Minimum Volume Embedding
Minimum Volume Embedding
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发表时间：
2007-03
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通讯作者：
B. Shaw;Tony Jebara
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作者：
B. Shaw;Tony Jebara
最小体积嵌入 (MVE) 是一种非线性降维算法，它使用半定规划 (SDP) 和矩阵分解来查找低维嵌入，该嵌入可以保留点之间的局部距离，同时以更少的维度表示数据集。 MVE 遵循类似于半定嵌入 (SDE) 等算法的方法，因为它在应用核主成分分析 (KPCA) 之前使用 SDP 学习核矩阵。然而，MVE 的目标函数直接优化数据的特征谱，以在可用于嵌入的少数维度内保留尽可能多的能量。同时，剩余的特征谱能量在与嵌入正交的方向上被最小化，从而将数据保持在所谓的最小体积流形中。我们展示了 MVE 如何在保留嵌入的体积和生成的特征谱方面改进 SDE，从而为各种合成和现实世界数据集（包括简单的玩具示例、人脸图像、手写数字、系统发育树和社交网络）提供更好的可视化效果。
Minimum Volume Embedding (MVE) is an algorithm for non-linear dimensionality reduction that uses semidefinite programming (SDP) and matrix factorization to find a low-dimensional embedding that preserves local distances between points while representing the dataset in many fewer dimensions. MVE follows an approach similar to algorithms such as Semidefinite Embedding (SDE), in that it learns a kernel matrix using an SDP before applying Kernel Principal Component Analysis (KPCA). However, the objective function for MVE directly optimizes the eigenspectrum of the data to preserve as much of its energy as possible within the few dimensions available to the embedding. Simultaneously, remaining eigenspectrum energy is minimized in directions orthogonal to the embedding thereby keeping data in a so-called minimum volume manifold. We show how MVE improves upon SDE in terms of the volume of the preserved embedding and the resulting eigenspectrum, producing better visualizations for a variety of synthetic and real-world datasets, including simple toy examples, face images, handwritten digits, phylogenetic trees, and social networks.