Information-Theoretic Limits and One-Bit Estimation of Sparse Relation Matrices Under Personalized Local Differential Privacy and Byzantine Contamination

We study estimation of globally sparse relation probability matrices from capped binary relation sets under personalized user-level local differential privacy and Byzantine response replacement. A cardinality-calibrated joint personalized relation sketch (JPRS) releases one bit per user. The decoder combines public channel-amplitude weighting with exact sparse feasible projection. We distinguish trimmed honest information, adversarial leverage, and a response-contamination indistinguishability radius. An explicit capped binary block construction yields a sampling lower bound with an explicit low-information truncation. A separate contamination construction gives a finite-sample error floor. For precommitted corrupted identities, the global decoder admits a uniform bound against adaptive valid-output attacks. In the strong-privacy, sufficiently informative regime, sampling and contamination bounds match up to logarithms on bounded-ratio budget classes, subject to the stated sample regularity and corruption-count conditions. The theory uses global sparsity rather than low-rank structure. Across six synthetic families, the global decoder has lower mean loss than the three tested private global baselines, but improves on zero only in the Zipf setting. Three public-data benchmarks further show that low aggregate error can be driven by a merged tail category. These results delimit practical utility; finite-budget attack searches do not certify worst-case performance.