Compressed Sensing with Quantized Tensor Trains (QTTs)
Abstract
The storage and recovery of a vector of length 2d can become prohibitively expensive as d grows, a manifestation of the curse of dimensionality. For certain structured functions and multiscale quantities, the resulting discretized vectors admit compact quantized tensor train (QTT) representations after binary tensorization: a length-2d vector is reshaped into a d-way binary tensor with low tensor-train (TT) rank. Motivated by this construction, we study compressed sensing of tensorized signals that combine bounded TT rank with sparse low-order interaction structure, using randomly sampled Walsh--Hadamard measurements. For this structured model class, we establish a uniform restricted isometry property and derive a measurement condition guaranteeing uniqueness of noiseless recovery. We also propose a structure-aware initializer for the alternating linear scheme (ALS), obtained by projecting the adjoint backprojection onto the low-order interaction space before applying TT-SVD. We prove a uniform initialization error bound whose measurement requirement, for fixed structural parameters, grows polynomially with the tensor order d rather than with the ambient dimension 2d. Numerical experiments support the predicted RIP and initialization scaling and demonstrate more reliable ALS recovery.
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BibTeX
@misc{shao2026compressed,
title = {{Compressed Sensing with Quantized Tensor Trains (QTTs)}},
author = {Jingchun Shao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36506},
eprint = {2609.36506},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36506},
}
RIS
TY - UNPB
TI - Compressed Sensing with Quantized Tensor Trains (QTTs)
AU - Jingchun Shao
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36506
UR - https://arxiv.org/abs/2609.36506
AB - The storage and recovery of a vector of length 2^d can become prohibitively expensive as d grows, a manifestation of the curse of dimensionality. For certain structured functions and multiscale quantities, the resulting discretized vectors admit compact quantized tensor train (QTT) representations after binary tensorization: a length-2^d vector is reshaped into a d-way binary tensor with low tensor-train (TT) rank. Motivated by this construction, we study compressed sensing of tensorized signals that combine bounded TT rank with sparse low-order interaction structure, using randomly sampled Walsh--Hadamard measurements. For this structured model class, we establish a uniform restricted isometry property and derive a measurement condition guaranteeing uniqueness of noiseless recovery. We also propose a structure-aware initializer for the alternating linear scheme (ALS), obtained by projecting the adjoint backprojection onto the low-order interaction space before applying TT-SVD. We prove a uniform initialization error bound whose measurement requirement, for fixed structural parameters, grows polynomially with the tensor order d rather than with the ambient dimension 2^d. Numerical experiments support the predicted RIP and initialization scaling and demonstrate more reliable ALS recovery.
ER -