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Machine Learning, Python

Convolutional hypercolumns in Python

If you are following some Machine Learning news, you certainly saw the work done by Ryan Dahl on Automatic Colorization (Hacker News comments, Reddit comments). This amazing work uses pixel hypercolumn information extracted from the VGG-16 network in order to colorize images. Samim also used the network to process Black & White video frames and produced the amazing video below:

https://www.youtube.com/watch?v=_MJU8VK2PI4

Colorizing Black&White Movies with Neural Networks (video by Samim, network by Ryan)

But how does this hypercolumns works ? How to extract them to use on such variety of pixel classification problems ? The main idea of this post is to use the VGG-16 pre-trained network together with Keras and Scikit-Learn in order to extract the pixel hypercolumns and take a superficial look at the information present on it. I’m writing this because I haven’t found anything in Python to do that and this may be really useful for others working on pixel classification, segmentation, etc.

Hypercolumns

Many algorithms using features from CNNs (Convolutional Neural Networks) usually use the last FC (fully-connected) layer features in order to extract information about certain input. However, the information in the last FC layer may be too coarse spatially to allow precise localization (due to sequences of maxpooling, etc.), on the other side, the first layers may be spatially precise but will lack semantic information. To get the best of both worlds, the authors of the hypercolumn paper define the hypercolumn of a pixel as the vector of activations of all CNN units “above” that pixel.

**Hypercolumn Extraction** (*by Hypercolumns for Object Segmentation and Fine-grained Localization*)

The first step on the extraction of the hypercolumns is to feed the image into the CNN (Convolutional Neural Network) and extract the feature map activations for each location of the image. The tricky part is when the feature maps are smaller than the input image, for instance after a pooling operation, the authors of the paper then do a bilinear upsampling of the feature map in order to keep the feature maps on the same size of the input. There are also the issue with the FC (fully-connected) layers, because you can’t isolate units semantically tied only to one pixel of the image, so the FC activations are seen as 1×1 feature maps, which means that all locations shares the same information regarding the FC part of the hypercolumn. All these activations are then concatenated to create the hypercolumn. For instance, if we take the VGG-16 architecture to use only the first 2 convolutional layers after the max pooling operations, we will have a hypercolumn with the size of:

64 filters (first conv layer before pooling)

128 filters (second conv layer before pooling ) = 192 features

This means that each pixel of the image will have a 192-dimension hypercolumn vector. This hypercolumn is really interesting because it will contain information about the first layers (where we have a lot of spatial information but little semantic) and also information about the final layers (with little spatial information and lots of semantics). Thus this hypercolumn will certainly help in a lot of pixel classification tasks such as the one mentioned earlier of automatic colorization, because each location hypercolumn carries the information about what this pixel semantically and spatially represents. This is also very helpful on segmentation tasks (you can see more about that on the original paper introducing the hypercolumn concept).

Everything sounds cool, but how do we extract hypercolumns in practice ?

VGG-16

Before being able to extract the hypercolumns, we’ll setup the VGG-16 pre-trained network, because you know, the price of a good GPU (I can’t even imagine many of them) here in Brazil is very expensive and I don’t want to sell my kidney to buy a GPU.

To setup a pretrained VGG-16 network on Keras, you’ll need to download the weights file from here (vgg16_weights.h5 file with approximately 500MB) and then setup the architecture and load the downloaded weights using Keras (more information about the weights file and architecture here):

from matplotlib import pyplot as plt

import theano
import cv2
import numpy as np
import scipy as sp

from keras.models import Sequential
from keras.layers.core import Flatten, Dense, Dropout
from keras.layers.convolutional import Convolution2D, MaxPooling2D
from keras.layers.convolutional import ZeroPadding2D
from keras.optimizers import SGD

from sklearn.manifold import TSNE
from sklearn import manifold
from sklearn import cluster
from sklearn.preprocessing import StandardScaler

def VGG_16(weights_path=None):
    model = Sequential()
    model.add(ZeroPadding2D((1,1),input_shape=(3,224,224)))
    model.add(Convolution2D(64, 3, 3, activation='relu'))
    model.add(ZeroPadding2D((1,1)))
    model.add(Convolution2D(64, 3, 3, activation='relu'))
    model.add(MaxPooling2D((2,2), stride=(2,2)))

    model.add(ZeroPadding2D((1,1)))
    model.add(Convolution2D(128, 3, 3, activation='relu'))
    model.add(ZeroPadding2D((1,1)))
    model.add(Convolution2D(128, 3, 3, activation='relu'))
    model.add(MaxPooling2D((2,2), stride=(2,2)))

    model.add(ZeroPadding2D((1,1)))
    model.add(Convolution2D(256, 3, 3, activation='relu'))
    model.add(ZeroPadding2D((1,1)))
    model.add(Convolution2D(256, 3, 3, activation='relu'))
    model.add(ZeroPadding2D((1,1)))
    model.add(Convolution2D(256, 3, 3, activation='relu'))
    model.add(MaxPooling2D((2,2), stride=(2,2)))

    model.add(ZeroPadding2D((1,1)))
    model.add(Convolution2D(512, 3, 3, activation='relu'))
    model.add(ZeroPadding2D((1,1)))
    model.add(Convolution2D(512, 3, 3, activation='relu'))
    model.add(ZeroPadding2D((1,1)))
    model.add(Convolution2D(512, 3, 3, activation='relu'))
    model.add(MaxPooling2D((2,2), stride=(2,2)))

    model.add(ZeroPadding2D((1,1)))
    model.add(Convolution2D(512, 3, 3, activation='relu'))
    model.add(ZeroPadding2D((1,1)))
    model.add(Convolution2D(512, 3, 3, activation='relu'))
    model.add(ZeroPadding2D((1,1)))
    model.add(Convolution2D(512, 3, 3, activation='relu'))
    model.add(MaxPooling2D((2,2), stride=(2,2)))

    model.add(Flatten())
    model.add(Dense(4096, activation='relu'))
    model.add(Dropout(0.5))
    model.add(Dense(4096, activation='relu'))
    model.add(Dropout(0.5))
    model.add(Dense(1000, activation='softmax'))

    if weights_path:
        model.load_weights(weights_path)

    return model

As you can see, this is a very simple code to declare the VGG16 architecture and load the pre-trained weights (together with Python imports for the required packages). After that we’ll compile the Keras model:

model = VGG_16('vgg16_weights.h5')
sgd = SGD(lr=0.1, decay=1e-6, momentum=0.9, nesterov=True)
model.compile(optimizer=sgd, loss='categorical_crossentropy')

Now let’s test the network using an image:

im_original = cv2.resize(cv2.imread('madruga.jpg'), (224, 224))
im = im_original.transpose((2,0,1))
im = np.expand_dims(im, axis=0)
im_converted = cv2.cvtColor(im_original, cv2.COLOR_BGR2RGB)
plt.imshow(im_converted)

Image used

As we can see, we loaded the image, fixed the axes and then we can now feed the image into the VGG-16 to get the predictions:

out = model.predict(im)
plt.plot(out.ravel())

As you can see, these are the final activations of the softmax layer, the class with the “jersey, T-shirt, tee shirt” category.

Extracting arbitrary feature maps

Now, to extract the feature map activations, we’ll have to being able to extract feature maps from arbitrary convolutional layers of the network. We can do that by compiling a Theano function using the get_output() method of Keras, like in the example below:

get_feature = theano.function([model.layers[0].input], model.layers[3].get_output(train=False), allow_input_downcast=False)
feat = get_feature(im)
plt.imshow(feat[0][2])

Feature Map

In the example above, I’m compiling a Theano function to get the 3 layer (a convolutional layer) feature map and then showing only the 3rd feature map. Here we can see the intensity of the activations. If we get feature maps of the activations from the final layers, we can see that the extracted features are more abstract, like eyes, etc. Look at this example below from the 15th convolutional layer:

get_feature = theano.function([model.layers[0].input], model.layers[15].get_output(train=False), allow_input_downcast=False)
feat = get_feature(im)
plt.imshow(feat[0][13])

More semantic feature maps.

As you can see, this second feature map is extracting more abstract features. And you can also note that the image seems to be more stretched when compared with the feature we saw earlier, that is because the the first feature maps has 224×224 size and this one has 56×56 due to the downscaling operations of the layers before the convolutional layer, and that is why we lose a lot of spatial information.

Extracting hypercolumns

Now finally let’s extract the hypercolumns of arbitrary set of layers. To do that, we will define a function to extract these hypercolumns:

def extract_hypercolumn(model, layer_indexes, instance):
    layers = [model.layers[li].get_output(train=False) for li in layer_indexes]
    get_feature = theano.function([model.layers[0].input], layers,
                                  allow_input_downcast=False)
    feature_maps = get_feature(instance)
    hypercolumns = []
    for convmap in feature_maps:
        for fmap in convmap[0]:
            upscaled = sp.misc.imresize(fmap, size=(224, 224),
                                        mode="F", interp='bilinear')
            hypercolumns.append(upscaled)

    return np.asarray(hypercolumns)

As we can see, this function will expect three parameters: the model itself, an list of layer indexes that will be used to extract the hypercolumn features and an image instance that will be used to extract the hypercolumns. Let’s now test the hypercolumn extraction for the first 2 convolutional layers:

layers_extract = [3, 8]
hc = extract_hypercolumn(model, layers_extract, im)

That’s it, we extracted the hypercolumn vectors for each pixel. The shape of this “hc” variable is: (192L, 224L, 224L), which means that we have a 192-dimensional hypercolumn for each one of the 224×224 pixel (a total of 50176 pixels with 192 hypercolumn feature each).

Let’s plot the average of the hypercolumns activations for each pixel:

ave = np.average(hc.transpose(1, 2, 0), axis=2)
plt.imshow(ave)

Ad you can see, those first hypercolumn activations are all looking like edge detectors, let’s see how these hypercolumns looks like for the layers 22 and 29:

layers_extract = [22, 29]
hc = extract_hypercolumn(model, layers_extract, im)
ave = np.average(hc.transpose(1, 2, 0), axis=2)
plt.imshow(ave)

Hypercolumn average for the layers 22 and 29.

As we can see now, the features are really more abstract and semantically interesting but with spatial information a little fuzzy.

Remember that you can extract the hypercolumns using all the initial layers and also the final layers, including the FC layers. Here I’m extracting them separately to show how they differ in the visualization plots.

Simple hypercolumn pixel clustering

Now, you can do a lot of things, you can use these hypercolumns to classify pixels for some task, to do automatic pixel colorization, segmentation, etc. What I’m going to do here just as an experiment, is to use the hypercolumns (from the VGG-16 layers 3, 8, 15, 22, 29) and then cluster it using KMeans with 2 clusters:

m = hc.transpose(1,2,0).reshape(50176, -1)
kmeans = cluster.KMeans(n_clusters=2, max_iter=300, n_jobs=5, precompute_distances=True)
cluster_labels = kmeans .fit_predict(m)

imcluster = np.zeros((224,224))
imcluster = imcluster.reshape((224*224,))
imcluster = cluster_labels

plt.imshow(imcluster.reshape(224, 224), cmap="hot")

Now you can imagine how useful hypercolumns can be to tasks like keypoints extraction, segmentation, etc. It’s a very elegant, simple and useful concept.

I hope you liked it !

– Christian S. Perone

Cite this article as: Christian S. Perone, "Convolutional hypercolumns in Python," in Terra Incognita, 11/01/2016, https://blog.christianperone.com/2016/01/convolutional-hypercolumns-in-python/.

11/01/201619/01/2020 by Christian S. Perone

Machine Learning, Math, Programming, Python

Deep learning – Convolutional neural networks and feature extraction with Python

Convolutional neural networks (or ConvNets) are biologically-inspired variants of MLPs, they have different kinds of layers and each different layer works different than the usual MLP layers. If you are interested in learning more about ConvNets, a good course is the CS231n – Convolutional Neural Newtorks for Visual Recognition. The architecture of the CNNs are shown in the images below:

A regular neural network. — A regular neural network (from CS231n website).

A ConvNet network achitecture (from CS231n website).

As you can see, the ConvNets works with 3D volumes and transformations of these 3D volumes. I won’t repeat in this post the entire CS231n tutorial, so if you’re really interested, please take time to read before continuing.

Lasagne and nolearn

One of the Python packages for deep learning that I really like to work with is Lasagne and nolearn. Lasagne is based on Theano so the GPU speedups will really make a great difference, and their declarative approach for the neural networks creation are really helpful. The nolearn libary is a collection of utilities around neural networks packages (including Lasagne) that can help us a lot during the creation of the neural network architecture, inspection of the layers, etc.

What I’m going to show in this post, is how to build a simple ConvNet architecture with some convolutional and pooling layers. I’m also going to show how you can use a ConvNet to train a feature extractor and then use it to extract features before feeding them into different models like SVM, Logistic Regression, etc. Many people use pre-trained ConvNet models and then remove the last output layer to extract the features from ConvNets that were trained on ImageNet datasets. This is usually called transfer learning because you can use layers from other ConvNets as feature extractors for different problems, since the first layer filters of the ConvNets works as edge detectors, they can be used as general feature detectors for other problems.

Loading the MNIST dataset

The MNIST dataset is one of the most traditional datasets for digits classification. We will use a pickled version of it for Python, but first, lets import the packages that we will need to use:

import matplotlib
import matplotlib.pyplot as plt
import matplotlib.cm as cm

from urllib import urlretrieve
import cPickle as pickle
import os
import gzip

import numpy as np
import theano

import lasagne
from lasagne import layers
from lasagne.updates import nesterov_momentum

from nolearn.lasagne import NeuralNet
from nolearn.lasagne import visualize

from sklearn.metrics import classification_report
from sklearn.metrics import confusion_matrix

As you can see, we are importing matplotlib for plotting some images, some native Python modules to download the MNIST dataset, numpy, theano, lasagne, nolearn and some scikit-learn functions for model evaluation.

After that, we define our MNIST loading function (this is pretty the same function used in the Lasagne tutorial):

def load_dataset():
    url = 'http://deeplearning.net/data/mnist/mnist.pkl.gz'
    filename = 'mnist.pkl.gz'
    if not os.path.exists(filename):
        print("Downloading MNIST dataset...")
        urlretrieve(url, filename)

    with gzip.open(filename, 'rb') as f:
        data = pickle.load(f)

    X_train, y_train = data[0]
    X_val, y_val = data[1]
    X_test, y_test = data[2]

    X_train = X_train.reshape((-1, 1, 28, 28))
    X_val = X_val.reshape((-1, 1, 28, 28))
    X_test = X_test.reshape((-1, 1, 28, 28))

    y_train = y_train.astype(np.uint8)
    y_val = y_val.astype(np.uint8)
    y_test = y_test.astype(np.uint8)

    return X_train, y_train, X_val, y_val, X_test, y_test

As you can see, we are downloading the MNIST pickled dataset and then unpacking it into the three different datasets: train, validation and test. After that we reshape the image contents to prepare them to input into the Lasagne input layer later and we also convert the numpy array types to uint8 due to the GPU/theano datatype restrictions.

After that, we’re ready to load the MNIST dataset and inspect it:

X_train, y_train, X_val, y_val, X_test, y_test = load_dataset()
plt.imshow(X_train[0][0], cmap=cm.binary)

This code above will output the following image (I’m using IPython Notebook):

An example of a MNIST digit (5 in the case).

ConvNet Architecture and Training

Now we can define our ConvNet architecture and then train it using a GPU/CPU (I have a very cheap GPU, but it helps a lot):

net1 = NeuralNet(
    layers=[('input', layers.InputLayer),
            ('conv2d1', layers.Conv2DLayer),
            ('maxpool1', layers.MaxPool2DLayer),
            ('conv2d2', layers.Conv2DLayer),
            ('maxpool2', layers.MaxPool2DLayer),
            ('dropout1', layers.DropoutLayer),
            ('dense', layers.DenseLayer),
            ('dropout2', layers.DropoutLayer),
            ('output', layers.DenseLayer),
            ],
    # input layer
    input_shape=(None, 1, 28, 28),
    # layer conv2d1
    conv2d1_num_filters=32,
    conv2d1_filter_size=(5, 5),
    conv2d1_nonlinearity=lasagne.nonlinearities.rectify,
    conv2d1_W=lasagne.init.GlorotUniform(),  
    # layer maxpool1
    maxpool1_pool_size=(2, 2),    
    # layer conv2d2
    conv2d2_num_filters=32,
    conv2d2_filter_size=(5, 5),
    conv2d2_nonlinearity=lasagne.nonlinearities.rectify,
    # layer maxpool2
    maxpool2_pool_size=(2, 2),
    # dropout1
    dropout1_p=0.5,    
    # dense
    dense_num_units=256,
    dense_nonlinearity=lasagne.nonlinearities.rectify,    
    # dropout2
    dropout2_p=0.5,    
    # output
    output_nonlinearity=lasagne.nonlinearities.softmax,
    output_num_units=10,
    # optimization method params
    update=nesterov_momentum,
    update_learning_rate=0.01,
    update_momentum=0.9,
    max_epochs=10,
    verbose=1,
    )

# Train the network
nn = net1.fit(X_train, y_train)

As you can see, in the parameter layers we’re defining a dictionary of tuples with the layer names/types and then we define the parameters for these layers. Our architecture here is using two convolutional layers with poolings and then a fully connected layer (dense layer) and the output layer. There are also dropouts between some layers, the dropout layer is a regularizer that randomly sets input values to zero to avoid overfitting (see the image below).

Dropout layer effect (from CS231n website).

After calling the train method, the nolearn package will show status of the learning process, in my machine with my humble GPU I got the results below:

# Neural Network with 160362 learnable parameters

## Layer information

  #  name      size
---  --------  --------
  0  input     1x28x28
  1  conv2d1   32x24x24
  2  maxpool1  32x12x12
  3  conv2d2   32x8x8
  4  maxpool2  32x4x4
  5  dropout1  32x4x4
  6  dense     256
  7  dropout2  256
  8  output    10

epoch   train loss    valid loss    train/val    valid acc  dur
------- ------------  ------------  -----------  ---------  ---
      1     0.85204   0.16707      5.09977      0.95174  33.71s
      2     0.27571   0.10732      2.56896      0.96825  33.34s
      3     0.20262   0.08567      2.36524      0.97488  33.51s
      4     0.16551   0.07695      2.15081      0.97705  33.50s
      5     0.14173   0.06803      2.08322      0.98061  34.38s
      6     0.12519   0.06067      2.06352      0.98239  34.02s
      7     0.11077   0.05532      2.00254      0.98427  33.78s
      8     0.10497   0.05771      1.81898      0.98248  34.17s
      9     0.09881   0.05159      1.91509      0.98407  33.80s
     10     0.09264   0.04958      1.86864      0.98526  33.40s

As you can see, the accuracy in the end was 0.98526, a pretty good performance for a 10 epochs training.

Prediction and Confusion Matrix

Now we can use the model to predict the entire testing dataset:

preds = net1.predict(X_test)

And we can also plot a confusion matrix to check the performance of the neural network classification:

cm = confusion_matrix(y_test, preds)
plt.matshow(cm)
plt.title('Confusion matrix')
plt.colorbar()
plt.ylabel('True label')
plt.xlabel('Predicted label')
plt.show()

The code above will plot the following confusion matrix:

As you can see, the diagonal is where the classification is more dense, showing the good performance of our classifier.

Filters Visualization

We can also visualize the 32 filters from the first convolutional layer:

visualize.plot_conv_weights(net1.layers_['conv2d1'])

The code above will plot the following filters below:

As you can see, the nolearn plot_conv_weights plots all the filters present in the layer we specified.

Theano layer functions and Feature Extraction

Now it is time to create theano-compiled functions that will feed-forward the input data into the architecture up to the layer you’re interested. I’m going to get the functions for the output layer and also for the dense layer before the output layer:

dense_layer = layers.get_output(net1.layers_['dense'], deterministic=True)
output_layer = layers.get_output(net1.layers_['output'], deterministic=True)
input_var = net1.layers_['input'].input_var

f_output = theano.function([input_var], output_layer)
f_dense = theano.function([input_var], dense_layer)

As you can see, we have now two theano functions called f_output and f_dense (for the output and dense layers). Please note that in order to get the layers here we are using a extra parameter called “deterministic“, this is to avoid the dropout layers affecting our feed-forward pass.

We can now convert an example instance to the input format and then feed it into the theano function for the output layer:

instance = X_test[0][None, :, :]
%timeit -n 500 f_output(instance)

500 loops, best of 3: 858 µs per loop

As you can see, the f_output function takes an average of 858 µs. We can also plot the output layer activations for the instance:

pred = f_output(instance)
N = pred.shape[1]
plt.bar(range(N), pred.ravel())

The code above will create the following plot:

As you can see, the digit was recognized as the digit 7. The fact that you can create theano functions for any layer of the network is very useful because you can create a function (like we did before) to get the activations for the dense layer (the one before the output layer) and you can use these activations as features and use your neural network not as classifier but as a feature extractor. Let’s plot now the 256 unit activations for the dense layer:

pred = f_dense(instance)
N = pred.shape[1]
plt.bar(range(N), pred.ravel())

The code above will create the following plot below:

You can now use the output of the these 256 activations as features on a linear classifier like Logistic Regression or SVM.

I hope you enjoyed the tutorial !

Cite this article as: Christian S. Perone, "Deep learning – Convolutional neural networks and feature extraction with Python," in Terra Incognita, 19/08/2015, https://blog.christianperone.com/2015/08/convolutional-neural-networks-and-feature-extraction-with-python/.

19/08/201519/01/2020 by Christian S. Perone

Machine Learning, Python

Machine Learning :: Text feature extraction (tf-idf) – Part II

Read the first part of this tutorial: Text feature extraction (tf-idf) – Part I.

This post is a continuation of the first part where we started to learn the theory and practice about text feature extraction and vector space model representation. I really recommend you to read the first part of the post series in order to follow this second post.

Since a lot of people liked the first part of this tutorial, this second part is a little longer than the first.

Introduction

In the first post, we learned how to use the term-frequency to represent textual information in the vector space. However, the main problem with the term-frequency approach is that it scales up frequent terms and scales down rare terms which are empirically more informative than the high frequency terms. The basic intuition is that a term that occurs frequently in many documents is not a good discriminator, and really makes sense (at least in many experimental tests); the important question here is: why would you, in a classification problem for instance, emphasize a term which is almost present in the entire corpus of your documents ?

The tf-idf weight comes to solve this problem. What tf-idf gives is how important is a word to a document in a collection, and that’s why tf-idf incorporates local and global parameters, because it takes in consideration not only the isolated term but also the term within the document collection. What tf-idf then does to solve that problem, is to scale down the frequent terms while scaling up the rare terms; a term that occurs 10 times more than another isn’t 10 times more important than it, that’s why tf-idf uses the logarithmic scale to do that.

But let’s go back to our definition of the $\mathrm{tf}(t,d)$ which is actually the term count of the term $t$ in the document $d$ . The use of this simple term frequency could lead us to problems like keyword spamming, which is when we have a repeated term in a document with the purpose of improving its ranking on an IR (Information Retrieval) system or even create a bias towards long documents, making them look more important than they are just because of the high frequency of the term in the document.

To overcome this problem, the term frequency $\mathrm{tf}(t,d)$ of a document on a vector space is usually also normalized. Let’s see how we normalize this vector.

Vector normalization

Suppose we are going to normalize the term-frequency vector $\vec{v_{d_4}}$ that we have calculated in the first part of this tutorial. The document $d4$ from the first part of this tutorial had this textual representation:

d4: We can see the shining sun, the bright sun.

And the vector space representation using the non-normalized term-frequency of that document was:

$\vec{v_{d_4}} = (0,2,1,0)$

To normalize the vector, is the same as calculating the Unit Vector of the vector, and they are denoted using the “hat” notation: $\hat{v}$ . The definition of the unit vector $\hat{v}$ of a vector $\vec{v}$ is:

$\displaystyle \hat{v} = \frac{\vec{v}}{\|\vec{v}\|_p}$

Where the $\hat{v}$ is the unit vector, or the normalized vector, the $\vec{v}$ is the vector going to be normalized and the $\|\vec{v}\|_p$ is the norm (magnitude, length) of the vector $\vec{v}$ in the $L^p$ space (don’t worry, I’m going to explain it all).

The unit vector is actually nothing more than a normalized version of the vector, is a vector which the length is 1.

The normalization process (Source: http://processing.org/learning/pvector/)

But the important question here is how the length of the vector is calculated and to understand this, you must understand the motivation of the $L^p$ spaces, also called Lebesgue spaces.

Lebesgue spaces

How long is this vector ? (Source: Source: http://processing.org/learning/pvector/)

Usually, the length of a vector $\vec{u} = (u_1, u_2, u_3, \ldots, u_n)$ is calculated using the Euclidean norm – a norm is a function that assigns a strictly positive length or size to all vectors in a vector space -, which is defined by:

(Source: http://processing.org/learning/pvector/)

$\|\vec{u}\| = \sqrt{u^2_1 + u^2_2 + u^2_3 + \ldots + u^2_n}$

But this isn’t the only way to define length, and that’s why you see (sometimes) a number $p$ together with the norm notation, like in $\|\vec{u}\|_p$ . That’s because it could be generalized as:

$\displaystyle \|\vec{u}\|_p = ( \left|u_1\right|^p + \left|u_2\right|^p + \left|u_3\right|^p + \ldots + \left|u_n\right|^p )^\frac{1}{p}$

and simplified as:

$\displaystyle \|\vec{u}\|_p = (\sum\limits_{i=1}^{n}\left|\vec{u}_i\right|^p)^\frac{1}{p}$

So when you read about a L2-norm, you’re reading about the Euclidean norm, a norm with $p=2$ , the most common norm used to measure the length of a vector, typically called “magnitude”; actually, when you have an unqualified length measure (without the $p$ number), you have the L2-norm (Euclidean norm).

When you read about a L1-norm, you’re reading about the norm with $p=1$ , defined as:

$\displaystyle \|\vec{u}\|_1 = ( \left|u_1\right| + \left|u_2\right| + \left|u_3\right| + \ldots + \left|u_n\right|)$

Which is nothing more than a simple sum of the components of the vector, also known as Taxicab distance, also called Manhattan distance.

Taxicab geometry versus Euclidean distance: In taxicab geometry all three pictured lines have the same length (12) for the same route. In Euclidean geometry, the green line has length $6 \times \sqrt{2} \approx 8.48$ , and is the unique shortest path.
Source: Wikipedia :: Taxicab Geometry

Note that you can also use any norm to normalize the vector, but we’re going to use the most common norm, the L2-Norm, which is also the default in the 0.9 release of the scikits.learn. You can also find papers comparing the performance of the two approaches among other methods to normalize the document vector, actually you can use any other method, but you have to be concise, once you’ve used a norm, you have to use it for the whole process directly involving the norm (a unit vector that used a L1-norm isn’t going to have the length 1 if you’re going to take its L2-norm later).

Back to vector normalization

Now that you know what the vector normalization process is, we can try a concrete example, the process of using the L2-norm (we’ll use the right terms now) to normalize our vector $\vec{v_{d_4}} = (0,2,1,0)$ in order to get its unit vector $\hat{v_{d_4}}$ . To do that, we’ll simple plug it into the definition of the unit vector to evaluate it:

$\hat{v} = \frac{\vec{v}}{\|\vec{v}\|_p} \\ \\ \hat{v_{d_4}} = \frac{\vec{v_{d_4}}}{||\vec{v_{d_4}}||_2} \\ \\ \\ \hat{v_{d_4}} = \frac{(0,2,1,0)}{\sqrt{0^2 + 2^2 + 1^2 + 0^2}} \\ \\ \hat{v_{d_4}} = \frac{(0,2,1,0)}{\sqrt{5}} \\ \\ \small \hat{v_{d_4}} = (0.0, 0.89442719, 0.4472136, 0.0)$

And that is it ! Our normalized vector $\hat{v_{d_4}}$ has now a L2-norm $\|\hat{v_{d_4}}\|_2 = 1.0$ .

Note that here we have normalized our term frequency document vector, but later we’re going to do that after the calculation of the tf-idf.

The term frequency – inverse document frequency (tf-idf) weight

Now you have understood how the vector normalization works in theory and practice, let’s continue our tutorial. Suppose you have the following documents in your collection (taken from the first part of tutorial):

Train Document Set:

d1: The sky is blue.
d2: The sun is bright.

Test Document Set:

d3: The sun in the sky is bright.
d4: We can see the shining sun, the bright sun.

Your document space can be defined then as $D = \{ d_1, d_2, \ldots, d_n \}$ where $n$ is the number of documents in your corpus, and in our case as $D_{train} = \{d_1, d_2\}$ and $D_{test} = \{d_3, d_4\}$ . The cardinality of our document space is defined by $\left|{D_{train}}\right| = 2$ and $\left|{D_{test}}\right| = 2$ , since we have only 2 two documents for training and testing, but they obviously don’t need to have the same cardinality.

Let’s see now, how idf (inverse document frequency) is then defined:

$\displaystyle \mathrm{idf}(t) = \log{\frac{\left|D\right|}{1+\left|\{d : t \in d\}\right|}}$

where $\left|\{d : t \in d\}\right|$ is the number of documents where the term $t$ appears, when the term-frequency function satisfies $\mathrm{tf}(t,d) \neq 0$ , we’re only adding 1 into the formula to avoid zero-division.

The formula for the tf-idf is then:

$\mathrm{tf\mbox{-}idf}(t) = \mathrm{tf}(t, d) \times \mathrm{idf}(t)$

and this formula has an important consequence: a high weight of the tf-idf calculation is reached when you have a high term frequency (tf) in the given document (local parameter) and a low document frequency of the term in the whole collection (global parameter).

Now let’s calculate the idf for each feature present in the feature matrix with the term frequency we have calculated in the first tutorial:

$M_{train} = \begin{bmatrix} 0 & 1 & 1 & 1\\ 0 & 2 & 1 & 0 \end{bmatrix}$

Since we have 4 features, we have to calculate $\mathrm{idf}(t_1)$ , $\mathrm{idf}(t_2)$ , $\mathrm{idf}(t_3)$ , $\mathrm{idf}(t_4)$ :

$\mathrm{idf}(t_1) = \log{\frac{\left|D\right|}{1+\left|\{d : t_1 \in d\}\right|}} = \log{\frac{2}{1}} = 0.69314718$

$\mathrm{idf}(t_2) = \log{\frac{\left|D\right|}{1+\left|\{d : t_2 \in d\}\right|}} = \log{\frac{2}{3}} = -0.40546511$

$\mathrm{idf}(t_3) = \log{\frac{\left|D\right|}{1+\left|\{d : t_3 \in d\}\right|}} = \log{\frac{2}{3}} = -0.40546511$

$\mathrm{idf}(t_4) = \log{\frac{\left|D\right|}{1+\left|\{d : t_4 \in d\}\right|}} = \log{\frac{2}{2}} = 0.0$

These idf weights can be represented by a vector as:

$\vec{idf_{train}} = (0.69314718, -0.40546511, -0.40546511, 0.0)$

Now that we have our matrix with the term frequency ( $M_{train}$ ) and the vector representing the idf for each feature of our matrix ( $\vec{idf_{train}}$ ), we can calculate our tf-idf weights. What we have to do is a simple multiplication of each column of the matrix $M_{train}$ with the respective $\vec{idf_{train}}$ vector dimension. To do that, we can create a square diagonal matrix called $M_{idf}$ with both the vertical and horizontal dimensions equal to the vector $\vec{idf_{train}}$ dimension:

$M_{idf} = \begin{bmatrix} 0.69314718 & 0 & 0 & 0\\ 0 & -0.40546511 & 0 & 0\\ 0 & 0 & -0.40546511 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}$

and then multiply it to the term frequency matrix, so the final result can be defined then as:

$M_{tf\mbox{-}idf} = M_{train} \times M_{idf}$

Please note that the matrix multiplication isn’t commutative, the result of $A \times B$ will be different than the result of the $B \times A$ , and this is why the $M_{idf}$ is on the right side of the multiplication, to accomplish the desired effect of multiplying each idf value to its corresponding feature:

$\begin{bmatrix} \mathrm{tf}(t_1, d_1) & \mathrm{tf}(t_2, d_1) & \mathrm{tf}(t_3, d_1) & \mathrm{tf}(t_4, d_1)\\ \mathrm{tf}(t_1, d_2) & \mathrm{tf}(t_2, d_2) & \mathrm{tf}(t_3, d_2) & \mathrm{tf}(t_4, d_2) \end{bmatrix} \times \begin{bmatrix} \mathrm{idf}(t_1) & 0 & 0 & 0\\ 0 & \mathrm{idf}(t_2) & 0 & 0\\ 0 & 0 & \mathrm{idf}(t_3) & 0\\ 0 & 0 & 0 & \mathrm{idf}(t_4) \end{bmatrix} \\ = \begin{bmatrix} \mathrm{tf}(t_1, d_1) \times \mathrm{idf}(t_1) & \mathrm{tf}(t_2, d_1) \times \mathrm{idf}(t_2) & \mathrm{tf}(t_3, d_1) \times \mathrm{idf}(t_3) & \mathrm{tf}(t_4, d_1) \times \mathrm{idf}(t_4)\\ \mathrm{tf}(t_1, d_2) \times \mathrm{idf}(t_1) & \mathrm{tf}(t_2, d_2) \times \mathrm{idf}(t_2) & \mathrm{tf}(t_3, d_2) \times \mathrm{idf}(t_3) & \mathrm{tf}(t_4, d_2) \times \mathrm{idf}(t_4) \end{bmatrix}$

Let’s see now a concrete example of this multiplication:

$M_{tf\mbox{-}idf} = M_{train} \times M_{idf} = \\ \begin{bmatrix} 0 & 1 & 1 & 1\\ 0 & 2 & 1 & 0 \end{bmatrix} \times \begin{bmatrix} 0.69314718 & 0 & 0 & 0\\ 0 & -0.40546511 & 0 & 0\\ 0 & 0 & -0.40546511 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \\ = \begin{bmatrix} 0 & -0.40546511 & -0.40546511 & 0\\ 0 & -0.81093022 & -0.40546511 & 0 \end{bmatrix}$

And finally, we can apply our L2 normalization process to the $M_{tf\mbox{-}idf}$ matrix. Please note that this normalization is “row-wise” because we’re going to handle each row of the matrix as a separated vector to be normalized, and not the matrix as a whole:

$M_{tf\mbox{-}idf} = \frac{M_{tf\mbox{-}idf}}{\|M_{tf\mbox{-}idf}\|_2}$ $= \begin{bmatrix} 0 & -0.70710678 & -0.70710678 & 0\\ 0 & -0.89442719 & -0.4472136 & 0 \end{bmatrix}$

And that is our pretty normalized tf-idf weight of our testing document set, which is actually a collection of unit vectors. If you take the L2-norm of each row of the matrix, you’ll see that they all have a L2-norm of 1.

Python practice

Environment Used: Python v.2.7.2, Numpy 1.6.1, Scipy v.0.9.0, Sklearn (Scikits.learn) v.0.9.

Now the section you were waiting for ! In this section I’ll use Python to show each step of the tf-idf calculation using the Scikit.learn feature extraction module.

The first step is to create our training and testing document set and computing the term frequency matrix:

from sklearn.feature_extraction.text import CountVectorizer

train_set = ("The sky is blue.", "The sun is bright.")
test_set = ("The sun in the sky is bright.",
"We can see the shining sun, the bright sun.")

count_vectorizer = CountVectorizer()
count_vectorizer.fit_transform(train_set)
print "Vocabulary:", count_vectorizer.vocabulary

# Vocabulary: {'blue': 0, 'sun': 1, 'bright': 2, 'sky': 3}

freq_term_matrix = count_vectorizer.transform(test_set)
print freq_term_matrix.todense()

#[[0 1 1 1]
#[0 2 1 0]]

Now that we have the frequency term matrix (called freq_term_matrix), we can instantiate the TfidfTransformer, which is going to be responsible to calculate the tf-idf weights for our term frequency matrix:

from sklearn.feature_extraction.text import TfidfTransformer

tfidf = TfidfTransformer(norm="l2")
tfidf.fit(freq_term_matrix)

print "IDF:", tfidf.idf_

# IDF: [ 0.69314718 -0.40546511 -0.40546511  0.        ]

Note that I’ve specified the norm as L2, this is optional (actually the default is L2-norm), but I’ve added the parameter to make it explicit to you that it it’s going to use the L2-norm. Also note that you can see the calculated idf weight by accessing the internal attribute called idf_. Now that fit() method has calculated the idf for the matrix, let’s transform the freq_term_matrix to the tf-idf weight matrix:

tf_idf_matrix = tfidf.transform(freq_term_matrix)
print tf_idf_matrix.todense()

# [[ 0.         -0.70710678 -0.70710678  0.        ]
# [ 0.         -0.89442719 -0.4472136   0.        ]]

And that is it, the tf_idf_matrix is actually our previous $M_{tf\mbox{-}idf}$ matrix. You can accomplish the same effect by using the Vectorizer class of the Scikit.learn which is a vectorizer that automatically combines the CountVectorizer and the TfidfTransformer to you. See this example to know how to use it for the text classification process.

I really hope you liked the post, I tried to make it simple as possible even for people without the required mathematical background of linear algebra, etc. In the next Machine Learning post I’m expecting to show how you can use the tf-idf to calculate the cosine similarity.

If you liked it, feel free to comment and make suggestions, corrections, etc.

Cite this article as: Christian S. Perone, "Machine Learning :: Text feature extraction (tf-idf) – Part II," in Terra Incognita, 03/10/2011, https://blog.christianperone.com/2011/10/machine-learning-text-feature-extraction-tf-idf-part-ii/.

References

Understanding Inverse Document Frequency: on theoretical arguments for IDF

Wikipedia :: tf-idf

The classic Vector Space Model

Sklearn text feature extraction code

Updates

13 Mar 2015 – Formating, fixed images issues.
03 Oct 2011 – Added the info about the environment used for Python examples

03/10/201119/01/2020 by Christian S. Perone

Tag: scikit.learn