提出结构化损失度量，揭示张量近似中的几何失真问题。
Structural Loss Metrics for Tensor Approximation via Matrix Low-Rank Approximation
- 引入方向损失与交互损失，量化张量近似中的几何偏差。
- 实验显示相同重建误差下方向损失差可达4.6倍，影响视觉质量。
- 适合关注张量分解结构保真度的研究者使用。
通过SVD的矩阵低秩近似是张量分解的标准替代方法，但逐元素重建误差无法捕捉多维几何退化。在正交Tucker模型下，我们提出两种度量：跨模态方向损失，用于衡量秩截断和噪声旋转导致的子空间偏离；交互损失，用于量化核心张量中多线性交互的扭曲。我们证明平方相对重建误差可正交分解为交互损失与非子空间能量之和，并推导出类似Wedin的界，保证了方向损失估计器的稳定性。在合成数据与高光谱数据集上的实验表明，几乎相同的重建误差可能对应显著不同的结构损失特征；高光谱图像块在相似重建误差下，方向损失最高相差4.6倍，且与严重视觉模糊密切相关。
原文摘要 · Abstract (English)
Matricized low-rank approximation via SVD is a standard surrogate for tensor decompositions, but entry-wise reconstruction error fails to capture multiway geometric degradation. Under an orthogonal Tucker model, we characterize this degradation using two metrics: cross-mode Direction Loss, measuring geometric subspace deviation from rank truncation and noise rotation, and Interaction Loss, quantifying multilinear interaction distortion in the core tensor. We prove that squared relative reconstruction error orthogonally decomposes into interaction loss and out-of-subspace energy, and derive a Wedin-type bound establishing the stability of a plug-in Direction Loss estimator. Experiments on synthetic and hyperspectral datasets demonstrate that nearly identical reconstruction errors can yield markedly different structural-loss profiles; hyperspectral patches with comparable reconstruction errors exhibit up to a 4.6-fold difference in Direction Loss, correlating with severe visual blurring.
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