We introduce Non-Euclidean-MDS (Neuc-MDS), an extension of classical Multidimensional Scaling (MDS) that accommodates non-Euclidean and non-metric inputs. The main idea is to generalize the standard inner product to symmetric bilinear forms to utilize the negative eigenvalues of dissimilarity Gram matrices. Neuc-MDS efficiently optimizes the choice of (both positive and negative) eigenvalues of the dissimilarity Gram matrix to reduce STRESS, the sum of squared pairwise error. We provide an in-depth error analysis and proofs of the optimality in minimizing lower bounds of STRESS. We demonstrate Neuc-MDS's ability to address limitations of classical MDS raised by prior research, and test it on various synthetic and real-world datasets in comparison with both linear and non-linear dimension reduction methods.

翻译：本文提出了非欧几里得多维缩放（Neuc-MDS），它是经典多维缩放（MDS）的一种扩展，能够处理非欧几里得及非度量的输入数据。其核心思想是将标准内积推广至对称双线性形式，从而利用相异性Gram矩阵的负特征值。Neuc-MDS通过优化相异性Gram矩阵（包括正负）特征值的选择，有效降低了STRESS（即两两误差的平方和）。我们提供了详细的误差分析，并证明了该方法在最小化STRESS下界方面的最优性。实验表明，Neuc-MDS能够解决先前研究指出的经典MDS的局限性，并在多种合成与真实数据集上，与线性和非线性降维方法进行了对比测试。