Spectral-Residual Continuous Greedy for Tensor Sampling
Abstract
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.
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BibTeX
@misc{li2026spectralresidual,
title = {{Spectral-Residual Continuous Greedy for Tensor Sampling}},
author = {Hao Li and Jie Xu and Zheng Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14346},
eprint = {2609.14346},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14346},
}
RIS
TY - UNPB
TI - Spectral-Residual Continuous Greedy for Tensor Sampling
AU - Hao Li
AU - Jie Xu
AU - Zheng Xie
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.14346
UR - https://arxiv.org/abs/2609.14346
AB - 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.
ER -