【论文翻译】Nonlinear Dimensionality Reduction by Locally Linear Embedding

【论文题目】：Nonlinear Dimensionality Reduction by Locally Linear Embedding

【论文来源】：Nonlinear Dimensionality Reduction by Locally Linear Embedding

【翻译人】：BDML@CQUT实验室

Science Nonlinear Dimensionality Reduction by Locally Linear EmbeddingSam T. Roweis and Lawrence K. Saul

Abstract

Many areas of science depend on exploratory data analysis and visualization. The need to analyze large amounts of multivariate data raises the fundamental problem of dimensionality reduction: how to discover compact representations of high-dimensional data. Here, we introduce locally linear embedding (LLE), an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs. Unlike clustering methods for local dimensionality reduction, LLE maps its inputs into a single global coordinate system of lower dimensionality, and its optimizations do not involve local minima. By exploiting the local symmetries of linear reconstructions, LLE is able to learn the global structure of nonlinear manifolds, such as those generated by images of faces or documents of text.

摘要

许多科学领域依赖于探索性数据分析和可视化。分析大量多元数据的需要提出了降维的基本问题：如何发现高维数据的紧凑表示。在这里，我们介绍了局部线性嵌入(LLE)，一种无监督的学习算法，它计算高维输入的低维的、保持邻域的嵌入。与局部降维的聚类方法不同，LLE将其输入映射到一个低维的单一全局坐标系统中，其优化不涉及局部极小值。通过利用线性重建的局部对称性，LLE能够学习非线性流形的全局结构，例如那些由人脸图像或文本文档生成的流形。

正文

How do we judge similarity? Our mental representations of the world are formed by processing large numbers of sensory inputs—including, for example, the pixel intensities of images, the power spectra of sounds, and the joint angles of articulated bodies. While complex stimuli of this form can be represented by points in a high-dimensional vector space, they typically have a much more compact description. Coherent structure in the world leads to strong correlations between inputs (such as between neighboring pixels in images), generating observations that lie on or close to a smooth low-dimensional manifold. To compare and classify such observations—in effect, to reason about the world— depends crucially on modeling the nonlinear geometry of these low-dimensional manifolds.

我们如何判断相似性?我们对世界的心理表征是通过处理大量的感官输入而形成的——例如，包括图像的像素强度、声音的功率谱和关节体的关节角。虽然这种形式的复杂刺激可以用高维向量空间中的点来表示，但它们通常具有更紧凑的描述。世界上的相干结构导致输入之间的强烈相关性(例如图像中相邻像素之间的相关性)，从而产生位于或接近平滑的低维流形上的观察结果。要比较和分类这样的观测——实际上，是为了解释世界——至关重要的是要建立这些低维流形的非线性几何模型。

Scientists interested in exploratory analysis or visualization of multivariate data (1) face a similar problem in dimensionality reduction. The problem, as illustrated in Fig. 1, involves mapping high-dimensional inputs into a lowdimensional “description” space with as many coordinates as observed modes of variability.Previous approaches to this problem, based on multidimensional scaling (MDS) (2), have computed embeddings that attempt to preserve pairwise distances [or generalized disparities (3)] between data points; these distances are measured along straight lines or, in more sophisticated usages of MDS such as Isomap (4),along shortest paths confined to the manifold of observed inputs. Here, we take a different approach, called locally linear embedding (LLE), that eliminates the need to estimate pairwise distances between widely separated data points. Unlike previous methods, LLE recovers global nonlinear structure from locally linear fits.

对多变量数据的探索性分析或可视化感兴趣的科学家(1)在降维方面也面临着类似的问题。如图1所示，该问题涉及将高维输入映射到一个低维“描述”空间，该空间具有与观察到的可变性模式相同的坐标。以前解决这个问题的方法是基于多维尺度(MDS)(2)计算嵌入，试图保持数据点之间的成对距离[或广义差(3)];这些距离是沿着直线测量的，或者在更复杂的MDS应用中，如Isomap (4)，沿着限定于观察输入流形的最短路径。这里，我们采用了一种不同的方法，称为局部线性嵌入(LLE)，它消除了对相距很远的数据点之间的成对距离进行估计的需要。与以往的方法不同，LLE从局部线性拟合中恢复了全局非线性结构。

The LLE algorithm, summarized in Fig. 2, is based on simple geometric intuitions. Suppose the data consist of N real-valued vectorsWXi, each of dimensionality D, sampled from some underlying manifold. Provided there is sufficient data (such that the manifold is well-sampled), we expect each data point and its neighbors to lie on or close to a locally linear patch of the manifold. We characterize the local geometry of these patches by linear coefficients that reconstruct each data point from its neighbors. Reconstruction errors are measured by the cost function

which adds up the squared distances between all the data points and their reconstructions. The weights Wij summarize the contribution of the jth data point to the ith reconstruction. To compute the weights Wij, we minimize the cost function subject to two constraints: first, that each data point Xi is reconstructed only from its neighbors (5), enforcing Wij=0 if Xj does not belong to the set of neighbors of Xi; second, that the rows of the weight matrix sum to one: Wij=1. The optimal weights Wij subject to these constraints (6) are found by solving a least-squares problem (7).

LLE算法基于简单的几何直觉，如图2所示。假设数据由N个维数D的实值vectorsWXi组成，从某个底层流形采样。如果有足够的数据(这样的流形是很好的采样)，我们期望每个数据点和它的邻居躺在或接近流形的局部线性patch。我们通过线性系数来描述这些小块的局部几何特征，这些线性系数可以重构邻近数据点。重构误差由成本函数来衡量：

也就是把所有数据点的距离平方和它们的重构相加。权重Wij总结了第j个数据点对第i次重构的贡献。为了计算权重Wij，我们在两个约束条件下最小化代价函数：首先，每个数据点Xi仅从其邻居进行重构(5)，如果Xj不属于Xi的邻居集，则强制Wij=0;第二，权值矩阵的行和为1:Wij=1。通过求解最小二乘问题(7)，得到了约束条件(6)下的最优权值Wij。

The constrained weights that minimize these reconstruction errors obey an important symmetry: for any particular data point, they are invariant to rotations, rescalings, and translations of that data point and its neighbors. By symmetry, it follows that the reconstruction weights characterize intrinsic geometric properties of each neighborhood, as opposed to properties that depend on a particular frame of reference (8). Note that the invariance to translations is specifically enforced by the sum-to-one constraint on the rows of the weight matrix.

使重构误差最小化的受限权值遵循一个重要的对称性:对于任何特定的数据点，它们不受该数据点及其相邻数据点的旋转、重调和平移的影响。通过对称，重构权值表征了每个邻域的内在几何性质，而不是依赖于特定参考系(8)的性质。注意，平移的不变性是由权值矩阵行上的和对一约束特别实施的。

Suppose the data lie on or near a smooth nonlinear manifold of lower dimensionality d<<D. To a good approximation then, there exists a linear mapping— consisting of a translation, rotation, and rescaling—that maps the high-dimensional coordinates of each neighborhood to global internal coordinates on the manifold. By design, the reconstruction weights Wij reflect intrinsic geometric properties of the data that are invariant to exactly such transformations. We therefore expect their characterization of local geometry in the original data space to be equally valid for local patches on the manifold. In particular, the same weights Wij that reconstruct the ith data point in D dimensions should also reconstruct its embedded manifold coordinates in d dimensions.

假设数据位于或靠近一个光滑的低维非线性流形 d<<D。为了很好地近似，存在一个线性映射—包括平移、旋转和重新标定—它将各邻域的高维坐标映射到流形上的全局内部坐标。通过设计，重构权值Wij反映了数据内在的几何特性，这些特性对这种转换是不变的。因此，我们期望它们在原始数据空间中的局部几何特征对流形上的局部patch同样有效。特别是重构D维第i个数据点的权重Wij也应该重构其嵌入的D维流形坐标。

LLE constructs a neighborhood-preserving mapping based on the above idea. In the final step of the algorithm, each high-dimensional observation Xi is mapped to a low-dimensional vector Yi representing global internal coordinates on the manifold. This is done by choosing d-dimensional coordinates Yi to minimize the embedding cost function

This cost function, like the previous one, is based on locally linear reconstruction errors, but here we fix the weights Wij while optimizing the coordinates Yi. The embedding cost in Eg.2 defines a quadratic form in the vectors Yi. Subject to constraints that make the problem well-posed, it can be minimized by solving a sparse N*N eigenvalue problem (9), whose bottom d nonzero eigenvectors provide an ordered set of orthogonal coordinates centered on the origin.

LLE在此思想的基础上构造了一个邻域保持映射。在算法的最后一步，将每个高维观测Xi映射到表示流形上全局内部坐标的低维向量Yi。这是通过选择d维坐标Yi来最小化嵌入代价函数来实现的：

这个代价函数和前面的一样，是基于局部线性重构误差的，但是这里我们在优化坐标Yi的同时确定了权值Wij。第2节中的嵌入代价定义了向量Yi的二次形式。通过求解一个稀疏的N*N特征值问题(9)，该问题的底d个非零特征向量提供了一个以原点为中心的正交坐标的有序集合，在约束条件下使问题适定，可以将其最小化。

Implementation of the algorithm is straightforward. In our experiments, data points were reconstructed from their K nearest neighbors, as measured by Euclidean distance or normalized dot products. For such implementations of LLE, the algorithm has only one free parameter: the number of neighbors, K.Once neighbors are chosen, the optimal weights Wij and coordinates Yi are computed by standard methods in linear algebra.The algorithm involves a single pass through the three steps in Fig. 2 and finds global minima of the reconstruction and embedding costs in Egs. 1 and 2.

算法的实现很简单。在我们的实验中，数据点从它们的K个最近的邻居重建，以欧几里得距离或标准化点积度量。对于这样的LLE实现，算法只有一个自由参数:邻居的数量K。选择邻域后，用线性代数的标准方法计算出最优权值Wij和坐标Yi。该算法只需简单地通过图2中的三个步骤，就可以找到例1和例2中重构和嵌入代价的全局最小值。

In addition to the example in Fig. 1, for which the true manifold structure was known (10), we also applied LLE to images of faces (11) and vectors of word-document counts (12). Two-dimensional embeddings of faces and words are shown in Figs. 3 and 4. Note how the coordinates of these embedding spaces are related to meaningful attributes, such as the pose and expression of human faces and the semantic associations of words.

除了图1示例中已知的真实流形结构(10)之外，我们还将LLE应用于人脸图像(11)和单词文档计数向量(12)。平面和文字的二维嵌入如图3和图4所示。请注意这些嵌入空间的坐标是如何与有意义的属性相关联的，例如人脸的姿态和表情以及词语的语义关联。

Many popular learning algorithms for nonlinear dimensionality reduction do not share the favorable properties of LLE. Iterative hill-climbing methods for autoencoder neural networks (13, 14), self-organizing maps (15), and latent variable models (16) do not have the same guarantees of global optimality or convergence;they also tend to involve many more free parameters, such as learning rates, convergence criteria, and architectural specifications. Finally, whereas other nonlinear methods rely on deterministic annealing schemes (17) to avoid local minima, the optimizations of LLE are especially tractable.

许多流行的非线性降维学习算法都不具有线性降维算法的优点。自编码器神经网络(13,14)、自组织映射(15)和潜变量模型(16)的迭代爬山方法不能保证全局最优性或收敛性;它们还倾向于包含更多的自由参数，如学习率、收敛标准和架构规范。最后，虽然其他非线性方法依赖于确定性退火方案(17)来避免局部极小值，但LLE的优化特别容易处理。

LLE scales well with the intrinsic manifold dimensionality, d, and does not require a discretized gridding of the embedding space. As more dimensions are added to the embedding space, the existing ones do not change, so that LLE does not have to be rerun to compute higher dimensional embeddings. Unlike methods such as principal curves and surfaces (18) or additive component models (19), LLE is not limited in practice to manifolds of extremely low dimensionality or codimensionality. Also, the intrinsic value of d can itself be estimated by analyzing a reciprocal cost function, in which reconstruction weights derived from the embedding vectors Yi are applied to the data points Xi.

LLE能很好地利用其固有的流形维数d，且不需要对嵌入空间进行离散网格化。随着嵌入空间的维数增加，现有的维数不会改变，因此LLE不必重新运行来计算更高维数的嵌入。与主曲线和曲面(18)或可加成分模型(19)等方法不同，LLE在实践中并不局限于极低维数或多维度的流形。同样，d的内在值本身可以通过分析一个倒易代价函数来估计，其中从嵌入向量Yi推导出的重构权值被应用到数据点Xi。

LLE illustrates a general principle of manifold learning, elucidated by Martinetz and Schulten (20) and Tenenbaum (4), that overlapping local neighborhoods—collectively analyzed—can provide information about global geometry.Many virtues of LLE are shared by Tenenbaum’s algorithm, Isomap, which has been successfully applied to similar problems in nonlinear dimensionality reduction. Isomap’s embeddings, however, are optimized to preserve geodesic distances between general pairs of data points, which can only be estimated by computing shortest paths through large sublattices of data. LLE takes a different approach, analyzing local symmetries, linear coefficients, and reconstruction errors instead of global constraints, pairwise distances, and stress functions. It thus avoids the need to solve large dynamic programming problems, and it also tends to accumulate very sparse matrices, whose structure can be exploited for savings in time and space.

LLE阐明了由Martinetz和Schulten(20)和Tenenbaum(4)阐明的流形学习的一般原理，即重叠的局部邻居——集体分析——可以提供关于全局几何的信息。Isomap算法具有LLE的许多优点，并已成功地应用于类似的非线性降维问题。然而，Isomap的嵌入优化了一般数据点对之间的测地线距离，这只能通过计算通过大数据子格的最短路径来估计。LLE采用不同的方法，分析局部对称性、线性系数和重构误差，而不是全局约束、成对距离和应力函数。因此，它避免了求解大型动态规划问题的需要，而且它还倾向于积累非常稀疏的矩阵，利用它们的结构可以节省时间和空间。

LLE is likely to be even more useful in combination with other methods in data analysis and statistical learning. For example, a parametric mapping between the observation and embedding spaces could be learned by supervised neural networks (21) whose target values are generated by LLE. LLE can also be generalized to harder settings, such as the case of disjoint data manifolds (22), and specialized to simpler ones, such as the case of time-ordered observations (23).

在数据分析和统计学习中，LLE与其他方法结合起来可能会更有用。例如，监视神经网络(21)可以学习观察空间和嵌入空间之间的参数映射，目标值由LLE生成。LLE还可以推广到较难的设置，比如不相交数据流形的情况(22)，也可以专门用于较简单的情况，比如时间有序观察的情况(23)。

Perhaps the greatest potential lies in applying LLE to diverse problems beyond those considered here. Given the broad appeal of traditional methods, such as PCA and MDS, the algorithm should find widespread use in many areas of science.

也许最大的潜力在于将LLE应用于本文所讨论的各种问题。鉴于传统方法(如PCA和MDS)的广泛吸引力，该算法应该在许多科学领域得到广泛应用。