Merge pull request #163 from infinnie/patch-2

Update neural-networks-2.md

Author: ranjaykrishna

neural-networks-1.md

@@ -108,7 +108,7 @@ Every activation function (or *non-linearity*) takes a single number and perform
 - (+) Compared to tanh/sigmoid neurons that involve expensive operations (exponentials, etc.), the ReLU can be implemented by simply thresholding a matrix of activations at zero.
 - (-) Unfortunately, ReLU units can be fragile during training and can "die". For example, a large gradient flowing through a ReLU neuron could cause the weights to update in such a way that the neuron will never activate on any datapoint again. If this happens, then the gradient flowing through the unit will forever be zero from that point on. That is, the ReLU units can irreversibly die during training since they can get knocked off the data manifold. For example, you may find that as much as 40% of your network can be "dead" (i.e. neurons that never activate across the entire training dataset) if the learning rate is set too high. With a proper setting of the learning rate this is less frequently an issue.
 
-**Leaky ReLU.** Leaky ReLUs are one attempt to fix the "dying ReLU" problem. Instead of the function being zero when x < 0, a leaky ReLU will instead have a small negative slope (of 0.01, or so). That is, the function computes \\(f(x) = \mathbb{1}(x < 0) (\alpha x) + \mathbb{1}(x>=0) (x) \\) where \\(\alpha\\) is a small constant. Some people report success with this form of activation function, but the results are not always consistent. The slope in the negative region can also be made into a parameter of each neuron, as seen in PReLU neurons, introduced in [Delving Deep into Rectifiers](http://arxiv.org/abs/1502.01852), by Kaiming He et al., 2015. However, the consistency of the benefit across tasks is presently unclear.
+**Leaky ReLU.** Leaky ReLUs are one attempt to fix the "dying ReLU" problem. Instead of the function being zero when x < 0, a leaky ReLU will instead have a small positive slope (of 0.01, or so). That is, the function computes \\(f(x) = \mathbb{1}(x < 0) (\alpha x) + \mathbb{1}(x>=0) (x) \\) where \\(\alpha\\) is a small constant. Some people report success with this form of activation function, but the results are not always consistent. The slope in the negative region can also be made into a parameter of each neuron, as seen in PReLU neurons, introduced in [Delving Deep into Rectifiers](http://arxiv.org/abs/1502.01852), by Kaiming He et al., 2015. However, the consistency of the benefit across tasks is presently unclear.
 
 **Maxout**. Other types of units have been proposed that do not have the functional form \\(f(w^Tx + b)\\) where a non-linearity is applied on the dot product between the weights and the data. One relatively popular choice is the Maxout neuron (introduced recently by [Goodfellow et al.](https://arxiv.org/abs/1302.4389)) that generalizes the ReLU and its leaky version. The Maxout neuron computes the function \\(\max(w_1^Tx+b_1, w_2^Tx + b_2)\\). Notice that both ReLU and Leaky ReLU are a special case of this form (for example, for ReLU we have \\(w_1, b_1 = 0\\)). The Maxout neuron therefore enjoys all the benefits of a ReLU unit (linear regime of operation, no saturation) and does not have its drawbacks (dying ReLU). However, unlike the ReLU neurons it doubles the number of parameters for every single neuron, leading to a high total number of parameters.
 

neural-networks-2.md

@@ -55,7 +55,7 @@ where the columns of `U` are the eigenvectors and `S` is a 1-D array of the sing
 Xrot = np.dot(X, U) # decorrelate the data
 ```
 
-Notice that the columns of `U` are a set of orthonormal vectors (norm of 1, and orthogonal to each other), so they can be regarded as basis vectors. The projection therefore corresponds to a rotation of the data in `X` so that the new axes are the eigenvectors. If we were to compute the covariance matrix of `Xrot`, we would see that it is now diagonal. A nice property of `np.linalg.svd` is that in its returned value `U`, the eigenvector columns are sorted by their eigenvalues. We can use this to reduce the dimensionality of the data by only using the top few eigenvectors, and discarding the dimensions along which the data has no variance. This is also sometimes refereed to as [Principal Component Analysis (PCA)](http://en.wikipedia.org/wiki/Principal_component_analysis) dimensionality reduction:
+Notice that the columns of `U` are a set of orthonormal vectors (norm of 1, and orthogonal to each other), so they can be regarded as basis vectors. The projection therefore corresponds to a rotation of the data in `X` so that the new axes are the eigenvectors. If we were to compute the covariance matrix of `Xrot`, we would see that it is now diagonal. A nice property of `np.linalg.svd` is that in its returned value `U`, the eigenvector columns are sorted by their eigenvalues. We can use this to reduce the dimensionality of the data by only using the top few eigenvectors, and discarding the dimensions along which the data has no variance. This is also sometimes referred to as [Principal Component Analysis (PCA)](http://en.wikipedia.org/wiki/Principal_component_analysis) dimensionality reduction:
 
 ```python
 Xrot_reduced = np.dot(X, U[:,:100]) # Xrot_reduced becomes [N x 100]