# Word Mover’s Distance¶

Demonstrates using Gensim’s implemenation of the WMD.

Word Mover’s Distance (WMD) is a promising new tool in machine learning that allows us to submit a query and return the most relevant documents. This tutorial introduces WMD and shows how you can compute the WMD distance between two documents using `wmdistance`.

## WMD Basics¶

WMD enables us to assess the “distance” between two documents in a meaningful way even when they have no words in common. It uses word2vec [4] vector embeddings of words. It been shown to outperform many of the state-of-the-art methods in k-nearest neighbors classification [3].

WMD is illustrated below for two very similar sentences (illustration taken from Vlad Niculae’s blog). The sentences have no words in common, but by matching the relevant words, WMD is able to accurately measure the (dis)similarity between the two sentences. The method also uses the bag-of-words representation of the documents (simply put, the word’s frequencies in the documents), noted as \$d\$ in the figure below. The intuition behind the method is that we find the minimum “traveling distance” between documents, in other words the most efficient way to “move” the distribution of document 1 to the distribution of document 2.

```# Image from https://vene.ro/images/wmd-obama.png
import matplotlib.pyplot as plt
import matplotlib.image as mpimg
imgplot = plt.imshow(img)
plt.axis('off')
plt.show()
```

This method was introduced in the article “From Word Embeddings To Document Distances” by Matt Kusner et al. (link to PDF). It is inspired by the “Earth Mover’s Distance”, and employs a solver of the “transportation problem”.

In this tutorial, we will learn how to use Gensim’s WMD functionality, which consists of the `wmdistance` method for distance computation, and the `WmdSimilarity` class for corpus based similarity queries.

Important

If you use Gensim’s WMD functionality, please consider citing [1], [2] and [3].

## Computing the Word Mover’s Distance¶

To use WMD, you need some existing word embeddings. You could train your own Word2Vec model, but that is beyond the scope of this tutorial (check out Word2Vec Model if you’re interested). For this tutorial, we’ll be using an existing Word2Vec model.

Let’s take some sentences to compute the distance between.

```# Initialize logging.
import logging
logging.basicConfig(format='%(asctime)s : %(levelname)s : %(message)s', level=logging.INFO)

sentence_obama = 'Obama speaks to the media in Illinois'
sentence_president = 'The president greets the press in Chicago'
```

These sentences have very similar content, and as such the WMD should be low. Before we compute the WMD, we want to remove stopwords (“the”, “to”, etc.), as these do not contribute a lot to the information in the sentences.

```# Import and download stopwords from NLTK.
from nltk.corpus import stopwords
stop_words = stopwords.words('english')

def preprocess(sentence):
return [w for w in sentence.lower().split() if w not in stop_words]

sentence_obama = preprocess(sentence_obama)
sentence_president = preprocess(sentence_president)
```

Out:

```/home/witiko/.virtualenvs/gensim4/lib/python3.7/site-packages/sklearn/feature_extraction/image.py:167: DeprecationWarning: `np.int` is a deprecated alias for the builtin `int`. To silence this warning, use `int` by itself. Doing this will not modify any behavior and is safe. When replacing `np.int`, you may wish to use e.g. `np.int64` or `np.int32` to specify the precision. If you wish to review your current use, check the release note link for additional information.
Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations
dtype=np.int):
/home/witiko/.virtualenvs/gensim4/lib/python3.7/site-packages/sklearn/linear_model/least_angle.py:30: DeprecationWarning: `np.float` is a deprecated alias for the builtin `float`. To silence this warning, use `float` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.float64` here.
Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations
method='lar', copy_X=True, eps=np.finfo(np.float).eps,
/home/witiko/.virtualenvs/gensim4/lib/python3.7/site-packages/sklearn/linear_model/least_angle.py:167: DeprecationWarning: `np.float` is a deprecated alias for the builtin `float`. To silence this warning, use `float` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.float64` here.
Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations
method='lar', copy_X=True, eps=np.finfo(np.float).eps,
/home/witiko/.virtualenvs/gensim4/lib/python3.7/site-packages/sklearn/linear_model/least_angle.py:284: DeprecationWarning: `np.float` is a deprecated alias for the builtin `float`. To silence this warning, use `float` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.float64` here.
Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations
eps=np.finfo(np.float).eps, copy_Gram=True, verbose=0,
/home/witiko/.virtualenvs/gensim4/lib/python3.7/site-packages/sklearn/linear_model/least_angle.py:862: DeprecationWarning: `np.float` is a deprecated alias for the builtin `float`. To silence this warning, use `float` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.float64` here.
Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations
eps=np.finfo(np.float).eps, copy_X=True, fit_path=True,
/home/witiko/.virtualenvs/gensim4/lib/python3.7/site-packages/sklearn/linear_model/least_angle.py:1101: DeprecationWarning: `np.float` is a deprecated alias for the builtin `float`. To silence this warning, use `float` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.float64` here.
Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations
eps=np.finfo(np.float).eps, copy_X=True, fit_path=True,
/home/witiko/.virtualenvs/gensim4/lib/python3.7/site-packages/sklearn/linear_model/least_angle.py:1127: DeprecationWarning: `np.float` is a deprecated alias for the builtin `float`. To silence this warning, use `float` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.float64` here.
Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations
eps=np.finfo(np.float).eps, positive=False):
/home/witiko/.virtualenvs/gensim4/lib/python3.7/site-packages/sklearn/linear_model/least_angle.py:1362: DeprecationWarning: `np.float` is a deprecated alias for the builtin `float`. To silence this warning, use `float` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.float64` here.
Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations
max_n_alphas=1000, n_jobs=None, eps=np.finfo(np.float).eps,
/home/witiko/.virtualenvs/gensim4/lib/python3.7/site-packages/sklearn/linear_model/least_angle.py:1602: DeprecationWarning: `np.float` is a deprecated alias for the builtin `float`. To silence this warning, use `float` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.float64` here.
Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations
max_n_alphas=1000, n_jobs=None, eps=np.finfo(np.float).eps,
/home/witiko/.virtualenvs/gensim4/lib/python3.7/site-packages/sklearn/linear_model/least_angle.py:1738: DeprecationWarning: `np.float` is a deprecated alias for the builtin `float`. To silence this warning, use `float` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.float64` here.
Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations
eps=np.finfo(np.float).eps, copy_X=True, positive=False):
[nltk_data]   Package stopwords is already up-to-date!
```

Now, as mentioned earlier, we will be using some downloaded pre-trained embeddings. We load these into a Gensim Word2Vec model class.

Important

The embeddings we have chosen here require a lot of memory.

```import gensim.downloader as api
```

So let’s compute WMD using the `wmdistance` method.

```distance = model.wmdistance(sentence_obama, sentence_president)
print('distance = %.4f' % distance)
```

Out:

```distance = 1.0175
```

Let’s try the same thing with two completely unrelated sentences. Notice that the distance is larger.

```sentence_orange = preprocess('Oranges are my favorite fruit')
distance = model.wmdistance(sentence_obama, sentence_orange)
print('distance = %.4f' % distance)
```

Out:

```distance = 1.3663
```

## References¶

1. Ofir Pele and Michael Werman, A linear time histogram metric for improved SIFT matching, 2008.

2. Ofir Pele and Michael Werman, Fast and robust earth mover’s distances, 2009.

3. Matt Kusner et al. From Embeddings To Document Distances, 2015.

4. Tomáš Mikolov et al. Efficient Estimation of Word Representations in Vector Space, 2013.

Total running time of the script: ( 0 minutes 55.983 seconds)

Estimated memory usage: 7537 MB

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