Low dimensional nonlinear structure abounds in datasets across computer vision and machine learning. Kernelized matrix factorization techniques have recently been proposed to learn these nonlinear structures for denoising, classification, dictionary learning, and missing data imputation, by observing that the image of the matrix in a sufficiently large feature space is low-rank. However, these nonlinear methods fail in the presence of sparse noise or outliers. In this work, we propose a new robust nonlinear factorization method called Robust Non-Linear Matrix Factorization (RNLMF). RNLMF constructs a dictionary for the data space by factoring a kernelized feature space; a noisy matrix can then be decomposed as the sum of a sparse noise matrix and a clean data matrix that lies in a low dimensional nonlinear manifold. RNLMF is robust to sparse noise and outliers and scales to matrices with thousands of rows and columns. Empirically, RNLMF achieves noticeable improvements over baseline methods in denoising and clustering.

中文翻译：

---

用于字典学习，去噪和聚类的鲁棒非线性矩阵分解

在计算机视觉和机器学习中的数据集中，低维非线性结构比比皆是。最近提出了核化矩阵分解技术，以通过观察在足够大的特征空间中矩阵的图像是低秩的来学习这些非线性结构的降噪，分类，字典学习和缺失数据归因。但是，这些非线性方法在稀疏噪声或离群值的情况下会失败。在这项工作中，我们提出了一种新的鲁棒非线性分解方法，称为鲁棒非线性矩阵分解（RNLMF）。RNLMF通过分解内核化的特征空间为数据空间构造一个字典。然后，可以将噪声矩阵分解为稀疏噪声矩阵和位于低维非线性流形中的纯净数据矩阵的总和。RNLMF具有强大的鲁棒性，可以减少噪声和异常值，并可以缩放到具有数千行和列的矩阵。从经验上讲，RNNLF在去噪和聚类方面比基线方法取得了明显的改进。