压缩感知与量化张量列（QTTs）
Compressed Sensing with Quantized Tensor Trains (QTTs)
- Texas A&M University（德克萨斯农工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对高维向量压缩感知，提出利用量化张量列（QTT）表示结合稀疏低阶交互结构，采用Walsh-Hadamard测量，建立RIP和唯一性条件，并设计结构感知初始化器，使测量需求随阶数而非环境维度增长，数值实验验证了其有效性。
AI中文摘要：
随着维度 d 的增长，长度为 2^d 的向量的存储和恢复可能变得极其昂贵，这是维度灾难的一种表现。对于某些结构化函数和多尺度量，离散化后的向量在二进制张量化后能够支持紧凑的量化张量列（QTT）表示：一个长度为 2^d 的向量被重塑为一个 d 维二进制张量，其张量列（TT）秩较低。基于这一构造，我们研究了结合有界 TT 秩与稀疏低阶交互结构的张量化信号的压缩感知，使用随机采样的 Walsh-Hadamard 测量。对于这一结构化模型类别，我们建立了均匀受限等距性质，并推导出一个测量条件，确保无噪声恢复的唯一性。我们还提出了一种结构感知的初始化器，用于交替线性方案（ALS），该初始化器通过将伴随反投影投影到低阶交互空间上，然后应用 TT-SVD 来获得。我们证明了一个均匀初始化误差界，其测量需求，在固定结构参数下，随张量阶数 d 多项式增长，而非随环境维度 2^d 增长。数值实验支持所预测的 RIP 和初始化缩放，并展示了更可靠的 ALS 恢复。
英文摘要：
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.
