fixing bug

Author: karpathy

convolutional-networks.md

@@ -197,7 +197,7 @@ This approach has the downside that it can use a lot of memory, since some value
 
 **1x1 convolution**. As an aside, several papers use 1x1 convolutions, as first investigated by [Network in Network](http://arxiv.org/abs/1312.4400). Some people are at first confused to see 1x1 convolutions especially when they come from signal processing background. Normally signals are 2-dimensional so 1x1 convolutions do not make sense (it's just pointwise scaling). However, in ConvNets this is not the case because one must remember that we operate over 3-dimensional volumes, and that the filters always extend through the full depth of the input volume. For example, if the input is [32x32x3] then doing 1x1 convolutions would effectively be doing 3-dimensional dot products (since the input depth is 3 channels).
 
-**Dilated convolutions.** A recent development (e.g. see [paper by Fisher Yu and Vladlen Koltun](https://arxiv.org/abs/1511.07122)) is to introduce one more hyperparameter to the CONV layer called the *dilation*. So far we've only dicussed CONV filters that are contiguous. However, it's possible to have filters that have spaces between each cell, called dilation. As an example, in one dimension a filter `w` of size 3 would compute over input `x` the following: `w[0]*x[0] + w[1]*x[1] + w[2]*x[2]`. This is dilation of 0. For dilation 1 the filter would instead compute `w[0]*x[0] + w[1]*x[2] + w[2]*x[4]`; In other words there is a gap of 1 between the applications. This can be very useful in some settings to use in conjunction with 0-dilated filters because it allows you to merge spatial information across the inputs much more agressively with fewer layers. For example, if you stack two 3x3 CONV layers on top of each other than you can convince yourself that the neurons on the 2nd layer are a function of a 5x5 patch of the input (we would say that the *effective receptive field* of these neurons is 5x5). If we use dilated convolutions then this effective receptive field would grow much quicker (indeed, exponentially).
+**Dilated convolutions.** A recent development (e.g. see [paper by Fisher Yu and Vladlen Koltun](https://arxiv.org/abs/1511.07122)) is to introduce one more hyperparameter to the CONV layer called the *dilation*. So far we've only dicussed CONV filters that are contiguous. However, it's possible to have filters that have spaces between each cell, called dilation. As an example, in one dimension a filter `w` of size 3 would compute over input `x` the following: `w[0]*x[0] + w[1]*x[1] + w[2]*x[2]`. This is dilation of 0. For dilation 1 the filter would instead compute `w[0]*x[0] + w[1]*x[2] + w[2]*x[4]`; In other words there is a gap of 1 between the applications. This can be very useful in some settings to use in conjunction with 0-dilated filters because it allows you to merge spatial information across the inputs much more agressively with fewer layers. For example, if you stack two 3x3 CONV layers on top of each other than you can convince yourself that the neurons on the 2nd layer are a function of a 5x5 patch of the input (we would say that the *effective receptive field* of these neurons is 5x5). If we use dilated convolutions then this effective receptive field would grow much quicker.
 
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