【Scikit-Learn 中文文档】流形学习 - 监督学习 - 用户指南 | ApacheCN

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2.2. 流形学习

Look for the bare necessities
The simple bare necessities
Forget about your worries and your strife
I mean the bare necessities
Old Mother Nature’s recipes
That bring the bare necessities of life

– Baloo的歌 [奇幻森林]





流形学习是一种减少非线性维度的方法。 这个任务的算法基于许多数据集的维度只是人为导致的高。

2.2.1. 介绍

高维数据集可能非常难以可视化。 虽然可以绘制两维或三维数据来显示数据的固有结构，但等效的高维图不太直观。 为了帮助可视化数据集的结构，必须以某种方式减小维度。

通过对数据的随机投影来实现降维的最简单方法。 虽然这允许数据结构的一定程度的可视化，但是选择的随机性远远不够。 在随机投影中，数据中更有趣的结构很可能会丢失。



 



为了解决这一问题，设计了一些监督和无监督的线性维数降低框架，如主成分分析（PCA），独立成分分析，线性判别分析等。 这些算法定义了特定的标题来选择数据的“有趣”线性投影。 这些是强大的，但是经常会错过重要的非线性结构的数据。



 



流形可以被认为是将线性框架（如PCA）推广为对数据中的非线性结构敏感的尝试。 虽然存在监督变量，但是典型的流形学习问题是无监督的：它从数据本身学习数据的高维结构，而不使用预定的分类。

例子:
参见 Manifold
learning on handwritten digits: Locally Linear Embedding, Isomap… ,手写数字降维的例子。
参见 Comparison
of Manifold Learning methods ,玩具“S曲线”数据集的维度降低的一个例子。

以下概述了scikit学习中可用的流形学习实现

2.2.2. Isomap

流形学习的最早方法之一是 Isomap 算法，等距映射(Isometric Mapping)的缩写。 Isomap 可以被视为多维缩放（Multi-dimensional Scaling：MDS）或 Kernel PCA 的扩展。 Isomap 寻求一个维度较低的嵌入，它保持所有点之间的测量距离。 Isomap 可以与 

Isomap

 对象执行。

2.2.2.1. 复杂度

Isomap 算法包括三个阶段:
搜索最近的邻居. Isomap 使用 

sklearn.neighbors.BallTree

 进行有效的邻居搜索。
对于 

 维中 

 个点的 

 个最近邻，成本约为 


最短路径图搜索. 最有效的已知算法是 Dijkstra 算法，它的复杂度大约是 

 ，或
Floyd-Warshall 算法，它的复杂度是 

. 该算法可以由用户使用
isomap 的 path_method 关键字来选择。 如果未指定，则代码尝试为输入数据选择最佳算法。
部分特征值分解. 嵌入在与 

 isomap内核的 

 个最大特征值相对应的特征向量中进行编码。
对于密集求解器，成本约为 

 。 通常可以使用ARPACK求解器来提高这个成本。
用户可以使用isomap的path_method关键字来指定特征。 如果未指定，则代码尝试为输入数据选择最佳算法。

Isomap 的整体复杂度是 

.


 : 训练的数据节点数


 : 输入维度


 : 最近的邻居数


 : 输出维度

参考文献:
“A global geometric framework for nonlinear dimensionality
reduction” Tenenbaum, J.B.; De Silva, V.; & Langford, J.C. Science 290 (5500)

2.2.3. 局部线性嵌入

局部线性嵌入（LLE）寻求保留局部邻域内距离的数据的低维投影。 它可以被认为是一系列局部主成分分析，与整体相比，找到最好的非线性嵌入。

局部线性嵌入可以使用 

locally_linear_embedding

 函数或其面向对象的副本方法 

LocallyLinearEmbedding

 执行。

2.2.3.1. 复杂度

标准的 LLE 算法包括三个阶段:
搜索最近的邻居. 参见上述 Isomap 讨论。
权重矩阵构造. 

.
LLE 权重矩阵的构造涉及每 

 个局部邻域的 

 线性方程的解
部分特征值分解. 参见上述 Isomap 讨论。

标准 LLE 的整体复杂度是 

.


 : 训练的数据节点数


 : 输入维度


 : 最近的邻居数


 : 输出维度

参考文献:
“Nonlinear dimensionality reduction by locally linear
embedding” Roweis, S. & Saul, L. Science 290:2323 (2000)

2.2.4. Modified Locally Linear Embedding

One well-known issue with LLE is the regularization problem. When the number of neighbors is greater than the number of input dimensions, the matrix defining each local neighborhood is rank-deficient. To address this, standard LLE applies an arbitrary regularization
parameter 

, which is chosen relative to
the trace of the local weight matrix. Though it can be shown formally that as 

,
the solution converges to the desired embedding, there is no guarantee that the optimal solution will be found for 

.
This problem manifests itself in embeddings which distort the underlying geometry of the manifold.

One method to address the regularization problem is to use multiple weight vectors in each neighborhood. This is the essence of modified locally linear embedding (MLLE). MLLE can be performed with function 

locally_linear_embedding

or
its object-oriented counterpart 

LocallyLinearEmbedding

,
with the keyword 

method = 'modified'

.
It requires 

n_neighbors > n_components

.

2.2.4.1. Complexity

The MLLE algorithm comprises three stages:
Nearest Neighbors Search. Same as standard LLE
Weight Matrix Construction. Approximately 

.
The first term is exactly equivalent to that of standard LLE. The second term has to do with constructing the weight matrix from multiple weights. In practice, the added cost of constructing the MLLE weight matrix is relatively small compared to the cost of
steps 1 and 3.
Partial Eigenvalue Decomposition. Same as standard LLE

The overall complexity of MLLE is 

.


 : number of
training data points


 : input dimension


 : number of
nearest neighbors


 : output dimension

References:
“MLLE: Modified Locally Linear Embedding Using
Multiple Weights” Zhang, Z. & Wang, J.

2.2.5. Hessian Eigenmapping

Hessian Eigenmapping (also known as Hessian-based LLE: HLLE) is another method of solving the regularization problem of LLE. It revolves around a hessian-based quadratic form at each neighborhood which is used to recover the locally linear structure. Though
other implementations note its poor scaling with data size, 

sklearn

 implements
some algorithmic improvements which make its cost comparable to that of other LLE variants for small output dimension. HLLE can be performed with function 

locally_linear_embedding

 or
its object-oriented counterpart 

LocallyLinearEmbedding

,
with the keyword 

method = 'hessian'

.
It requires 

n_neighbors > n_components * (n_components + 3) / 2

.

2.2.5.1. Complexity

The HLLE algorithm comprises three stages:
Nearest Neighbors Search. Same as standard LLE
Weight Matrix Construction. Approximately 

.
The first term reflects a similar cost to that of standard LLE. The second term comes from a QR decomposition of the local hessian estimator.
Partial Eigenvalue Decomposition. Same as standard LLE

The overall complexity of standard HLLE is 

.


 : number of
training data points


 : input dimension


 : number of
nearest neighbors


 : output dimension

References:
“Hessian Eigenmaps: Locally linear embedding techniques for high-dimensional
data” Donoho, D. & Grimes, C. Proc Natl Acad Sci USA. 100:5591 (2003)

2.2.6. Spectral Embedding

Spectral Embedding is an approach to calculating a non-linear embedding. Scikit-learn implements Laplacian Eigenmaps, which finds a low dimensional representation of the data using a spectral decomposition of the graph Laplacian. The graph generated can be
considered as a discrete approximation of the low dimensional manifold in the high dimensional space. Minimization of a cost function based on the graph ensures that points close to each other on the manifold are mapped close to each other in the low dimensional
space, preserving local distances. Spectral embedding can be performed with the function 

spectral_embedding

 or
its object-oriented counterpart 

SpectralEmbedding

.

2.2.6.1. Complexity

The Spectral Embedding (Laplacian Eigenmaps) algorithm comprises three stages:
Weighted Graph Construction. Transform the raw input data into graph representation using affinity (adjacency) matrix representation.
Graph Laplacian Construction. unnormalized Graph Laplacian is constructed as 

 for
and normalized one as 

.
Partial Eigenvalue Decomposition. Eigenvalue decomposition is done on graph Laplacian

The overall complexity of spectral embedding is 

.


 : number of
training data points


 : input dimension


 : number of
nearest neighbors


 : output dimension

References:
“Laplacian Eigenmaps for Dimensionality Reduction
and Data Representation” M. Belkin, P. Niyogi, Neural Computation, June 2003; 15 (6):1373-1396

2.2.7. Local Tangent Space Alignment

Though not technically a variant of LLE, Local tangent space alignment (LTSA) is algorithmically similar enough to LLE that it can be put in this category. Rather than focusing on preserving neighborhood distances as in LLE, LTSA seeks to characterize the local
geometry at each neighborhood via its tangent space, and performs a global optimization to align these local tangent spaces to learn the embedding. LTSA can be performed with function 

locally_linear_embedding

 or
its object-oriented counterpart 

LocallyLinearEmbedding

,
with the keyword 

method = 'ltsa'

.

2.2.7.1. Complexity

The LTSA algorithm comprises three stages:
Nearest Neighbors Search. Same as standard LLE
Weight Matrix Construction. Approximately 

.
The first term reflects a similar cost to that of standard LLE.
Partial Eigenvalue Decomposition. Same as standard LLE

The overall complexity of standard LTSA is 

.


 : number of
training data points


 : input dimension


 : number of
nearest neighbors


 : output dimension

References:
“Principal manifolds and nonlinear dimensionality
reduction via tangent space alignment” Zhang, Z. & Zha, H. Journal of Shanghai Univ. 8:406 (2004)

2.2.8. Multi-dimensional Scaling (MDS)

Multidimensional scaling (

MDS

)
seeks a low-dimensional representation of the data in which the distances respect well the distances in the original high-dimensional space.

In general, is a technique used for analyzing similarity or dissimilarity data. 

MDS

 attempts
to model similarity or dissimilarity data as distances in a geometric spaces. The data can be ratings of similarity between objects, interaction frequencies of molecules, or trade indices between countries.

There exists two types of MDS algorithm: metric and non metric. In the scikit-learn, the class 

MDS

 implements
both. In Metric MDS, the input similarity matrix arises from a metric (and thus respects the triangular inequality), the distances between output two points are then set to be as close as possible to the similarity or dissimilarity data. In the non-metric
version, the algorithms will try to preserve the order of the distances, and hence seek for a monotonic relationship between the distances in the embedded space and the similarities/dissimilarities.





Let 

 be the similarity matrix, and 

 the
coordinates of the 

 input points. Disparities 

 are
transformation of the similarities chosen in some optimal ways. The objective, called the stress, is then defined by 

2.2.8.1. Metric MDS

The simplest metric 

MDS

 model,
called absolute MDS, disparities are defined by 

.
With absolute MDS, the value 

 should
then correspond exactly to the distance between point 

 and 

 in
the embedding point.

Most commonly, disparities are set to 

.

2.2.8.2. Nonmetric MDS

Non metric 

MDS

 focuses
on the ordination of the data. If 

,
then the embedding should enforce 

.
A simple algorithm to enforce that is to use a monotonic regression of 

 on 

,
yielding disparities 

 in the
same order as 

.

A trivial solution to this problem is to set all the points on the origin. In order to avoid that, the disparities 

 are
normalized.





References:
“Modern Multidimensional Scaling - Theory and Applications” Borg,
I.; Groenen P. Springer Series in Statistics (1997)
“Nonmetric multidimensional scaling: a numerical method” Kruskal,
J. Psychometrika, 29 (1964)
“Multidimensional scaling by optimizing goodness of
fit to a nonmetric hypothesis” Kruskal, J. Psychometrika, 29, (1964)

2.2.9. t-distributed Stochastic Neighbor Embedding (t-SNE)

t-SNE (

TSNE

)
converts affinities of data points to probabilities. The affinities in the original space are represented by Gaussian joint probabilities and the affinities in the embedded space are represented by Student’s t-distributions. This allows t-SNE to be particularly
sensitive to local structure and has a few other advantages over existing techniques:
Revealing the structure at many scales on a single map
Revealing data that lie in multiple, different, manifolds or clusters
Reducing the tendency to crowd points together at the center

While Isomap, LLE and variants are best suited to unfold a single continuous low dimensional manifold, t-SNE will focus on the local structure of the data and will tend to extract clustered local groups of samples as highlighted on the S-curve example. This
ability to group samples based on the local structure might be beneficial to visually disentangle a dataset that comprises several manifolds at once as is the case in the digits dataset.

The Kullback-Leibler (KL) divergence of the joint probabilities in the original space and the embedded space will be minimized by gradient descent. Note that the KL divergence is not convex, i.e. multiple restarts with different initializations will end up
in local minima of the KL divergence. Hence, it is sometimes useful to try different seeds and select the embedding with the lowest KL divergence.

The disadvantages to using t-SNE are roughly:
t-SNE is computationally expensive, and can take several hours on million-sample datasets where PCA will finish in seconds or minutes
The Barnes-Hut t-SNE method is limited to two or three dimensional embeddings.
The algorithm is stochastic and multiple restarts with different seeds can yield different embeddings. However, it is perfectly legitimate to pick the embedding with the least error.
Global structure is not explicitly preserved. This is problem is mitigated by initializing points with PCA (using init=’pca’).

2.2.9.1. Optimizing t-SNE

The main purpose of t-SNE is visualization of high-dimensional data. Hence, it works best when the data will be embedded on two or three dimensions.

Optimizing the KL divergence can be a little bit tricky sometimes. There are five parameters that control the optimization of t-SNE and therefore possibly the quality of the resulting embedding:
perplexity
early exaggeration factor
learning rate
maximum number of iterations
angle (not used in the exact method)

The perplexity is defined as 

 where 

 is
the Shannon entropy of the conditional probability distribution. The perplexity of a 

-sided
die is 

, so that 

 is
effectively the number of nearest neighbors t-SNE considers when generating the conditional probabilities. Larger perplexities lead to more nearest neighbors and less sensitive to small structure. Conversely a lower perplexity considers a smaller number of
neighbors, and thus ignores more global information in favour of the local neighborhood. As dataset sizes get larger more points will be required to get a reasonable sample of the local neighborhood, and hence larger perplexities may be required. Similarly
noisier datasets will require larger perplexity values to encompass enough local neighbors to see beyond the background noise.

The maximum number of iterations is usually high enough and does not need any tuning. The optimization consists of two phases: the early exaggeration phase and the final optimization. During early exaggeration the joint probabilities in the original space will
be artificially increased by multiplication with a given factor. Larger factors result in larger gaps between natural clusters in the data. If the factor is too high, the KL divergence could increase during this phase. Usually it does not have to be tuned.
A critical parameter is the learning rate. If it is too low gradient descent will get stuck in a bad local minimum. If it is too high the KL divergence will increase during optimization. More tips can be found in Laurens van der Maaten’s FAQ (see references).
The last parameter, angle, is a tradeoff between performance and accuracy. Larger angles imply that we can approximate larger regions by a single point,leading to better speed but less accurate results.

“How to Use t-SNE Effectively” provides a good discussion of the effects of the various parameters,
as well as interactive plots to explore the effects of different parameters.

2.2.9.2. Barnes-Hut t-SNE

The Barnes-Hut t-SNE that has been implemented here is usually much slower than other manifold learning algorithms. The optimization is quite difficult and the computation of the gradient is 

,
where 

 is the number of output dimensions
and 

 is the number of samples. The Barnes-Hut
method improves on the exact method where t-SNE complexity is 

,
but has several other notable differences:
The Barnes-Hut implementation only works when the target dimensionality is 3 or less. The 2D case is typical when building visualizations.
Barnes-Hut only works with dense input data. Sparse data matrices can only be embedded with the exact method or can be approximated by a dense low rank projection for instance using 

sklearn.decomposition.TruncatedSVD

Barnes-Hut is an approximation of the exact method. The approximation is parameterized with the angle parameter, therefore the angle parameter is unused when method=”exact”
Barnes-Hut is significantly more scalable. Barnes-Hut can be used to embed hundred of thousands of data points while the exact method can handle thousands of samples before becoming computationally intractable

For visualization purpose (which is the main use case of t-SNE), using the Barnes-Hut method is strongly recommended. The exact t-SNE method is useful for checking the theoretically properties of the embedding possibly in higher dimensional space but limit
to small datasets due to computational constraints.

Also note that the digits labels roughly match the natural grouping found by t-SNE while the linear 2D projection of the PCA model yields a representation where label regions largely overlap. This is a strong clue that this data can be well separated by non
linear methods that focus on the local structure (e.g. an SVM with a Gaussian RBF kernel). However, failing to visualize well separated homogeneously labeled groups with t-SNE in 2D does not necessarily implie that the data cannot be correctly classified by
a supervised model. It might be the case that 2 dimensions are not enough low to accurately represents the internal structure of the data.

References:
“Visualizing High-Dimensional Data Using t-SNE” van der
Maaten, L.J.P.; Hinton, G. Journal of Machine Learning Research (2008)
“t-Distributed Stochastic Neighbor Embedding” van der Maaten, L.J.P.
“Accelerating t-SNE using Tree-Based Algorithms.” L.J.P.
van der Maaten. Journal of Machine Learning Research 15(Oct):3221-3245, 2014.

2.2.10. Tips on practical use

Make sure the same scale is used over all features. Because manifold learning methods are based on a nearest-neighbor search, the algorithm may perform poorly otherwise. See StandardScaler for
convenient ways of scaling heterogeneous data.
The reconstruction error computed by each routine can be used to choose the optimal output dimension. For a 

-dimensional
manifold embedded in a 

-dimensional parameter
space, the reconstruction error will decrease as 

n_components

 is
increased until 

n_components == d

.
Note that noisy data can “short-circuit” the manifold, in essence acting as a bridge between parts of the manifold that would otherwise be well-separated. Manifold learning on noisy and/or incomplete data is an active area of research.
Certain input configurations can lead to singular weight matrices, for example when more than two points in the dataset are identical, or when the data is split into disjointed groups. In this case, 

solver='arpack'

 will
fail to find the null space. The easiest way to address this is to use 

solver='dense'

 which
will work on a singular matrix, though it may be very slow depending on the number of input points. Alternatively, one can attempt to understand the source of the singularity: if it is due to disjoint sets, increasing 

n_neighbors

 may
help. If it is due to identical points in the dataset, removing these points may help.

See also
   

完全随机树嵌入 can also
be useful to derive non-linear representations of feature space, also it does not perform dimensionality reduction.

中文文档: http://sklearn.apachecn.org/cn/stable/modules/manifold.html

英文文档: http://sklearn.apachecn.org/en/stable/modules/manifold.html









官方文档: http://scikit-learn.org/stable/

GitHub: https://github.com/apachecn/scikit-learn-doc-zh（觉得不错麻烦给个
Star，我们一直在努力）

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