SING-CH: Task-Scale-Agnostic Lifelong Cross-Modal Hashing on Statistical Manifolds

Lifelong cross-modal hashing demands incremental updates as new tasks arrive, yet existing regularization-based methods such as EWC and SI share a fundamental weakness: their effective regularization strength is entangled with task scale, as the empirical Fisher Information Matrix (FIM) scales linearly with sample size. This forces costly per-task hyperparameter tuning in streaming deployments to avoid over- or under-regularizing tasks of varying sizes. We propose \textsc{Sing-CH}, a lifelong hashing framework built on information geometry. At its core is a Gibbs probabilistic reconstruction of the pairwise objective, which gives the parameter space a statistical manifold structure and yields a natural $1/n_t^2$ normalization. We prove this normalization renders the FIM scale-agnostic: its spectral properties remain independent of task size even under diagonal and K-FAC approximations. Beyond this invariance, a Forgetting-Rigidity trade-off theorem shows that natural gradient applied to the composite objective reintroduces scale dependence. This makes our decoupled design, pairing the FIM as regularizer with Adam for optimization, a structural necessity rather than a convenience. Experiments on three benchmarks demonstrate that \textsc{Sing-CH} outperforms 11 baselines by 4.1\% average mAP without per-task tuning. Under extreme task-scale imbalance, degradation is only 2.0\% versus 3.9\% for the strongest baseline, providing the first empirically grounded solution to scale-agnostic lifelong retrieval.