Multidimensional Scaling (MDS) is a classical technique for embedding data in low dimensions, still in widespread use today. In this paper we study MDS in a modern setting - specifically, high dimensions and ambient measurement noise. We show that as the ambient noise level increases, MDS suffers a sharp breakdown that depends on the data dimension and noise level, and derive an explicit formula for this breakdown point in the case of white noise. We then introduce MDS+, a simple variant of MDS, which applies a shrinkage nonlinearity to the eigenvalues of the MDS similarity matrix. Under a natural loss function measuring the embedding quality, we prove that MDS+ is the unique, asymptotically optimal shrinkage function. MDS+ offers improved embedding, sometimes significantly so, compared with MDS. Importantly, MDS+ calculates the optimal embedding dimension, into which the data should be embedded.

多维缩放（MDS）是将数据嵌入低维的经典技术，至今仍在广泛使用。在本文中，我们研究了现代环境中的MDS-特别是高尺寸和环境测量噪声。我们表明，随着环境噪声水平的提高，MDS会受到取决于数据维度和噪声水平的急剧击穿，并在出现白噪声的情况下得出针对此击穿点的明确公式。然后，我们介绍MDS +，这是MDS的简单变体，它将收缩非线性应用于MDS相似矩阵的特征值。在测量嵌入质量的自然损失函数下，我们证明MDS +是唯一的，渐近最优的收缩函数。与MDS相比，MDS +提供了更好的嵌入效果，有时效果显着。重要的，