CS2. 31n Convolutional Neural Networks for Visual Recognition. Table of Contents Setting up the data and the model. In the previous section we introduced a model of a Neuron, which computes a dot product following a non linearity, and Neural Networks that arrange neurons into layers. Norm Activation Model' title='Norm Activation Model' />Together, these choices define the new form of the score function, which we have extended from the simple linear mapping that we have seen in the Linear Classification section. In particular, a Neural Network performs a sequence of linear mappings with interwoven non linearities. In this section we will discuss additional design choices regarding data preprocessing, weight initialization, and loss functions. Data Preprocessing. Norm Activation Model' title='Norm Activation Model' />There are three common forms of data preprocessing a data matrix X, where we will assume that X is of size N x D N is the number of data, D is their dimensionality. Mean subtraction is the most common form of preprocessing. It involves subtracting the mean across every individual feature in the data, and has the geometric interpretation of centering the cloud of data around the origin along every dimension. In numpy, this operation would be implemented as X np. X, axis 0. With images specifically, for convenience it can be common to subtract a single value from all pixels e. X np. meanX, or to do so separately across the three color channels. Normalization refers to normalizing the data dimensions so that they are of approximately the same scale. There are two common ways of achieving this normalization. Norm Activation Model' title='Norm Activation Model' />One is to divide each dimension by its standard deviation, once it has been zero centered X np. Le Peril Jeune on this page. X, axis 0. Another form of this preprocessing normalizes each dimension so that the min and max along the dimension is 1 and 1 respectively. It only makes sense to apply this preprocessing if you have a reason to believe that different input features have different scales or units, but they should be of approximately equal importance to the learning algorithm. In case of images, the relative scales of pixels are already approximately equal and in range from 0 to 2. Vista Panorama Software. Common data preprocessing pipeline. Left Original toy, 2 dimensional input data. Middle The data is zero centered by subtracting the mean in each dimension. The data cloud is now centered around the origin. Right Each dimension is additionally scaled by its standard deviation. If we want more evidencebased practice, we need more practicebased evidence. Meet Melanie Gaydos, 28YearOld Model With A Rare Genetic Disorder Who Broke All Fashion Stereotypes. The Apple iPhone 7 and 7 Plus come in 6 model numbers, including A1660, A1661, A1778, A1784, A1779, and A1785. Here are the differences between those models. The red lines indicate the extent of the data they are of unequal length in the middle, but of equal length on the right. PCA and Whitening is another form of preprocessing. In this process, the data is first centered as described above. Then, we can compute the covariance matrix that tells us about the correlation structure in the data Assume input data matrix X of size N x DX np. X,axis0 zero center the data importantcovnp. X. T,XX. shape0 get the data covariance matrix. Atm Font Installer For Windows 7 more. The i,j element of the data covariance matrix contains the covariance between i th and j th dimension of the data. In particular, the diagonal of this matrix contains the variances. Furthermore, the covariance matrix is symmetric and positive semi definite. We can compute the SVD factorization of the data covariance matrix where the columns of U are the eigenvectors and S is a 1 D array of the singular values. To decorrelate the data, we project the original but zero centered data into the eigenbasis Xrotnp. 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. 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 dimensionality reduction Xrotreducednp. X,U, 1. 00 Xrotreduced becomes N x 1. After this operation, we would have reduced the original dataset of size N x D to one of size N x 1. It is very often the case that you can get very good performance by training linear classifiers or neural networks on the PCA reduced datasets, obtaining savings in both space and time. The last transformation you may see in practice is whitening. The whitening operation takes the data in the eigenbasis and divides every dimension by the eigenvalue to normalize the scale. The geometric interpretation of this transformation is that if the input data is a multivariable gaussian, then the whitened data will be a gaussian with zero mean and identity covariance matrix. This step would take the form whiten the data divide by the eigenvalues which are square roots of the singular valuesXwhiteXrotnp. S1e 5Warning Exaggerating noise. Note that were adding 1e 5 or a small constant to prevent division by zero. One weakness of this transformation is that it can greatly exaggerate the noise in the data, since it stretches all dimensions including the irrelevant dimensions of tiny variance that are mostly noise to be of equal size in the input. This can in practice be mitigated by stronger smoothing i. PCA Whitening. Left Original toy, 2 dimensional input data. Middle After performing PCA. The data is centered at zero and then rotated into the eigenbasis of the data covariance matrix. This decorrelates the data the covariance matrix becomes diagonal. Right Each dimension is additionally scaled by the eigenvalues, transforming the data covariance matrix into the identity matrix. Geometrically, this corresponds to stretching and squeezing the data into an isotropic gaussian blob. We can also try to visualize these transformations with CIFAR 1. The training set of CIFAR 1. We can then compute the 3. SVD decomposition which can be relatively expensive. What do the computed eigenvectors look like visuallyAn image might help Left An example set of 4. Left The top 1. 44 out of 3. The top eigenvectors account for most of the variance in the data, and we can see that they correspond to lower frequencies in the images. Right The 4. 9 images reduced with PCA, using the 1. That is, instead of expressing every image as a 3. In order to visualize what image information has been retained in the 1. Since U is a rotation, this can be achieved by multiplying by U. You can see that the images are slightly blurrier, reflecting the fact that the top eigenvectors capture lower frequencies. However, most of the information is still preserved. Right Visualization of the white representation, where the variance along every one of the 1. Here, the whitened 1. U. transpose 1. The lower frequencies which accounted for most variance are now negligible, while the higher frequencies which account for relatively little variance originally become exaggerated. In practice. We mention PCAWhitening in these notes for completeness, but these transformations are not used with Convolutional Networks. However, it is very important to zero center the data, and it is common to see normalization of every pixel as well. Common pitfall. An important point to make about the preprocessing is that any preprocessing statistics e.