fixed a few typos.

classification.md

@@ -24,17 +24,17 @@ This is an introductory lecture designed to introduce people from outside of Com
 
 <div class="fig figcenter fighighlight">
   <img src="/assets/classify.png">
-  <div class="figcaption">The task in Image Classification is to predict a single label (or a distribution over labels as shown here to indicates our confidence) for a given image. Images are 3-dimensional arrays of integers from 0 to 255, of size Width x Height x 3. The 3 is due to the three color channels Red, Green, Blue.</div>
+  <div class="figcaption">The task in Image Classification is to predict a single label (or a distribution over labels as shown here to indicate our confidence) for a given image. Images are 3-dimensional arrays of integers from 0 to 255, of size Width x Height x 3. The 3 represents the three color channels Red, Green, Blue.</div>
 </div>
 
 **Challenges**. Since this task of recognizing a visual concept (e.g. cat) is relatively trivial for a human to perform, it is worth considering the challenges involved from the perspective of a Computer Vision algorithm. As we present (an inexhaustive) list of challenges below, keep in mind the raw representation of images as a 3-D array of brightness values:
 
 - **Viewpoint variation**. A single instance of an object can be oriented in many ways with respect to the camera.
-- **Scale variation**. Visual classes often exhibit variation in their size (size in the real world, not only in terms of their extent in the image)
-- **Deformation**. Many objects of interest are not rigid bodies and can be deformed in extreme ways
+- **Scale variation**. Visual classes often exhibit variation in their size (size in the real world, not only in terms of their extent in the image).
+- **Deformation**. Many objects of interest are not rigid bodies and can be deformed in extreme ways.
 - **Occlusion**. The objects of interest can be occluded. Sometimes only a small portion of an object (as little as few pixels) could be visible.
 - **Illumination conditions**. The effects of illumination are drastic on the pixel level.
-- **Background clutter**. The objects of interest may *blend* into their environment, making them hard to identify
+- **Background clutter**. The objects of interest may *blend* into their environment, making them hard to identify.
 - **Intra-class variation**. The classes of interest can often be relatively broad, such as *chair*. There are many different types of these objects, each with their own appearance.
 
 A good image classification model must be invariant to the cross product of all these variations, while simultaneously retaining sensitivity to the inter-class variations.
@@ -44,7 +44,7 @@ A good image classification model must be invariant to the cross product of all
   <div class="figcaption"></div>
 </div>
 
-**Data-driven approach**. How might we go about writing an algorithm that can classify images into distinct categories? Unlike writing an algorithm for, for example, sorting a list of numbers, it is not obvious how one might write an algorithm for identifying cats in images.Therefore, instead of trying to specify what every one of the categories of interest look like directly in code, the approach that we will take is not unlike one you would take with a child: we're going to provide the computer with many examples of each class and then develop learning algorithms that look at these examples and learn about the visual appearance of each class. This approach is referred to as a *data-driven approach*, since it relies on first accumulating a *training dataset* of labeled images. Here is an example of what such a dataset might look like:
+**Data-driven approach**. How might we go about writing an algorithm that can classify images into distinct categories? Unlike writing an algorithm for, for example, sorting a list of numbers, it is not obvious how one might write an algorithm for identifying cats in images. Therefore, instead of trying to specify what every one of the categories of interest look like directly in code, the approach that we will take is not unlike one you would take with a child: we're going to provide the computer with many examples of each class and then develop learning algorithms that look at these examples and learn about the visual appearance of each class. This approach is referred to as a *data-driven approach*, since it relies on first accumulating a *training dataset* of labeled images. Here is an example of what such a dataset might look like:
 
 <div class="fig figcenter fighighlight">
   <img src="/assets/trainset.jpg">
@@ -53,9 +53,9 @@ A good image classification model must be invariant to the cross product of all
 
 **The image classification pipeline**. We've seen that the task in Image Classification is to take an array of pixels that represents a single image and assign a label to it. Our complete pipeline can be formalized as follows:
 
-- **Input:**. Our input consists of a set of *N* images, each labeled with one of *K* different classes. We refer to this data as the *training set*.
-- **Learning:**. Our task is to use the training set to learn what every one of the classes looks like. We refer to this step as *training a classifier*, or *learning a model*.
-- **Evaluation:**. In the end, we evaluate the quality of the classifier by asking it to predict labels for a new set of images that it has never seen before. We will then compare the true labels of these images to the ones predicted by the classifier. Intuitively, we're hoping that a lot of the predictions match up with the true answers  (which we call the *ground truth*).
+- **Input:** Our input consists of a set of *N* images, each labeled with one of *K* different classes. We refer to this data as the *training set*.
+- **Learning:** Our task is to use the training set to learn what every one of the classes looks like. We refer to this step as *training a classifier*, or *learning a model*.
+- **Evaluation:** In the end, we evaluate the quality of the classifier by asking it to predict labels for a new set of images that it has never seen before. We will then compare the true labels of these images to the ones predicted by the classifier. Intuitively, we're hoping that a lot of the predictions match up with the true answers  (which we call the *ground truth*).
 
 <a name='nn'></a>
 ### Nearest Neighbor Classifier
@@ -68,7 +68,7 @@ As our first approach, we will develop what we call a **Nearest Neighbor Classif
   <div class="figcaption">Left: Example images from the <a href="http://www.cs.toronto.edu/~kriz/cifar.html">CIFAR-10 dataset</a>. Right: first column shows a few test images and next to each we show the top 10 nearest neighbors in the training set according to pixel-wise difference.</div>
 </div>
 
-Suppose now that we are given the CIFAR-10 training set of 50,000 images (5,000 images for every one of the labels), and we wish to label the remaining 10,000. The nearest neighbor classifier will take a test image, compare it to every single one of the training images, and predict the label of the closest training image. In the image above and on the right you can see an example result of such procedure for 10 example test images. Notice that in only about 3 out of 10 examples an image of the same class is retrieved, while in the other 7 examples this is not the case. For example, in the 8th row the nearest training image to the horse head is a red car, presumably due to the strong black background. As a result, this image of a horse would in this case be mislabeled as a car.
+Suppose now that we are given the CIFAR-10 training set of 50,000 images (5,000 images for every one of the labels), and we wish to label the remaining 10,000. The nearest neighbor classifier will take a test image, compare it to every single one of the training images, and predict the label of the closest training image. In the image above and on the right you can see an example result of such a procedure for 10 example test images. Notice that in only about 3 out of 10 examples an image of the same class is retrieved, while in the other 7 examples this is not the case. For example, in the 8th row the nearest training image to the horse head is a red car, presumably due to the strong black background. As a result, this image of a horse would in this case be mislabeled as a car.
 
 You may have noticed that we left unspecified the details of exactly how we compare two images, which in this case are just two blocks of 32 x 32 x 3. One of the simplest possibilities is to compare the images pixel by pixel and add up all the differences. In other words, given two images and representing them as vectors \\( I\_1, I\_2 \\) , a reasonable choice for comparing them might be the **L1 distance**:
 
@@ -83,7 +83,7 @@ Where the sum is taken over all pixels. Here is the procedure visualized:
   <div class="figcaption">An example of using pixel-wise differences to compare two images with L1 distance (for one color channel in this example). Two images are subtracted elementwise and then all differences are added up to a single number. If two images are identical the result will be zero. But if the images are very different the result will be large.</div>
 </div>
 
-Lets also look at how we might implement the classifier in code. First, lets load the CIFAR-10 data into memory as 4 arrays: the training data/labels and the test data/labels. In the code below, `Xtr` (of size 50,000 x 32 x 32 x 3) holds all the images in the training set, and a corresponding 1-dimensional array `Ytr` (of length 50,000) holds the training labels (from 0 to 9):
+Let's also look at how we might implement the classifier in code. First, let's load the CIFAR-10 data into memory as 4 arrays: the training data/labels and the test data/labels. In the code below, `Xtr` (of size 50,000 x 32 x 32 x 3) holds all the images in the training set, and a corresponding 1-dimensional array `Ytr` (of length 50,000) holds the training labels (from 0 to 9):
 
 ```python
 Xtr, Ytr, Xte, Yte = load_CIFAR10('data/cifar10/') # a magic function we provide
@@ -103,7 +103,7 @@ Yte_predict = nn.predict(Xte_rows) # predict labels on the test images
 print 'accuracy: %f' % ( np.mean(Yte_predict == Yte) )
 ```
 
-Notice that as an evaluation criterion, it is common to use the **accuracy**, which measures the fraction of predictions that were correct. Notice that all classifiers we will build satisfy this one common API: they have a `train(X,y)` function that takes the data and the labels to learn from. Internally, the class should build some kind of model of the labels and how they can be predicted from the data. And then there is a `predict(X)` function which takes new data and predicts the labels. Of course, we've left out the meat of things - the actual classifier itself. Here is an implementation of a simple Nearest Neighbor classifier with the L1 distance that satisfies this template:
+Notice that as an evaluation criterion, it is common to use the **accuracy**, which measures the fraction of predictions that were correct. Notice that all classifiers we will build satisfy this one common API: they have a `train(X,y)` function that takes the data and the labels to learn from. Internally, the class should build some kind of model of the labels and how they can be predicted from the data. And then there is a `predict(X)` function, which takes new data and predicts the labels. Of course, we've left out the meat of things - the actual classifier itself. Here is an implementation of a simple Nearest Neighbor classifier with the L1 distance that satisfies this template:
 
 ```python
 import numpy as np
@@ -135,7 +135,7 @@ class NearestNeighbor:
     return Ypred
 ```
 
-If you ran this code you would see that this classifier only achieves **38.6%** on CIFAR-10. That's more impressive than guessing at random (which would give 10% accuracy since there are 10 classes), but nowhere near human performance (which is [estimated at about 94%](http://karpathy.github.io/2011/04/27/manually-classifying-cifar10/)) or near state-of-the-art Convolutional Neural Networks that achieve about 95%, matching human accuracy (see the [leaderboard](http://www.kaggle.com/c/cifar-10/leaderboard) of a recent Kaggle competition on CIFAR-10).
+If you ran this code, you would see that this classifier only achieves **38.6%** on CIFAR-10. That's more impressive than guessing at random (which would give 10% accuracy since there are 10 classes), but nowhere near human performance (which is [estimated at about 94%](http://karpathy.github.io/2011/04/27/manually-classifying-cifar10/)) or near state-of-the-art Convolutional Neural Networks that achieve about 95%, matching human accuracy (see the [leaderboard](http://www.kaggle.com/c/cifar-10/leaderboard) of a recent Kaggle competition on CIFAR-10).
 
 **The choice of distance.** 
 There are many other ways of computing distances between vectors. Another common choice could be to instead use the **L2 distance**, which has the geometric interpretation of computing the euclidean distance between two vectors. The distance takes the form: