谱残差连续贪心算法用于张量采样
Spectral-Residual Continuous Greedy for Tensor Sampling
- College of Science, National University of Defense Technology（国防科技大学理学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对张量采样中跨模态预算分配难题，提出谱残差连续贪心算法，结合方向认证与路径引导交换，实现更优框架势设计与更低重建误差。
AI中文摘要：
从有限测量中采样多域张量是结构化线性逆问题的基础。Kronecker结构采样避免了完整的高维感知矩阵，但设计仍然困难：顺序离散方法可能过早地承诺跨模态预算，而标准连续贪心算法避免了这种过早承诺，但代价是重复进行依赖于状态的梯度评估。我们提出了用于框架势（FP）张量采样的谱残差连续贪心算法（\alg）。\alg在舍入之前维持一个依赖于状态的分数分配。逐模态的Gram矩阵为方向认证提供了安全的梯度区间，而Shapley值则优先处理未解决的梯度查询。确定性舍入之后是路径引导交换（PGX），它只接受精确的FP递减交换。我们建立了一个有限步近似保证，当方向认证变得精确且有限步残差误差消失时，该保证趋近于经典的$1-1/e$因子。实验表明，\alg需要更少的精确梯度评估并产生更好的FP设计，在跨模态预算分配难以确定的实例上，其优势更为明显。\alg在所有测试的噪声水平下也实现了比Greedy-FP更低的平均归一化均方误差（NMSE），尽管重建增益小于FP增益。
英文摘要：
Sampling a multidomain tensor from limited measurements is fundamental in structured linear inverse problems. Kronecker-structured sampling avoids the full high-dimensional sensing matrix, but design remains difficult: sequential discrete methods can commit the cross-mode budget too early, whereas standard continuous greedy avoids such early commitment at the cost of repeated state-dependent gradient evaluations. We propose spectral-residual continuous greedy (\alg) for frame-potential (FP) tensor sampling. \alg maintains a state-dependent fractional allocation before rounding. Mode-wise Gram matrices provide safe gradient intervals for direction certification, while Shapley values prioritize unresolved gradient queries. SR-CG then applies deterministic rounding followed by path-guided exchange (PGX), which reuses the final fractional state to restrict candidate swaps and accepts only exact FP-decreasing exchanges. We establish a finite-step approximation guarantee that approaches the classical $1-1/e$ factor as direction certification becomes exact and finite-step residual error vanishes. Experiments show fewer exact gradient evaluations and better FP designs, with clearer gains on instances where the cross-mode budget allocation is difficult to determine. \alg also achieves lower average normalized mean-squared error (NMSE) than Greedy-FP at all tested noise levels, although the reconstruction gain is smaller than the FP gain.
