text) are necessary to substantiate the claim of generality.\n    *   Within the IQA validation, there is a lack of experiments on real-world images with complex, mixed distortions. This is crucial for verifying the method's ability to assess information changes caused by complicated, realistic impairments.\n*   **Lack of Comparative Analysis:** A critical weakness is the absence of a comparative analysis between IEM and other established information-theoretic distance measures like Kullback-Leibler Divergence (KLD), Jensen-Shannon Divergence (JSD), and Earth Mover's Distance (EMD). Such a comparison is essential for demonstrating IEM's potential superiority or unique advantages.\n*   **Inadequate Explanations:**\n    *   The significant performance discrepancy of `IEM_sq` (poor on photo quality assessment but good on TQD) lacks a thorough explanation from either a theoretical or experimental analysis perspective.\n    *   The notation `IEM_fw` is introduced without a clear explanation.\n*   **Parameter Sensitivity:** The IEM demonstrates high sensitivity to the parameter `Γ` (Gamma), with the \"optimal\" value varying drastically (e.g., from `1/4` to `10^6`) between different IQA tasks (photographic vs. texture images). This instability is a practical concern that needs addressing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xWGGYNCmKT", "reviewer": "Reviewer_dR9Y", "summary": "This paper innovatively proposes the Information-Estimation Metric (IEM), a distance measure derived from the local geometric properties of probability distributions (denoising errors and score vector fields). It breaks through the limitation that traditional supervised perceptual metrics rely on manual annotations, validates its unsupervised learning capability on the ImageNet dataset, and outperforms most existing methods in image quality assessment tasks. The overall theoretical derivation is rigorous and the experimental design is comprehensive.", "questions": "More experiments on different datasets should be more convincing.\nCould you provide more evidence of the demand for IEM?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.Solid Mathematical Foundation. The paper reliably demonstrates the mathematical significance of its work.\n2.Comprehensive Experimental Validation and Credible Conclusions. The paper verifies the advantages of IEM over other metrics through experiments and tests.", "weaknesses": "1. The drawback of low computational efficiency of the IEM. \n2.The experimental scope is relatively limited, with insufficient validation of the IEM’s performance across a wide range of noise types.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)