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 $τ$. 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 $\arccos(τ)+2ρ$, where $ρ$ 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.

翻译：我们研究了在固定阈值$τ$下，如何生成可被面部验证器判定为同一身份对匹配、不同身份对不匹配的合成身份数量。针对由生成器与归一化识别映射（输出位于单位超球面）构成的生成式人脸识别流程，我们将此问题形式化。将可区分身份生成容量定义为：其诱导的嵌入分布满足预定同身份与异身份验证约束的最大潜在身份数量。在确定性视角不变场景中，我们证明该容量可由可实现的嵌入集合上的球面编码问题表征，并在完全角度表达性假设下退化为经典球面编码量。对于随机身份生成，我们引入中心化模型并推导出充分可容许性条件：身份中心之间的所需间隔为$\arccos(τ)+2ρ$，其中$ρ$为身份内集中半径。在完全角度表达性条件下，该条件给出基于球面编码的可达下界，以及随嵌入维度指数增长率的正向渐近下界。我们还引入先验约束随机编码容量（潜在身份从给定先验分布中独立采样），并根据诱导身份中心对分离失效概率推导出高概率下界。在更强的全支撑模型下，我们获得逆定理与精确的球面编码表征。