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word2gauss

Gaussian word embeddings

Python/Cython implementation of Luke Vilnis and Andrew McCallum <i>Word Representations via Gaussian Embedding</i>, ICLR 2015 that represents each word as a multivariate Gaussian. Scales to (relatively) large corpora using Cython extensions and threading with asynchronous stochastic gradient descent (Adagrad).

Getting started

Installing

1. Install the dependencies: numpy, scipy, the packages in requirements.txt The Travis CI provisioning script installs these packages provision.sh and may be useful as a starting point.

2. Build/install word2gauss: sudo make install

3. Finally it's a good idea to run the test suite: make test

NOTE: The performance sensitive parts of the code have been carefully written in a way that allows gcc to auto-vectorize all the important loops. Accordingly we recommend using gcc to compile and setting these flags for building:

export CFLAGS="-ftree-vectorizer-verbose=2 -O3 -ffast-math"
sudo -E bash -c "make install"

If you are using a Mac, gcc compiled code runs approximately 2.5X faster than the default clang compiler. You can force the build to use gcc instead of clang with:

# change these to the location of gcc -- note that /usr/bin/gcc is really
# clang in a default XCode installation
export CC=/usr/local/bin/gcc
export CXX=/usr/local/bin/g++
export CFLAGS="-ftree-vectorizer-verbose=2 -O3 -ffast-math"
sudo -E bash -c "make install"

Code overview

GaussianEmbedding

The GaussianEmbedding class is the main workhorse for most tasks. It stores the model data, deals with serialization to/from files and learns the parameters. To allow embedding of non-word types like hierarchies and entailment relations, GaussianEmbedding has no knowledge of any vocabulary and operates only on unique IDs. Each ID is a uint32 from 0 .. N-1 with -1 signifying an OOV token.

Vocabulary

For learning word embeddings, the token - id mapping is off-loaded to a Vocabulary class. This class bundles together a string tokenizer, a token - id map, and a random token id generator (used for the negative sampling in training, see below). This allows us to translate streams of documents into training examples.

The class needs this interface:

    .word2id: given a token, return the id or raise KeyError if not in the vocab
    .id2word: given a token id, return the token or raise IndexError if invalid
    .tokenize: given a string, tokenize it using the tokenizer and then
    remove all OOV tokens
    .tokenize_ids: given a string, tokenize and return the token ids
    .random_ids: given an integer, return a numpy array of random token ids

There is a simple implementation of a vocabulary class (word2gauss.words.Vocabulary) that uses a simple uniform random from the token_id space for the negative samples.

Alternatively, you can use https://github.com/seomoz/vocab that uses a sample based on the token counts, or provide your own implementation.

Learning embeddings

To learn embeddings, you will need a suitable corpus and an implementation of the vocab class.

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

from gzip import GzipFile

from word2gauss import GaussianEmbedding, iter_pairs
from vocab import Vocabulary

# load the vocabulary
vocab = Vocabulary(...)

# create the embedding to train
# use 100 dimensional spherical Gaussian with KL-divergence as energy function
embed = GaussianEmbedding(len(vocab), 100,
    covariance_type='spherical', energy_type='KL')

# open the corpus and train with 8 threads
# the corpus is just an iterator of documents, here a new line separated
# gzip file for example
with GzipFile('location_of_corpus', 'r') as corpus:
    embed.train(iter_pairs(corpus, vocab), n_workers=8)

# save the model for later
embed.save('model_file_location', vocab=vocab.id2word, full=True)

Examining trained models

from word2gauss import GaussianEmbedding
from vocab import Vocabulary

# load in a previously trained model and the vocab
vocab = Vocabulary(...)
embed = GaussianEmbedding.load('model_file_location')

# find nearest neighbors to 'rock'
embed.nearest_neighbors('rock', vocab=vocab)

# find nearest neighbors to 'rock' sorted by covariance
embed.nearest_neighbors('rock', num=100, vocab=vocab, sort_order='sigma')

# solve king + woman - man = ??
embed.nearest_neighbors([['king', 'woman'], ['man']], num=10, vocab=vocab)

Background details

Instead of representing a word as a vector as in word2vec, word2gauss represents each word as a multivariate Gaussian. Assuming some dictionary of known tokens w[i], i = 0 .. N-1, each word is represented as a probability P[i], a K dimensional Gaussian parameterized by

   P[i] ~ N(x; mu[i], Sigma[i])

Here, mu[i] and Sigma[i] are the mean and co-variance matrix for word i. The mean is a vector of length K and in the most general case Sigma[i] is a (K, K) matrix. The paper makes one of two approximations to simplify Sigma[i]:

• 'diagonal' in which case Sigma[i] is a vector length K
• 'spherical' in which case Sigma[i] is a single float

To learn the probabilities, first define an energy function E(P[i], P[j]) that returns a similarity like measure of the two probabilities. Both the symmetric Expected Likelihood Inner Product and asymmetric KL-divergence are implemented.

Given a pair of "positive" and "negative" indices, define Delta E = E(P[posi], P[posj]) - E(P[negi], P[negj]). Intuitively the training process optimizes the parameters to make Delta E positive. Formally, use a max-margin loss:

    loss = max(0, Closs - Delta E)

and optimize the parameters to minimize the sum of the loss over the entire training set of positive/negative pairs.

To generate the training pairs, use co-occuring words as the positive examples and randomly sampled words as the negative examples. Since the energy function is potentially asymmetric, for each co-occuring word pair randomly sample both the left and right tokens for negative examples. In addition, we allow the option to generate several sets of training pairs from each word. In pseudo-code:

for sentence in corpus:
    for i in len(sentence):
        for k in 1..window_size:
            for nsample in 1..number_of_samples_per_word:
                positive pair = (left, right) = (sentence[i], sentence[i + k])
                negative pairs = [(left, random ID), (random ID, right)]
                update model weights