Kohli Dhruv, Cloninger Alexander, Mishne Gal
Department of Mathematics, University of California San Diego, CA 92093, USA.
Halicioğlu Data Science Institute, University of California San Diego, CA 92093, USA.
J Mach Learn Res. 2021 Jan-Dec;22.
We present Low Distortion Local Eigenmaps (LDLE), a manifold learning technique which constructs a set of low distortion local views of a data set in lower dimension and registers them to obtain a global embedding. The local views are constructed using the global eigenvectors of the graph Laplacian and are registered using Procrustes analysis. The choice of these eigenvectors may vary across the regions. In contrast to existing techniques, LDLE can embed closed and non-orientable manifolds into their intrinsic dimension by tearing them apart. It also provides gluing instruction on the boundary of the torn embedding to help identify the topology of the original manifold. Our experimental results will show that LDLE largely preserved distances up to a constant scale while other techniques produced higher distortion. We also demonstrate that LDLE produces high quality embeddings even when the data is noisy or sparse.
我们提出了低失真局部特征映射（LDLE），这是一种流形学习技术，它在低维空间中构建数据集的一组低失真局部视图，并将它们对齐以获得全局嵌入。局部视图使用图拉普拉斯算子的全局特征向量构建，并使用普罗克汝斯分析进行对齐。这些特征向量的选择可能因区域而异。与现有技术相比，LDLE可以通过将封闭和不可定向流形撕开，将它们嵌入到其固有维度中。它还在撕开的嵌入边界上提供粘合指令，以帮助识别原始流形的拓扑结构。我们的实验结果将表明，LDLE在恒定尺度下很大程度上保留了距离，而其他技术产生的失真更高。我们还证明，即使数据有噪声或稀疏，LDLE也能产生高质量的嵌入。
