On the Capacity of Distinguishable Synthetic Identity Generation under Face Verification

Behrooz Razeghi

arxiv: 2604.10641 · v1 · submitted 2026-04-12 · 💻 cs.IT · cs.IR· math.IT· math.PR· stat.AP

On the Capacity of Distinguishable Synthetic Identity Generation under Face Verification

Behrooz Razeghi This is my paper

Pith reviewed 2026-05-10 15:56 UTC · model grok-4.3

classification 💻 cs.IT cs.IRmath.ITmath.PRstat.AP

keywords synthetic identity generationface verificationcapacityspherical codesembedding distributionsadmissibility conditioninformation theoretic bounds

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The pith

The maximum number of synthetic identities distinguishable by a face verifier at fixed threshold τ is given by a spherical code problem on the unit hypersphere.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the capacity of distinguishable synthetic identity generation as the largest set of latent identities whose induced embedding distributions meet prescribed same-identity match and different-identity non-match rules under a fixed verification threshold τ. In the deterministic view-invariant regime this capacity equals the size of the largest spherical code packable in the realizable embedding set, and equals the classical spherical-code quantity once full angular expressivity is assumed. For stochastic generation the authors introduce a centered model and give a sufficient admissibility condition requiring identity centers to be separated by arccos(τ) + 2ρ, where ρ is the within-identity concentration radius; this yields achievable lower bounds and a positive exponential growth rate with embedding dimension.

Core claim

The capacity is the largest number of latent identities whose embedding distributions satisfy the verification constraints at threshold τ. In the deterministic case this capacity is characterized by a spherical-code problem over the realizable embedding set and reduces to the classical spherical-code quantity under full angular expressivity. For stochastic generation under a centered model with concentration radius ρ, a sufficient condition is that identity centers are separated by at least arccos(τ) + 2ρ; under full angular expressivity this produces spherical-code-based achievable rates and a positive asymptotic lower bound on the exponential growth rate. A prior-constrained random-code 0.

What carries the argument

The spherical-code characterization of capacity in embedding space, which counts the largest set of points whose minimum angular separation satisfies the same-identity and different-identity verification rules at threshold τ.

If this is right

The number of usable synthetic identities grows exponentially with embedding dimension at a rate given by the spherical code exponent.
Generators must enforce a minimum center separation of arccos(τ) + 2ρ to guarantee that different-identity pairs are rejected.
Prior-constrained random sampling of identities yields high-probability lower bounds based on pairwise separation failure probabilities.
Under a stronger full-cap-support model an exact spherical-code characterization of capacity holds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same packing view could be applied to evaluate how many synthetic samples can be added to training sets without creating duplicates that a verifier would treat as matches.
If real generators produce non-centered or non-concentrated embeddings, the actual number of distinguishable identities would be smaller than the derived bounds.
The framework connects to classical packing problems in other high-dimensional biometric spaces such as voice or gait embeddings.

Load-bearing premise

The assumption that embeddings achieve full angular expressivity on the sphere, together with the centered stochastic model and fixed within-identity radius ρ.

What would settle it

Generate synthetic identities at scale and count whether the largest set satisfying the match/non-match conditions at threshold τ exceeds the size of the largest spherical code with minimum angular distance arccos(τ) in the same embedding dimension.

read the original abstract

We study how many synthetic identities can be generated so that a face verifier declares same-identity pairs as matches and different-identity pairs as non-matches at a fixed threshold $\tau$. We formalize this question for a generative face-recognition pipeline consisting of a generator followed by a normalized recognition map with outputs on the unit hypersphere. We define the capacity of distinguishable identity generation as the largest number of latent identities whose induced embedding distributions satisfy prescribed same-identity and different-identity verification constraints. In the deterministic view-invariant regime, we show that this capacity is characterized by a spherical-code problem over the realizable set of embeddings, and reduces to the classical spherical-code quantity under a full angular expressivity assumption. For stochastic identity generation, we introduce a centered model and derive a sufficient admissibility condition in which the required separation between identity centers is $\arccos(\tau)+2\rho$, where $\rho$ is a within-identity concentration radius. Under full angular expressivity, this yields spherical-code-based achievable lower bounds and a positive asymptotic lower bound on the exponential growth rate with embedding dimension. We also introduce a prior-constrained random-code capacity, in which latent identities are sampled independently from a given prior, and derive high-probability lower bounds in terms of pairwise separation-failure probabilities of the induced identity centers. Under a stronger full-cap-support model, we obtain a converse and an exact spherical-code characterization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper maps synthetic identity capacity under face verification to spherical codes with some clean derivations, but the reductions rest on strong unverified assumptions about embedding coverage.

read the letter

The main thing to know is that this work defines a capacity for the largest number of synthetic identities you can generate while satisfying fixed-threshold face verification constraints, then characterizes it via spherical coding problems on the unit sphere for both deterministic and stochastic generators. It sets up a pipeline model with a generator followed by a normalized embedding map and derives sufficient conditions plus some asymptotic growth bounds. That formalization is the core new piece, and it handles the deterministic view-invariant case by reducing to a packing problem over realizable embeddings, plus a stochastic centered model that gives a center-separation rule of arccos(τ) + 2ρ. The prior-constrained random-code extension and the converse under full support are also straightforward additions. The derivations look internally consistent and draw on standard coding arguments without obvious algebraic slips. The paper does a decent job keeping the models explicit and separating the cases clearly. The soft spots sit mostly in the assumptions. The reduction to classical spherical-code quantities requires full angular expressivity over the realizable set, but nothing in the work checks whether real generators plus normalized maps actually produce dense enough coverage or whether the embeddings stay in a lower-dimensional subset. The centered stochastic model with fixed concentration radius ρ is another modeling choice that may not match empirical generator outputs, so the separation condition and lower bounds become sufficient rather than tight. There are no numerical validations or comparisons to existing generators, which leaves the practical reach unclear. This is for information theorists or biometric-security researchers who already know spherical codes and want to see them applied to verification constraints. A reader comfortable with coding theory on spheres could extract the framework and try relaxing the assumptions themselves. It has enough structure and honest engagement with the literature to deserve peer review, though the referees will need to press on whether the assumptions can be justified or weakened.

Referee Report

3 major / 2 minor

Summary. The paper formalizes the capacity of distinguishable synthetic identity generation for face verification at fixed threshold τ in a pipeline consisting of a generator followed by a normalized embedding map on the unit hypersphere. Capacity is defined as the largest number of latent identities whose induced embedding distributions satisfy the same-identity match and different-identity non-match constraints. In the deterministic view-invariant regime, this capacity is characterized as a spherical-code problem over the realizable set of embeddings and reduces to the classical spherical-code quantity under a full angular expressivity assumption. For stochastic generation, a centered model yields a sufficient admissibility condition requiring identity-center separation arccos(τ)+2ρ (with ρ the within-identity concentration radius), producing spherical-code-based achievable lower bounds and a positive asymptotic lower bound on the exponential growth rate with embedding dimension. The paper also analyzes a prior-constrained random-code capacity with high-probability lower bounds and, under a stronger full-cap-support model, a converse with exact spherical-code characterization.

Significance. If the stated assumptions hold, the work supplies a clean information-theoretic reduction of a practical biometric generation problem to spherical coding, with explicit sufficient conditions and asymptotic growth-rate results. The derivation of positive lower bounds on exponential growth with dimension and the treatment of both deterministic and stochastic regimes (including random-code variants) are strengths; the explicit statement of the full-angular-expressivity and centered-model assumptions allows readers to assess applicability directly.

major comments (3)

[deterministic view-invariant regime characterization] Deterministic regime (as stated in the abstract and the characterization paragraph): the reduction of capacity to the classical spherical-code quantity is load-bearing and rests entirely on the full angular expressivity assumption over the realizable embedding set. The manuscript provides no quantitative bound on how densely the generator-plus-normalized-map composition fills the sphere, nor any deviation analysis when the realizable set is a proper subset; without this, the claimed exact characterization does not hold and the capacity is instead the (unknown) maximum packing number inside that subset.
[stochastic identity generation section] Stochastic identity generation (sufficient admissibility condition): the separation requirement arccos(τ)+2ρ is derived under the centered model whose support lies inside spherical caps of fixed radius ρ. The paper does not analyze robustness when within-identity distributions deviate from this centered-cap geometry (e.g., multimodal or anisotropic concentrations), which would invalidate the sufficient condition and the subsequent lower bounds on achievable rate.
[prior-constrained random-code capacity] Prior-constrained random-code capacity: the high-probability lower bounds are expressed in terms of pairwise separation-failure probabilities, yet the manuscript supplies neither concrete priors nor numerical evaluation of those probabilities for typical face-embedding generators, leaving the practical tightness of the bounds unassessed.

minor comments (2)

The term 'full angular expressivity' should be given a precise mathematical definition (e.g., a statement that the image of the generator-plus-normalized-map is dense in the sphere or satisfies a covering-radius bound) rather than left as an informal assumption.
Notation for the within-identity concentration radius ρ and the verification threshold τ should be introduced once with explicit units (angles in radians) and reused consistently in all subsequent equations and statements.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating the revisions we will incorporate.

read point-by-point responses

Referee: [deterministic view-invariant regime characterization] Deterministic regime (as stated in the abstract and the characterization paragraph): the reduction of capacity to the classical spherical-code quantity is load-bearing and rests entirely on the full angular expressivity assumption over the realizable embedding set. The manuscript provides no quantitative bound on how densely the generator-plus-normalized-map composition fills the sphere, nor any deviation analysis when the realizable set is a proper subset; without this, the claimed exact characterization does not hold and the capacity is instead the (unknown) maximum packing number inside that subset.

Authors: We agree that the exact reduction to the classical spherical-code quantity holds only under the full angular expressivity assumption, which is explicitly stated in the manuscript. In the absence of this assumption, the capacity is the maximum packing number over the realizable embedding set, as already characterized in the deterministic regime section. We will revise the abstract and the relevant characterization paragraph to emphasize this distinction more prominently and add a clarifying remark that quantitative density bounds on the realizable set would require generator-specific analysis outside the current information-theoretic scope. revision: yes
Referee: [stochastic identity generation section] Stochastic identity generation (sufficient admissibility condition): the separation requirement arccos(τ)+2ρ is derived under the centered model whose support lies inside spherical caps of fixed radius ρ. The paper does not analyze robustness when within-identity distributions deviate from this centered-cap geometry (e.g., multimodal or anisotropic concentrations), which would invalidate the sufficient condition and the subsequent lower bounds on achievable rate.

Authors: The separation requirement arccos(τ)+2ρ and the resulting lower bounds are derived as a sufficient condition under the centered model with fixed-radius spherical-cap support. This model is introduced to obtain tractable admissibility and rate results. We acknowledge that deviations such as multimodal or anisotropic distributions could invalidate the condition. We will add an explicit discussion in the stochastic generation section noting that the condition is sufficient rather than necessary and identifying robustness to non-centered geometries as an open direction for future work. revision: partial
Referee: [prior-constrained random-code capacity] Prior-constrained random-code capacity: the high-probability lower bounds are expressed in terms of pairwise separation-failure probabilities, yet the manuscript supplies neither concrete priors nor numerical evaluation of those probabilities for typical face-embedding generators, leaving the practical tightness of the bounds unassessed.

Authors: The prior-constrained random-code capacity is formulated for an arbitrary prior, with the high-probability bounds expressed directly in terms of the induced pairwise separation-failure probabilities. This generality permits evaluation for any chosen prior or generator. We will add a short illustrative example in the revised manuscript using a simple isotropic Gaussian prior on the sphere to demonstrate numerical evaluation of the probabilities. A comprehensive assessment for state-of-the-art face-embedding generators would require additional empirical computation and is noted as future work. revision: partial

Circularity Check

0 steps flagged

No circularity: capacity mapped to spherical-code problems under explicit assumptions with no self-referential reductions or fitted predictions

full rationale

The paper defines capacity as the largest number of latent identities whose induced embeddings satisfy verification constraints at threshold τ. It then shows this is characterized by a spherical-code problem over the realizable embedding set, reducing to the classical quantity only under the stated 'full angular expressivity assumption.' For stochastic generation a centered model is introduced by definition, yielding the sufficient separation arccos(τ)+2ρ. All bounds and converses follow from these models and the assumption; no parameter is fitted to data and renamed as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled via prior work. The derivation therefore remains self-contained against external spherical-coding benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The claims rest on modeling choices for the recognition map and embedding distributions rather than on new physical entities or heavily fitted constants.

free parameters (2)

verification threshold τ
Fixed input parameter that defines the match/non-match boundary; not fitted to data in the paper.
within-identity concentration radius ρ
Model parameter controlling stochastic variation around each identity center; introduced to derive the separation condition.

axioms (2)

domain assumption full angular expressivity assumption
Allows the realizable embedding set to be treated as the full sphere, reducing the capacity to the classical spherical-code quantity.
domain assumption centered stochastic model for identity generation
Assumes each identity's embeddings are concentrated around a center with radius ρ.

pith-pipeline@v0.9.0 · 5550 in / 1473 out tokens · 59355 ms · 2026-05-10T15:56:58.622736+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 1 internal anchor

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Synface: Face recognition with synthetic data,

H. Qiu, B. Yu, D. Gong, Z. Li, W. Liu, and D. Tao, “Synface: Face recognition with synthetic data,” in Proceedings of the IEEE/CVF international conference on computer vision, 2021, pp. 10880–10890

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Z. Sun, S. Song, I. Patras, and G. Tzimiropoulos, “Cemiface: Center-based semi-hard synthetic face generation for face recognition,”Advances in Neural Information Processing Systems, vol. 37, pp. 35612– 35638, 2024

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Schaeffer, V

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B. R. Gálvez, A. Blaas, P. Rodríguez, A. Golinski, X. Suau, J. Ramapuram, D. Busbridge, and L. Zappella, “The role of entropy and reconstruction in multi-view self-supervised learning,” inInternational conference on machine learning. PMLR, 2023, pp. 29143–29160

work page 2023

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Representation Learning with Contrastive Predictive Coding

A. v. d. Oord, Y. Li, and O. Vinyals, “Representation learning with contrastive predictive coding,”arXiv preprint arXiv:1807.03748, 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018

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Learning representations by maximizing mutual information across views,

P. Bachman, R. D. Hjelm, and W. Buchwalter, “Learning representations by maximizing mutual information across views,”Advances in neural information processing systems, vol. 32, 2019

work page 2019

[16] [16]

Understanding contrastive representation learning through alignment and uniformity on the hypersphere,

T. Wang and P. Isola, “Understanding contrastive representation learning through alignment and uniformity on the hypersphere,” inInternational conference on machine learning. PMLR, 2020, pp. 9929–9939

work page 2020

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arXiv preprint arXiv:2005.13149(2020)

M. Wu, C. Zhuang, M. Mosse, D. Yamins, and N. Goodman, “On mutual information in contrastive learning for visual representations,”arXiv preprint arXiv:2005.13149, 2020

work page arXiv 2005

[18] [18]

An information theory perspective on variance-invariance-covariance regularization,

R. Shwartz-Ziv, R. Balestriero, K. Kawaguchi, T. G. Rudner, and Y. LeCun, “An information theory perspective on variance-invariance-covariance regularization,”Advances in neural information processing systems, vol. 36, pp. 33965–33998, 2023

work page 2023

[19] [19]

On the biometric capacity of generative face models,

V. N. Boddeti, G. Sreekumar, and A. Ross, “On the biometric capacity of generative face models,”arXiv preprint arXiv:2308.02065, 2023

work page arXiv 2023

[20] [20]

The closest packing of spherical caps in n dimensions,

R. A. Rankin, “The closest packing of spherical caps in n dimensions,”Glasgow Mathematical Journal, vol. 2, no. 3, pp. 139–144, 1955

work page 1955

[21] [21]

J. H. Conway and N. J. A. Sloane,Sphere packings, lattices and groups. Springer Science & Business Media, 2013, vol. 290

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On bounds for packings on a sphere and in space,

G. A. Kabatiansky and V. I. Levenshtein, “On bounds for packings on a sphere and in space,”Problemy peredachi informatsii, vol. 14, no. 1, pp. 3–25, 1978. 19 arXiv preprint A Detailed Proofs of the Main Results A.1 Proof of Proposition 5 Proof.Suppose that Pr h {PCi}M i=1 is(τ, ε in, εout)-admissible i ≥1−δ.(56) Since δ < 1, this probability is strictly p...

work page 1978