A few updates in the text.

Author: rossant

main.md

@@ -10,7 +10,7 @@ The answer lies in the observation that many real-world datasets have a low intr
 
 This is the topic of [**manifold learning**](http://en.wikipedia.org/wiki/Nonlinear_dimensionality_reduction), also called **nonlinear dimensionality reduction**, a branch of machine learning (more specifically, _unsupervised learning_). It is still an active area of research today to develop algorithms that can automatically recover a hidden structure in a high-dimensional dataset.
 
-This post is an introduction to a popular dimensonality reduction algorithm: [**t-distributed stochastic neighbor embedding (t-SNE)**](http://en.wikipedia.org/wiki/T-distributed_stochastic_neighbor_embedding). Developed by [Laurens van der Maaten](http://lvdmaaten.github.io/) and [Geoffrey Hinton](http://www.cs.toronto.edu/~hinton/), this algorithm has been successfully applied to many real-world datasets. Here, we'll follow the original paper and describe the key mathematical concepts of the method, when applied to a toy dataset (handwritten digits). We'll use Python and the [scikit-learn](http://scikit-learn.org/stable/index.html) library.
+This post is an introduction to a popular dimensonality reduction algorithm: [**t-distributed stochastic neighbor embedding (t-SNE)**](http://en.wikipedia.org/wiki/T-distributed_stochastic_neighbor_embedding). Developed by [Laurens van der Maaten](http://lvdmaaten.github.io/) and [Geoffrey Hinton](http://www.cs.toronto.edu/~hinton/) (see the [original paper here](http://jmlr.csail.mit.edu/papers/volume9/vandermaaten08a/vandermaaten08a.pdf)), this algorithm has been successfully applied to many real-world datasets. Here, we'll follow the original paper and describe the key mathematical concepts of the method, when applied to a toy dataset (handwritten digits). We'll use Python and the [scikit-learn](http://scikit-learn.org/stable/index.html) library.
 
 ## Visualizing handwritten digits
 
@@ -256,6 +256,8 @@ Here is an illustration of a dynamic graph layout based on a similar idea. Nodes
 
 
 
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+
 ## Algorithm
 
 Remarkably, this physical analogy stems naturally from the mathematical algorithm. It corresponds to minimizing the [Kullback-Leiber](http://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) divergence between the two distributions <span class="math-tex" data-type="tex">\\(\big(p_{ij}\big)\\)</span> and <span class="math-tex" data-type="tex">\\(\big(q_{ij}\big)\\)</span>:
@@ -394,7 +396,7 @@ animation.write_gif("images/animation_matrix.gif", fps=20)
 
 ## The t-Student distribution
 
-Let's now explain the choice of the t-Student distribution for the map points, while a normal distribution is used for the data points. It is well known that the volume of the <span class="math-tex" data-type="tex">\\(N\\)</span>-dimensional ball of radius <span class="math-tex" data-type="tex">\\(r\\)</span> scales as <span class="math-tex" data-type="tex">\\(r^N\\)</span>. When <span class="math-tex" data-type="tex">\\(N\\)</span> is large, if we pick random points uniformly in the ball, most points will be close to the surface, and very few will be near the center.
+Let's now explain the choice of the t-Student distribution for the map points, while a normal distribution is used for the data points. [It is well known that](http://en.wikipedia.org/wiki/Volume_of_an_n-ball) the volume of the <span class="math-tex" data-type="tex">\\(N\\)</span>-dimensional ball of radius <span class="math-tex" data-type="tex">\\(r\\)</span> scales as <span class="math-tex" data-type="tex">\\(r^N\\)</span>. When <span class="math-tex" data-type="tex">\\(N\\)</span> is large, if we pick random points uniformly in the ball, most points will be close to the surface, and very few will be near the center.
 
 This is illustrated by the following simulation, showing the distribution of the distances of these points, for different dimensions.
 
@@ -425,9 +427,9 @@ plt.savefig('images/spheres.png', dpi=100, bbox_inches='tight')
 
 ![Spheres](images/spheres.png)
 
-When reducing the dimensionality of a dataset, if we used the same Gaussian distribution for the data points and the map points, we could get an _imbalance_ among the neighbors of a given point. This imbalance would lead to an excess of attraction forces and a sometimes unappealing mapping. This is actually what happens in the original SNE algorithm, by Hinton and Roweis (2002).
+When reducing the dimensionality of a dataset, if we used the same Gaussian distribution for the data points and the map points, we would get an _imbalance_ in the distribution of the distances of a point's neighbors. This is because the distribution of the distances is so different between a high-dimensional space and a low-dimensional space. Yet, the algorithm tries to reproduce the same distances in the two spaces. This imbalance would lead to an excess of attraction forces and a sometimes unappealing mapping. This is actually what happens in the original SNE algorithm, by [Hinton and Roweis (2002)](http://www.cs.toronto.edu/~fritz/absps/sne.pdf).
 
-The t-SNE algorithm works around this problem by using a t-Student with one degree of freedom (or Cauchy) distribution for the map points. This distribution has a much heavier tail than the Gaussian distribution, which _compensates_ the original imbalance. For a given data similarity between two data points, the two corresponding map points will need to be much further apart in order for their similarity to match the data similarity. This is can be seen in the following plot.
+The t-SNE algorithm works around this problem by using a t-Student with one degree of freedom (or Cauchy) distribution for the map points. This distribution has a much heavier tail than the Gaussian distribution, which _compensates_ the original imbalance. For a given similarity between two data points, the two corresponding map points will need to be much further apart in order for their similarity to match the data similarity. This can be seen in the following plot.
 
 <pre data-code-language="python"
      data-executable="true"
@@ -443,11 +445,13 @@ plt.savefig('images/distributions.png', dpi=100)
 
 ![Gaussian and Cauchy distributions](images/distributions.png)
 
+Using this distribution leads to more effective data visualizations, where clusters of points are more distinctly separated.
+
 ## Conclusion
 
 The t-SNE algorithm provides an effective method to visualize a complex dataset. It successfully uncovers hidden structures in the data, exposing natural clusters and smooth nonlinear variations along the dimensions. It has been implemented in many languages, including Python, and it can be easily used thanks to the scikit-learn library.
 
-The references below link to some optimizations and improvements that can be made to the algorithm and implementations. In particular, the algorithm described here is quadratic in the number of samples, which makes it unscalable to large datasets. One could for example obtain an <span class="math-tex" data-type="tex">\\(O(N \log N)\\)</span> complexity by using the Barnes-Hut algorithm to accelerate the N-body simulation via a quadtree or an octree.
+The references below describe some optimizations and improvements that can be made to the algorithm and implementations. In particular, the algorithm described here is quadratic in the number of samples, which makes it unscalable to large datasets. One could for example obtain an <span class="math-tex" data-type="tex">\\(O(N \log N)\\)</span> complexity by using the Barnes-Hut algorithm to accelerate the N-body simulation via a quadtree or an octree.
 
 ## References