Multidimensional scaling (MDS) is a family of methods that embed a given set
of points into a simple, usually flat, domain. The points are assumed to be
sampled from some metric space, and the mapping attempts to preserve the
distances between each pair of points in the set. Distances in the target space
can be computed analytically in this setting. Generalized MDS is an extension
that allows mapping one metric space into another, that is, multidimensional
scaling into target spaces in which distances are evaluated numerically rather
than analytically. Here, we propose an efficient approach for computing such
mappings between surfaces based on their natural spectral decomposition, where
the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS
procedure enables efficient embedding by implicitly incorporating smoothness of
the mapping into the problem, thereby substantially reducing the complexity
involved in its solution while practically overcoming its non-convex nature.
The method is compared to existing techniques that compute dense correspondence
between shapes. Numerical experiments of the proposed method demonstrate its
efficiency and accuracy compared to state-of-the-art approaches.

该研究提出了一种基于表面的谱广义多维尺度映射 (Spectral-GMDS) 方法，用于计算两个度量空间之间的映射关系，该方法在解决问题时隐含地将映射的平滑性纳入，从而实现了高效的嵌入，并在实验中取得了很好的效果。

频谱广义多维尺度缩放

Spectral Generalized Multi-Dimensional Scaling

This work provides a computationally efficient and statistically consistent
moment-based estimator for mixtures of spherical Gaussians. Under the condition
that component means are in general position, a simple spectral decomposition
technique yields consistent parameter estimates from low-order observable
moments, without additional minimum separation assumptions needed by previous
computationally efficient estimation procedures. Thus computational and
information-theoretic barriers to efficient estimation in mixture models are
precluded when the mixture components have means in general position and
spherical covariances. Some connections are made to estimation problems related
to independent component analysis.

本研究提供了一种计算效率高、统计一致的基于矩的混合高斯估计算法，通过简单的谱分解技术，无需额外的最小分离条件，在一般位置上获取成分均值的低阶观测矩，从而通过排除计算和信息理论方面的障碍以便混合模型的高效估计，当混合成分具有一般位置和球形协方差时，与独立成分分析相关的估计问题得到了一些联系。

学习球形高斯混合模型：矩方法与谱分解

Learning mixtures of spherical Gaussians: moment methods and spectral  decompositions

The problem of topic modeling can be seen as a generalization of the
clustering problem, in that it posits that observations are generated due to
multiple latent factors (e.g., the words in each document are generated as a
mixture of several active topics, as opposed to just one). This increased
representational power comes at the cost of a more challenging unsupervised
learning problem of estimating the topic probability vectors (the distributions
over words for each topic), when only the words are observed and the
corresponding topics are hidden.
We provide a simple and efficient learning procedure that is guaranteed to
recover the parameters for a wide class of mixture models, including the
popular latent Dirichlet allocation (LDA) model. For LDA, the procedure
correctly recovers both the topic probability vectors and the prior over the
topics, using only trigram statistics (i.e., third order moments, which may be
estimated with documents containing just three words). The method, termed
Excess Correlation Analysis (ECA), is based on a spectral decomposition of low
order moments (third and fourth order) via two singular value decompositions
(SVDs). Moreover, the algorithm is scalable since the SVD operations are
carried out on $k\times k$ matrices, where $k$ is the number of latent factors
(e.g. the number of topics), rather than in the $d$-dimensional observed space
(typically $d \gg k$).

提出了一种名为 ECA 的学习流程，基于低阶矩的光谱分解，它可以恢复多种混合模型的参数，并在仅使用三元统计时正确恢复主题概率向量和主题优先级，因此具有可扩展性。