提出可证明正确的凸模型，解决噪声下平滑可分离NMF的因子恢复问题。
A Provably-Correct and Robust Convex Model for Smooth Separable NMF
- 基于平滑可分离假设，构建凸优化模型，提升鲁棒性。
- 在含噪条件下仍能准确恢复原始因子，理论保证可靠。
- 适用于高光谱解混等实际场景，性能优于当前主流方法。
非负矩阵分解（NMF）是一种用于非负数据的线性降维技术，广泛应用于高光谱解混和主题建模等领域。由于一般情况下NMF是NP难问题且解不唯一，通常需引入额外约束或假设。其中，可分离性假设指出基向量等于输入矩阵的某些列，此时称为可分离NMF（SNMF），可在多项式时间内求解并保证解的唯一性和鲁棒性。然而在真实场景中，因噪声或变化，多个数据点可能靠近基向量，而传统SNMF未加以利用。本文采用平滑可分离性假设，即每个基向量接近多个数据点，提出平滑可分离NMF（SSNMF）的凸模型，并证明其在噪声存在时仍能准确恢复目标因子。进一步地，结合已有快速梯度法求解该凸模型，在合成数据和高光谱数据集上均表现优于当前先进方法。
原文摘要 · Abstract (English)
Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for nonnegative data, with applications such as hyperspectral unmixing and topic modeling. NMF is a difficult problem in general (NP-hard), and its solutions are typically not unique. To address these two issues, additional constraints or assumptions are often used. In particular, separability assumes that the basis vectors in the NMF are equal to some columns of the input matrix. In that case, the problem is referred to as separable NMF (SNMF) and can be solved in polynomial-time with robustness guarantees, while identifying a unique solution. However, in real-world scenarios, due to noise or variability, multiple data points may lie near the basis vectors, which SNMF does not leverage. In this work, we rely on the smooth separability assumption, which assumes that each basis vector is close to multiple data points. We explore the properties of the corresponding problem, referred to as smooth SNMF (SSNMF), and examine how it relates to SNMF and orthogonal NMF. We then propose a convex model for SSNMF and show that it provably recovers the sought-after factors, even in the presence of noise. We finally adapt an existing fast gradient method to solve this convex model for SSNMF, and show that it compares favorably with state-of-the-art methods on both synthetic and hyperspectral datasets.
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