Computable Distortion Index

• Computable Distortion Index is a measure that maps structured inputs to scalar values or curves to quantify distortion, used in fields like rate–distortion theory and dimensionality reduction.
• It distinguishes between optimal value computability and the non-computability of the corresponding optimizer, ensuring reproducible performance metrics even with finite procedures.
• Applications span information theory, model-extraction detection in MLaaS, class geometry preservation in dimensionality reduction, and speech intelligibility prediction via auditory-envelope analysis.

Searching arXiv for the cited papers and closely related work on computable distortion indices. A computable distortion index is a distortion-related quantity that can be evaluated algorithmically from a formal specification of a source, representation, estimator, or signal pair. In the cited literature, the term is not confined to a single construction. It appears as the rate–distortion value $R(D)$ or its finite-blocklength feed-forward analogue $R_n(D)$, as a distortion–perception tradeoff function $D(P)$, as the Class Angular Distortion Index (CADI) for dimensionality reduction, as the Feature Distortion Index (FDI) for model-extraction detection, and as the Gammachirp Envelope Distortion Index (GEDI) for speech intelligibility prediction. A recurring distinction is that scalar performance values can be computable even when the exact optimizing object that attains them is not (Huffmann et al., 12 Nov 2025).

1. Conceptual scope and recurring forms

In the cited work, a computable distortion index is always an explicit map from structured inputs to either a scalar or a curve. The inputs vary by domain: a source law and distortion matrix in rate–distortion theory, a joint law $p_{X,Y}$ and a perceptual metric in distortion–perception analysis, a labeled high-dimensional dataset and its embedding in dimensionality reduction, a query and anchor features in MLaaS security, or a clean/enhanced speech pair in auditory modeling. What is common is that the index is specified by an algorithmic pipeline and is intended to be reproducible from the available mathematical or signal representation.

Domain	Index	Computable object
Rate–distortion theory	$R(D)$	Minimum achievable rate at distortion $D$
Feed-forward source coding	$R_n(D)$	Finite-$n$ directed-information rate
Distortion–perception theory	$D(P)$	Minimum distortion under perception constraint
Dimensionality reduction	CADI	Mean squared cosine-angle distortion over between-class triplets
MLaaS security	FDI	Layerwise feature-distance vector to class anchors
Speech intelligibility	GEDI / mr-GEDI	Envelope-domain signal-to-distortion ratio

This range of usages indicates that computability can refer either to a value function produced by optimization or to a directly evaluated residual/error functional. In some settings the output is a single scalar, as with CADI or GEDI. In others it is a parametric tradeoff curve, as with $R(D)$, $R_n(D)$0, or $R_n(D)$1. The cited papers also show that computability is domain-specific: in finite-alphabet information theory it is formulated through computable analysis, while in applied settings it typically means that the index can be evaluated with explicit finite procedures, sampling schemes, or convex programs (Freirich et al., 2021).

2. Information-theoretic computability and the optimizer gap

For a discrete memoryless source over finite alphabets, with computable source law $R_n(D)$2, computable single-letter distortion $R_n(D)$3, and computable distortion level $R_n(D)$4, the rate–distortion function is

$R_n(D)$5

The corresponding optimizing test channel is any

$R_n(D)$6

The paper "Computability of the Optimizer for Rate Distortion Functions" establishes a precise asymmetry: $R_n(D)$7 is computable, but the optimizer is in general non-computable, even for simple finite distortion measures including Hamming-type settings (Huffmann et al., 12 Nov 2025).

The computability framework used there is computable analysis in the sense of Pour-El and Richards and Weihrauch. A computable real is represented by an effectively convergent rational sequence; Banach–Mazur computability and uniform effective continuity give the stronger function notion on compact domains. Within that framework, a classic minimum-versus-argmin dichotomy is recalled: the minimum of a computable function on a compact set is computable, but an argmin need not be. The rate–distortion problem inherits exactly that pattern. Mutual information is continuous and computable in $R_n(D)$8 and $R_n(D)$9, the feasible set $D(P)$0 is compact and effectively specified, and therefore the value $D(P)$1 is computable. By contrast, there is no universal Banach–Mazur, hence no universal Turing-computable, selector

$D(P)$2

for all computable instances with finite alphabets $D(P)$3, $D(P)$4, and nontrivial $D(P)$5 (Huffmann et al., 12 Nov 2025).

The negative result is strengthened in two directions. First, even if $D(P)$6 is fixed and only the distortion measure varies, a computable optimizer map need not exist. Second, even if the distortion measure is fixed—including binary weighted Hamming distortion and more general normal distortion measures—there is no general computable map from $D(P)$7 to an exact optimal test channel. The constructions encode a recursively enumerable but non-recursive set into the structure of the optimizer, using explicit distortion matrices and source perturbations. The paper also states that there is not even a uniformly computable approximator that recovers the optimizer to arbitrary precision in total variation distance (Huffmann et al., 12 Nov 2025).

A related but operationally distinct construction appears in source coding with feed-forward. For stationary, ergodic, discrete-time sources, the finite-blocklength quantity

$D(P)$8

is achievable for any $D(P)$9, and $p_{X,Y}$0 is the feed-forward rate–distortion function (Naiss et al., 2011). Here the distortion index is not the optimizer but the sequence of computable rates themselves. The paper provides an alternating minimization procedure, derived as an extended Blahut–Arimoto algorithm, and also a dual formulation that becomes a geometric programming problem. This yields a sequence of achievable and computable rates converging to the operational limit (Naiss et al., 2011).

Taken together, these results separate value computability from structure computability. This suggests that, in information-theoretic usage, a computable distortion index is safest when defined through optimal values such as $p_{X,Y}$1, $p_{X,Y}$2, or directly computable thresholds such as

$p_{X,Y}$3

rather than through exact optimal test channels or induced codebook distributions (Huffmann et al., 12 Nov 2025).

3. Distortion–perception functions as computable indices

A second major line of work defines a computable distortion index as a distortion–perception tradeoff function. For random vectors $p_{X,Y}$4 and $p_{X,Y}$5, with estimator $p_{X,Y}$6 satisfying the Markov condition $p_{X,Y}$7, the general distortion–perception function is

$p_{X,Y}$8

In the MSE–Wasserstein-2 setting, with

$p_{X,Y}$9

the paper "A Theory of the Distortion-Perception Tradeoff in Wasserstein Space" proves that the DP function is always quadratic: $R(D)$0 where $R(D)$1 and $R(D)$2 with $R(D)$3 (Freirich et al., 2021).

That result is geometric rather than merely variational. The minimizers on the DP curve form a Wasserstein-2 geodesic between the distribution of the signal and that of the minimum-MSE estimator. If $R(D)$4 is an optimal perfect-perception estimator satisfying $R(D)$5, then every optimal estimator for $R(D)$6 can be written as

$R(D)$7

In the Gaussian case, the paper gives closed-form optimal estimators and a computable lower bound via the Gelbrich distance

$R(D)$8

with equality for jointly Gaussian $R(D)$9 (Freirich et al., 2021).

For finite-alphabet channels, the same idea becomes a linear-programming object. The paper "Characterization of the Distortion-Perception Tradeoff for Finite Channels with Arbitrary Metrics" studies arbitrary distortion matrices and Wasserstein-$D$0 perception induced by an arbitrary metric on a finite source alphabet. In that setting, computing both the DP function and optimal reconstructions is equivalent to solving linear programs. The distortion criterion is affine in the stochastic decoder $D$1, and the Wasserstein-$D$2 perception constraint is encoded through an explicit coupling $D$3. The resulting DP function is non-increasing, convex, and piecewise linear in the perception index $D$4 (Freirich et al., 2024).

The finite-alphabet paper also derives a dual characterization in which

$D$5

so the curve is the upper envelope of finitely many affine functions. In the binary case, it provides closed-form breakpoints, slopes, and optimal deterministic threshold reconstructions, with randomized interpolation between adjacent breakpoints. This makes the distortion index fully computable in the sense of explicit LP evaluation or explicit closed form, depending on the alphabet size (Freirich et al., 2024).

These two DP frameworks reveal two distinct computable geometries. In Wasserstein-2 with MSE, the tradeoff is quadratic and geodesic. In finite alphabets with Wasserstein-1 and arbitrary metrics, it is polyhedral and piecewise linear. In both cases, the index is the optimal value function, not a heuristic proxy.

4. Angular distortion and class geometry in dimensionality reduction

In dimensionality reduction, the computable distortion index can be defined directly on the geometry of labeled point sets rather than through an optimization over encoders or decoders. The Class Angular Distortion Index (CADI) is defined for high-dimensional data

$D$6

a low-dimensional embedding

$D$7

and a class partition $D$8. For between-class triplets with reference point $D$9 and unordered pair $R_n(D)$0, CADI is

$R_n(D)$1

The cosines are computed from inner products and norms of the corresponding edge vectors in the original and embedded spaces (Gunaratne et al., 1 May 2026).

CADI measures distortion at the level of class-to-class geometry. Because it uses between-class triplets, it is sensitive to how one class is seen from another: nested shells, rings, interlinked manifolds, anisotropic spreads, and non-globular cluster shapes. The paper positions it against metrics such as Silhouette Score, Davies–Bouldin Index, NMI, ARI, centroid-based Cluster Distance Score, Label Trustworthiness/Continuity, and Steadiness & Cohesiveness. Its stated role is not separability alone, but the preservation of inter-class manifold geometry (Gunaratne et al., 1 May 2026).

The naïve computation is at least $R_n(D)$2, since the number of between-class triplets is on that order. To make the index computably practical, the paper uniformly samples

$R_n(D)$3

between-class triplets. It reports that $R_n(D)$4 already gives stable estimates to approximately three decimal digits, and uses $R_n(D)$5 in its evaluations. With fixed dimensionalities, this makes runtime linear in $R_n(D)$6, and the paper reports evaluation times around a second for its largest datasets on a consumer laptop (Gunaratne et al., 1 May 2026).

CADI is also differentiable because each triplet loss

$R_n(D)$7

depends smoothly on dot products and norms of the embedded coordinates, except at degenerate zero-norm cases, which are handled explicitly. This enables AngleEmbedding, a supervised dimensionality reduction method that minimizes CADI with a feedforward MLP and Adam at learning rate $R_n(D)$8, using sampled triplets during training (Gunaratne et al., 1 May 2026).

Several limitations are explicit. CADI is not a separability metric; it can assign low distortion to embeddings with overlap if the class geometry is preserved. It requires class labels or a clustering. Larger classes contribute more triplets by design, because the normalization is by sampled triplet count rather than by class size. It is invariant to translation, rotation, and uniform scaling, but sensitive to non-uniform feature scaling and nonlinear transforms (Gunaratne et al., 1 May 2026).

5. Feature-space distortion indices in MLaaS security

In MLaaS model-extraction defense, the Feature Distortion Index (FDI) instantiates a computable distortion index as a layerwise deviation between the internal feature representation of an incoming query and anchor features derived from the victim model’s training set. The setting is a private DNN $R_n(D)$9 deployed behind a prediction API. For each class $n$0, the defender selects $n$1 high-confidence anchor samples from the training set and precomputes their features at $n$2 layers. For a query $n$3, the victim’s predicted class $n$4 determines which anchor set is used (Yao et al., 2023).

The per-layer, per-anchor index is

$n$5

with $n$6 chosen as the $n$7 norm, so concretely

$n$8

The full FDI vector is the concatenation over all anchors $n$9 and considered layers $D(P)$0: $D(P)$1 For VGG19 on CIFAR-10, the paper uses $D(P)$2 and $D(P)$3, giving a 500-dimensional FDI vector (Yao et al., 2023).

This index is not a divergence between explicitly estimated continuous distributions. Instead, the distributional claim is operationalized through distances to class-specific anchor features. The anchor set approximates the center of each class cluster in feature space, and the FDI vector records how much an incoming query deviates from that benign feature distribution. FDINet uses these vectors to train a binary detector with architecture Linear$D(P)$4, Linear$D(P)$5, Linear$D(P)$6, followed by Softmax, optimized by cross-entropy on benign training data and an auxiliary out-of-distribution positive set (Yao et al., 2023).

Detection is performed over batches rather than single queries. For a batch of size $D(P)$7, each FDI vector is passed through the classifier, converted to a binary malicious/benign decision by $D(P)$8, and averaged: $D(P)$9 A threshold $R(D)$0, estimated from benign validation batches, is used to flag malicious clients. The same FDI representation is also used for colluding-adversary detection via two-sample $R(D)$1-tests on FDI statistics across clients, with a significance threshold $R(D)$2 such as $R(D)$3 (Yao et al., 2023).

The empirical claims are specific. FDINet achieves 100% detection accuracy on DFME and DaST, uses 50 queries to raise an extraction alarm with an average confidence of 96.08% for GTSRB, identifies colluding adversaries with accuracy exceeding 91%, and detects two adaptive attacks, Dummy Query and Feature Correction, though with some degradation on attacks whose queries are closer to benign data such as Knockoff and ActiveThief (Yao et al., 2023).

As a distortion index, FDI is fully computable online because anchor features are precomputed and the per-query computation reduces to forward feature extraction and $R(D)$4 Euclidean distances. Its limitations are also domain-specific: it depends on label-conditioned anchors, it was evaluated in computer vision rather than in other modalities, and similarity-based colluding detection is weaker when distinct attacks induce very similar feature distributions (Yao et al., 2023).

6. Auditory-envelope distortion indices and cross-domain synthesis

GEDI and mr-GEDI define a computable distortion index for speech intelligibility prediction by operating in the temporal envelope after auditory and modulation filtering. The pipeline begins with a dynamic compressive gammachirp filterbank (dcGC-FB) with $R(D)$5 channels, equally spaced in ERB$R(D)$6-number from 100 to 6000 Hz. For each auditory channel, the clean and enhanced signals are transformed into Hilbert envelopes and low-pass filtered at 150 Hz. Envelope distortion is then defined, with $R(D)$7, as

$R(D)$8

This produces an explicit distortion signal in the auditory-envelope domain (Yamamoto et al., 2019).

GEDI then analyzes modulation power. For clean and distortion envelopes, modulation-filter outputs are summarized as powers $R(D)$9 and $R_n(D)$00, with an ERB$R_n(D)$01-based weighting

$R_n(D)$02

For each modulation channel $R_n(D)$03, the weighted envelope-domain signal-to-distortion ratio is

$R_n(D)$04

and the global GEDI core quantity is

$R_n(D)$05

This scalar is then mapped, via an ideal-observer model, to predicted intelligibility (Yamamoto et al., 2019).

Mr-GEDI extends the same construction to non-stationary noise by using multi-resolution temporal analysis. Instead of a single FFT-based modulation analysis, it employs one third-order low-pass and eight second-order band-pass IIR modulation filters, with frame lengths inversely matched to modulation center frequency. It computes framewise powers and framewise

$R_n(D)$06

averages them over frames, and then pools across modulation bands. The paper reports that mr-GEDI predicts intelligibility curves better than STOI, ESTOI, and HASPI under pink-noise conditions, and better than HASPI under babble-noise conditions. It further states that mr-GEDI does not present an overestimation tendency and is considered a more conservative approach than STOI and ESTOI (Yamamoto et al., 2019).

Across the cited literature, several misconceptions are explicitly corrected. First, computable does not imply that the exact optimizer is computable; in finite-alphabet rate–distortion theory, the opposite is shown for $R_n(D)$07 (Huffmann et al., 12 Nov 2025). Second, a low-distortion geometric index need not mean strong class separability; CADI is designed to preserve class geometry, not to optimize blob-like separation (Gunaratne et al., 1 May 2026). Third, an index can be practically computable without being a universal exact solver; the feed-forward $R_n(D)$08 sequence, LP-based finite-channel DP functions, sampled CADI, and FDI all rely on explicit finite procedures rather than inaccessible analytic formulas (Naiss et al., 2011).

A plausible synthesis is that computable distortion indices fall into two broad categories. The first consists of optimal-value functions, such as $R_n(D)$09, $R_n(D)$10, and $R_n(D)$11, where computability attaches to the optimum value of a convex or linear program. The second consists of direct residual measures, such as CADI, FDI, and GEDI, where computability attaches to an explicitly evaluated discrepancy in a chosen representation. The cited papers jointly show that the design choice is consequential: computable values are often robustly definable, whereas exact optimizing structures may be non-computable, non-unique, or only practically approximable.