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In the last section we introduced the problem of Image Classification, which is the task of assigning a single label to an image from a fixed set of categories. Morever, we described the k-Nearest Neighbor kNN classifier which labels images by comparing them to annotated images from the training set. As we saw, kNN has a number of disadvantages:. We are now going to develop a more powerful approach to image classification that we will eventually naturally extend to entire Neural 2311d and Convolutional Neural Networks.

The approach will have two major components: We will then cast this as an optimization problem in which we will minimize the loss function with respect to the parameters of the score function.

The first component of this approach is to define the score function that maps the pixel values of an image to confidence scores for each class. We will develop the approach with a concrete example. That is, we have N examples each with a dimensionality D and K distinct categories. In this module we will start out with arguably the simplest possible function, a linear mapping:.

The matrix W of size [K x D]and the vector b of size [K x 1] are the parameters of the function.

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However, you will often hear people use the terms weights and parameters interchangeably. Convolutional Nax Networks will map image pixels to scores exactly as shown above, but the mapping f will be more complex and will contain more 23d.

Notice that a ma classifier computes the score of a class as a weighted sum of all of its pixel values across all 3 of its color channels. Depending on precisely what values we set for these weights, the function has the capacity to like or dislike depending on the sign of each weight certain colors at certain positions in the image.

Analogy of images as high-dimensional points. Since the images are stretched into high-dimensional column vectors, we can interpret each image as a single point in this space e.

Analogously, the entire dataset is a labeled set of points.

Since we defined the score of each class as a weighted 2231d of all image pixels, each class score is a linear function over this space. We cannot visualize dimensional spaces, but if we imagine squashing all those dimensions into only two dimensions, then we can try to visualize what the classifier might be doing:.

Interpretation of linear classifiers as template matching. With this terminology, the linear classifier is doing template matching, where the templates are learned. Another way to think of it is that we are still effectively doing Nearest Neighbor, but instead mxa having thousands of training images we are only using a single image per class although we will learn it, and it does not necessarily have to be one of the images in the training setand we use the negative inner product as the distance instead of the L1 or L2 distance.

Additionally, note that the horse template seems to contain a two-headed horse, which is due to both left and right facing horses in the dataset. The linear classifier merges these two modes of horses in the data into a single template.

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Similarly, the car classifier seems to have merged several modes into a single template which has to identify cars from all sides, and of all colors. In particular, this template ended up being red, which hints that there are mxa red cars in the CIFAR dataset than of any other color.

The linear classifier is too weak to properly account for different-colored cars, but as we will see later neural networks will allow us to perform this task. Looking ahead a bit, a neural network 231r be able to develop intermediate neurons in its hidden layers that could detect specific car types e. Recall that we defined the score function as:. With the extra dimension, the new score function will simplify to a single matrix multiply:. An illustration might help clarify:.

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As a quick note, in the examples above we used the raw pixel values which range from [0…]. In Machine Learning, it is a very common practice to always perform normalization of your input features in the case of images, every pixel is thought of as a feature.

In particular, it is important to center your data by subtracting the mean from every feature. In the case of images, this corresponds to computing a mean image across the training images and subtracting it from every image to get images where the pixels range from approximately [ … ]. Further common preprocessing is to scale each input feature so that its values range from [-1, 1]. Of these, zero mean centering is arguably more important but we will have to wait for its justification until we understand the dynamics of gradient descent.

We fed in the pixels that depict a cat but the cat score came out very low We are going to measure our unhappiness with outcomes such as this one with a loss function or sometimes also referred to as the cost function or the objective.

There are several ways to define the details of the loss function. For example, the score for the j-th class is the j-th element: The Multiclass SVM loss for the i-th example is then formalized as follows:.

Lets unpack this with an example 23d see how it works. We get zero loss for this pair because the correct class score 13 was greater than the incorrect class score -7 by at kax the margin In fact the difference was 20, which is much greater than 10 but the SVM only cares that the difference is at least 10; Any additional difference above the margin is clamped at zero with the max operation. The difference was only 2, mac is why the loss comes out to 8 i. If this is not the case, we will accumulate loss.

The unsquared version is more standard, but in some datasets mxx squared hinge loss can work better. This can be determined during cross-validation. There is one bug with the loss function we presented above.

Suppose that we have a dataset and a set of parameters W that correctly classify every example i. The issue is that this set of W is not necessarily unique: For example, if the difference in scores between a correct class and a nearest incorrect class was 15, then multiplying all elements of W by 2 would make the new difference In other words, we wish to encode some preference for a certain set of weights W over others to remove this ambiguity.

The most common regularization penalty is the L2 norm that discourages large weights through an elementwise quadratic penalty over all parameters:. Notice that the regularization function is not a function of the data, it is only based on the weights.

Including the regularization penalty completes the full Multiclass Support Vector Machine loss, which is made up of two components: That is, the full Multiclass SVM loss becomes:. There is no simple way of setting this hyperparameter and it is usually determined by cross-validation. In addition to the motivation we provided above there are many desirable properties to include the regularization penalty, many of which we will come back to in later sections.

For example, it turns out that including the L2 penalty leads to the appealing max margin property in SVMs See CS lecture notes for full details if you are interested.

The most appealing property is that penalizing large weights tends to improve generalization, because it means that no input dimension can have a very large influence on the scores all by itself. Since the L2 penalty prefers smaller and more diffuse weight vectors, the final classifier is encouraged to take into account all input dimensions to small amounts rather than a few input dimensions and very strongly.

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As we will see later in the class, this effect can improve the generalization performance of the classifiers on test images and lead to less overfitting. Note that biases do not have the same effect since, unlike the weights, they do not control the strength of influence of an input dimension.

However, in practice this often turns out to have a negligible effect. Lastly, note that due to the regularization penalty we can never achieve loss of exactly 0. Here is the loss function without regularization implemented in Python, in both unvectorized and half-vectorized form:.

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The takeaway from this section is that the SVM loss takes one particular approach to measuring how consistent the predictions on training data are with the ground truth labels. Additionally, making good predictions on the training set is equivalent to minimizing the loss. What value should it be set to, and do we have to cross-validate it? The tradeoff between the data loss and the regularization loss in the objective.

Therefore, the exact value of the margin between the scores e. Relation to Binary Support Vector Machine. You may be coming to this class with previous experience with Binary Support Vector Machines, where the loss for the i-th example can be written as:.

You can convince yourself that the formulation we presented in this section contains the binary SVM as a special case when there are only two classes.

That is, if we only had two classes then the loss reduces to the binary SVM shown above. In this class as is the case with Neural Networks in general we will always work with the optimization objectives in their unconstrained primal form. Many of these objectives are technically not differentiable e.

Other Multiclass SVM formulations. Our formulation follows the Weston and Watkins pdf version, which is a more powerful version than OVA in the sense that you can construct multiclass datasets where this version can achieve zero data loss, but OVA cannot. See details in the paper if interested. The last formulation you may see is a Structured SVMwhich maximizes the margin between the score of the correct class and the score of the highest-scoring incorrect runner-up class.

Understanding the differences between these formulations is outside of the scope of the class. The version presented in these notes is a safe bet to use in practice, but the arguably simplest OVA strategy is likely to work just as well as also argued by Rikin et al.

It turns out that the SVM is one of two commonly seen classifiers. The other popular choice is the Softmax classifierwhich has a different loss function. In other words, the cross-entropy objective wants the predicted distribution to have all of its mass on the correct answer.

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Exponentiating these quantities therefore gives the unnormalized probabilities, and the division performs the normalization so that the probabilities sum to one. In the probabilistic interpretation, we are therefore minimizing the negative log likelihood of the correct class, which can be interpreted as performing Maximum Likelihood Estimation MLE.

We mention these interpretations to help your intuitions, but the full details of this derivation are beyond the scope of this class. Dividing large numbers can be numerically unstable, so it is important to use a normalization trick. This will not change any of the results, but we can use this value to improve the numerical stability of the computation. Possibly confusing naming conventions. To be precise, the SVM classifier uses the hinge lossor also sometimes called the max-margin loss.

The Softmax classifier uses the cross-entropy loss. The Softmax classifier gets its name from the softmax functionwhich is used to squash the raw class scores into normalized positive values that sum to one, so that the cross-entropy loss can be applied.

For example, given an image the SVM classifier might give you scores [