t-SNE算法详解：高维数据可视化新法

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t-SNE算法教程深入讲解了一种强大的数据可视化工具，由LaurensvanderMaaten和GeoffreyHinton在2008年的《JournalofMachineLearningResearch》上发表。该方法旨在将高维数据映射到二维或三维空间，使得复杂的数据结构变得直观易懂。与Stochastic Neighbor Embedding（SNE）的早期版本不同，t-SNE优化过程更为便捷，特别在处理多尺度结构时表现出色。 t-SNE的核心在于它通过减少数据点聚集在地图中心的趋势，避免了传统可视化技术可能出现的“crowding”问题，即在低维度映射中，相似的数据点密集堆叠在一起。这使得t-SNE非常适合用于高维数据，如图像数据，其中可能包含多个对象类别，且这些类别之间的关系是多角度和层次的。例如，一张图片可能从不同的视角展示同一个物体，这些图片在高维空间中形成多个相关的低维嵌入。该算法的优势在于其能够揭示数据中的复杂模式和潜在结构，无论这些结构是在全局还是局部层次上。这使得t-SNE在许多领域，如机器学习、计算机视觉、生物信息学等，被广泛应用于数据探索、特征可视化、异常检测以及降维等领域。通过t-SNE生成的地图，研究者和数据分析师可以直观地识别出数据集中的集群、趋势或者潜在的群组结构，这对于理解数据的内在联系和特征分布至关重要。 t-SNE算法是一种强大的工具，它简化了高维数据的可视化过程，使得非专业人士也能快速理解和解释复杂的高维数据集。掌握并应用t-SNE不仅有助于提升数据分析的质量，也促进了科学研究和商业决策的可视化表达能力。

VISUALIZING DATA USING T-SNE

where Y

(t)

indicates the solution at iteration t, η indicates the learning rate, and α(t) represents the

momentum at iteration t.

In addition, in the early stages of the optimization, Gaussian noise is added to the map points after

each iteration. Gradually reducing the variance of this noise performs a type of simulated annealing

that helps the optimization to escape from poor local minima in the cost function. If the variance

of the noise changes very slowly at the critical point at which the global structure of the map starts

to form, SNE tends to ﬁnd maps with a better global organization. Unfortunately, this requires

sensible choices of the initial amount of Gaussian noise and the rate at which it decays. Moreover,

these choices interact with the amount of momentum and the step size that are employed in the

gradient descent. It is therefore common to run the optimization several times on a dataset to ﬁnd

appropriate values for the parameters

. In this respect, SNE is inferior to methods that allow convex

optimization and it would be useful to ﬁnd an optimization method that gives good results without

requiring the extra computation time and parameter choices introduced by the simulated annealing.

3. t-Distributed Stochastic Neighbor Embedding

Section 2 discussed SNE as it was presented by Hinton and Roweis (2002). Although SNE con-

structs reasonably good visualizations, it is hampered by a cost function that is difﬁcult to optimize

and by a problem we refer to as the “crowding problem”. In this section, we present a new technique

called “t-Distributed Stochastic Neighbor Embedding” or “t-SNE” that aims to alleviate these prob-

lems. The cost function used by t-SNE differs from the one used by SNE in two ways: (1) it uses

a symmetrized version of the SNE cost function with simpler gradients that was brieﬂy introduced

by Cook et al. (2007) and (2) it uses a Student-t distribution rather than a Gaussian to compute the

similarity between two points in the low-dimensional space. t-SNE employs a heavy-tailed distri-

bution in the low-dimensional space to alleviate both the crowding problem and the optimization

problems of SNE.

In this section, we ﬁrst discuss the symmetric version of SNE (subsection 3.1). Subsequently, we

discuss the crowding problem (subsection 3.2), and the use of heavy-tailed distributions to address

this problem (subsection 3.3). We conclude the section by describing our approach to the optimiza-

tion of the t-SNE cost function (subsection 3.4).

3.1 Symmetric SNE

As an alternative to minimizing the sum of the Kullback-Leibler divergences between the condi-

tional probabilities p

j|i

and q

j|i

, it is also possible to minimize a single Kullback-Leibler divergence

between a joint probability distribution, P , in the high-dimensional space and a joint probability

distribution, Q, in the low-dimensional space:

C = KL(P ||Q) =

log

. (8)

where again, we set p

and q

to zero. We refer to this type of SNE as symmetric SNE, because it

has the property that p

= p

and q

= q

for ∀i, j. In symmetric SNE, the pairwise similarities

4. Picking the best map after several runs as a visualization of the data is not nearly as problematic as picking the model

that does best on a test set during supervised learning. In visualization, the aim is to see the structure in the training

data, not to generalize to held out test data.

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