On the Capacity of Distinguishable Synthetic Identity Generation under Face Verification

Abstract

Synthetic face generators can produce many nominal identities, but nominal count does not determine how many are jointly distinguishable under a specified verification rule. We define finite-dimensional capacity as the supremum of codebook sizes over distinct latent identity codes whose induced identity-conditional embedding distributions satisfy per-identity genuine acceptance and pairwise impostor non-match constraints. For deterministic view-invariant pipelines, fixed-code capacity equals the spherical-code cardinality over the realizable embedding set and reduces to the classical spherical-code cardinality when every sphere direction is realizable. For stochastic identity-conditional embedding distributions concentrated with probability at least $1-η$ in spherical caps of angular radius $ρ$, we derive a sufficient center-separation condition, spherical-code capacity lower bounds under full angular expressivity, and positive asymptotic lower-bound exponents for dimension-indexed pipeline families. We also derive prior-constrained random-code lower bounds from pairwise center-separation failure probabilities. When each identity-conditional embedding distribution has support equal to a spherical cap of angular radius $ρ$, we derive necessary zero-error geometric conditions and, for $2ρ<\arccos(τ)$ under full $ρ$-cap angular expressivity, show that the restricted zero-error capacity equals the classical spherical-code cardinality at minimum angle $\arccos(τ)+2ρ$. For finite repeated-view samples, a maximum clique in the resulting compatibility graph identifies the largest sampled subset satisfying all empirical genuine and pairwise impostor constraints. We evaluate this sample-restricted quantity on a deterministically selected DigiFace-1M subset under three fixed recognizers with identity-disjoint in-domain threshold calibration.