Cube Rounding for Matrix Sketching: Complete-Path Risk and Efficient Gram-Aware Updates

Cube sampling preserves prescribed inclusion probabilities, yet minimizing the variance of individual rounding steps need not minimize the error of the final matrix sketch. We study this gap through the complete flight-and-landing process and develop an efficient implementation of a Gram-aware sampling rule. For a specified class of legal actions and landing procedures, we show how residual balance infeasibility forces terminal mutual exclusion, yielding policy-invariant covariance structure and conditions for exact optimality and quantitative near-optimality. Conversely, an explicit six-unit construction exhibits a unique locally optimal update with strictly positive complete-path regret, revealing the role of subsequent branching and landing. For rank-one positive semidefinite contributions, the resulting risk analysis connects Gram approximation to PCA reconstruction and subspace error. We then introduce Incremental Gram-Cube (IGC), which combines cross-step Gram reuse, an endpoint nullspace basis, and direct two-dimensional direction evaluation to accelerate a directional-energy surrogate on four-unit buffers. Under matched tie conventions, the implementation preserves the sampling rule in exact arithmetic. On 24 development cases spanning synthetic populations, real-data subsets, and larger synthetic inputs, IGC reduces end-to-end sampling-and-estimation time by 7.30–12.20% across the three groups relative to an already cached implementation of the same rule. All 9,216 paired evaluation draws produce identical selected indices and Frobenius errors. These results characterize when local Cube decisions suffice and demonstrate how matrix-aware updates can be computed with less overhead.