Neural-Network Representation of Quantum Many-Body States

There are many embeddings algorithm for representations. Sammon embedding is the oldest one, and we have Word2Vec, GloVe, FastText etc. for word-embedding algorithms. Embeddings are useful for dimensionality reduction.

Traditionally, quantum many-body states are represented by Fock states, which is useful when the excitations of quasi-particles are the concern. But to capture the quantum entanglement between many solitons or particles in a statistical systems, it is important not to lose the topological correlation between the states. It has been known that restricted Boltzmann machines (RBM) have been used to represent such states, but it has its limitation, which Xun Gao and Lu-Ming Duan have stated in their article published in Nature Communications:

There exist states, which can be generated by a constant-depth quantum circuit or expressed as PEPS (projected entangled pair states) or ground states of gapped Hamiltonians, but cannot be efficiently represented by any RBM unless the polynomial hierarchy collapses in the computational complexity theory.

PEPS is a generalization of matrix product states (MPS) to higher dimensions. (See this.)

However, Gao and Duan were able to prove that deep Boltzmann machine (DBM) can bridge the loophole of RBM, as stated in their article:

Any quantum state of n qubits generated by a quantum circuit of depth T can be represented exactly by a sparse DBM with O(nT) neurons.

41467_2017_705_fig3_html

(diagram adapted from Gao and Duan’s article)

Continue reading “Neural-Network Representation of Quantum Many-Body States”

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Release of shorttext 0.5.4

The Python package for text mining shorttext has a new release: 0.5.4. It can be installed by typing in the command line:

pip install -U shorttext

For some people, you may need to install it from “root”, i.e., adding sudo in front of the command. Since the version 0.5 (including releases 0.5.1 and 0.5.4), there have been substantial addition of functionality, mostly about comparisons between short phrases without running a supervised or unsupervised machine learning algorithm, but calculating the “similarity” with various metrics, including:

  • soft Jaccard score (the same kind of fuzzy scores based on edit distance in SOCcer),
  • Word Mover’s distance (WMD, detailedly described in a previous post), and
  • Jaccard index due to word-embedding model.

For the soft Jaccard score due to edit distance, we can call it by:

>>> from shorttext.metrics.dynprog import soft_jaccard_score
>>> soft_jaccard_score(['book', 'seller'], ['blok', 'sellers'])     # gives 0.6716417910447762
>>> soft_jaccard_score(['police', 'station'], ['policeman'])        # gives 0.2857142857142858

The core of this code was written in C, and interfaced to Python using SWIG.

For the Word Mover’s Distance (WMD), while the source codes are the same as my previous post, it can now be called directly. First, load the modules and the word-embedding model:

>>> from shorttext.metrics.wasserstein import word_mover_distance
>>> from shorttext.utils import load_word2vec_model
>>> wvmodel = load_word2vec_model('/path/to/model_file.bin')

And compute the WMD with a single function:

>>> word_mover_distance(['police', 'station'], ['policeman'], wvmodel)                      # gives 3.060708999633789
>>> word_mover_distance(['physician', 'assistant'], ['doctor', 'assistants'], wvmodel)      # gives 2.276337146759033

And the Jaccard index due to cosine distance in Word-embedding model can be called like this:

>>> from shorttext.metrics.embedfuzzy import jaccardscore_sents
>>> jaccardscore_sents('doctor', 'physician', wvmodel)   # gives 0.6401538990056869
>>> jaccardscore_sents('chief executive', 'computer cluster', wvmodel)   # gives 0.0022515450768836143
>>> jaccardscore_sents('topological data', 'data of topology', wvmodel)   # gives 0.67588977344632573

Most new functions can be found in this tutorial.

And there are some minor bugs fixed.

Continue reading “Release of shorttext 0.5.4”

Word Mover’s Distance as a Linear Programming Problem

Much about the use of word-embedding models such as Word2Vec and GloVe have been covered. However, how to measure the similarity between phrases or documents? One natural choice is the cosine similarity, as I have toyed with in a previous post. However, it smoothed out the influence of each word. Two years ago, a group in Washington University in St. Louis proposed the Word Mover’s Distance (WMD) in a PMLR paper that captures the relations between words, not simply by distance, but also the “transportation” from one phrase to another conveyed by each word. This Word Mover’s Distance (WMD) can be seen as a special case of Earth Mover’s Distance (EMD), or Wasserstein distance, the one people talked about in Wasserstein GAN. This is better than bag-of-words (BOW) model in a way that the word vectors capture the semantic similarities between words.

Word Mover’s Distance (WMD)

The formulation of WMD is beautiful. Consider the embedded word vectors \mathbf{X} \in R^{d \times n}, where d is the dimension of the embeddings, and n is the number of words. For each phrase, there is a normalized BOW vector d \in R^n, and d_i = \frac{c_i}{\sum_i c_i}, where i‘s denote the word tokens. The distance between words are the Euclidean distance of their embedded word vectors, denoted by c(i, j) = || \mathbf{x}_i - \mathbf{x}_j ||_2, where i and j denote word tokens. The document distance, which is WMD here, is defined by \sum_{i, j} \mathbf{T}_{i j} c(i, j), where \mathbf{T} is a n \times n matrix. Each element \mathbf{T}_{ij} \geq 0 denote how nuch of word i in the first document (denoted by \mathbf{d}) travels to word j in the new document (denoted by \mathbf{d}').

Then the problem becomes the minimization of the document distance, or the WMD, and is formulated as:

\text{min}_{\mathbf{T} \geq 0} \sum_{i, j=1}^n \mathbf{T}_{ij} c(i, j),

given the constraints:

\sum_{j=1}^n \mathbf{T}_{ij} = d_i, and

\sum_{i=1}^n \mathbf{T}_{ij} = d_j'.

This is essentially a simplified case of the Earth Mover’s distance (EMD), or the Wasserstein distance. (See the review by Gibbs and Su.)

Using PuLP

The WMD is essentially a linear optimization problem. There are many optimization packages on the market, and my stance is that, for those common ones, there are no packages that are superior than others. In my job, I happened to handle a missing data problem, in turn becoming a non-linear optimization problem with linear constraints, and I chose limSolve, after I shop around. But I actually like a lot of other packages too. For WMD problem, I first tried out cvxopt first, which should actually solve the exact same problem, but the indexing is hard to maintain. Because I am dealing with words, it is good to have a direct hash map, or a dictionary. I can use the Dictionary class in gensim. But I later found out I should use PuLP, as it allows indices with words as a hash map (dict in Python), and WMD is a linear programming problem, making PuLP is a perfect choice, considering code efficiency.

An example of using PuLP can be demonstrated by the British 1997 UG Exam, as in the first problem of this link, with the Jupyter Notebook demonstrating this.

Implementation of WMD using PuLP

The demonstration can be found in the Jupyter Notebook.

Load the necessary packages:

from itertools import product
from collections import defaultdict

import numpy as np
from scipy.spatial.distance import euclidean
import pulp
import gensim

Then define the functions the gives the BOW document vectors:

def tokens_to_fracdict(tokens):
    cntdict = defaultdict(lambda : 0)
    for token in tokens:
        cntdict[token] += 1
    totalcnt = sum(cntdict.values())
    return {token: float(cnt)/totalcnt for token, cnt in cntdict.items()}

Then implement the core calculation. Note that PuLP is actually a symbolic computing package. This function return a pulp.LpProblem class:

def word_mover_distance_probspec(first_sent_tokens, second_sent_tokens, wvmodel, lpFile=None):
    all_tokens = list(set(first_sent_tokens+second_sent_tokens))
    wordvecs = {token: wvmodel[token] for token in all_tokens}

    first_sent_buckets = tokens_to_fracdict(first_sent_tokens)
    second_sent_buckets = tokens_to_fracdict(second_sent_tokens)

    T = pulp.LpVariable.dicts('T_matrix', list(product(all_tokens, all_tokens)), lowBound=0)

    prob = pulp.LpProblem('WMD', sense=pulp.LpMinimize)
    prob += pulp.lpSum([T[token1, token2]*euclidean(wordvecs[token1], wordvecs[token2])
                        for token1, token2 in product(all_tokens, all_tokens)])
    for token2 in second_sent_buckets:
        prob += pulp.lpSum([T[token1, token2] for token1 in first_sent_buckets])==second_sent_buckets[token2]
    for token1 in first_sent_buckets:
        prob += pulp.lpSum([T[token1, token2] for token2 in second_sent_buckets])==first_sent_buckets[token1]

    if lpFile!=None:
        prob.writeLP(lpFile)

    prob.solve()

    return prob

To extract the value, just run pulp.value(prob.objective)

We use Google Word2Vec. Refer the \mathbf{T} matrices in the Jupyter Notebook. Running this by a few examples:

  1. document1 = President, talk, Chicago
    document2 = President, speech, Illinois
    WMD = 2.88587622936
  2. document1 = physician, assistant
    document2 = doctor
    WMD = 2.8760048151
  3. document1 = physician, assistant
    document2 = doctor, assistant
    WMD = 1.00465738773
    (compare with example 2!)
  4. document1 = doctors, assistant
    document2 = doctor, assistant
    WMD = 1.02825379372
    (compare with example 3!)
  5. document1 = doctor, assistant
    document2 = doctor, assistant
    WMD = 0.0
    (totally identical; compare with example 3!)

There are more examples in the notebook.

Conclusion

WMD is a good metric comparing two documents or sentences, by capturing the semantic meanings of the words. It is more powerful than BOW model as it captures the meaning similarities; it is more powerful than the cosine distance between average word vectors, as the transfer of meaning using words from one document to another is considered. But it is not immune to the problem of misspelling.

This algorithm works well for short texts. However, when the documents become large, this formulation will be computationally expensive. The author actually suggested a few modifications, such as the removal of constraints, and word centroid distances.

Example codes can be found in my Github repository: stephenhky/PyWMD.

Continue reading “Word Mover’s Distance as a Linear Programming Problem”

Sammon Embedding with Tensorflow

Embedding algorithms, especially word-embedding algorithms, have been one of the recurrent themes of this blog. Word2Vec has been mentioned in a few entries (see this); LDA2Vec has been covered (see this); the mathematical principle of GloVe has been elaborated (see this); I haven’t even covered Facebook’s fasttext; and I have not explained the widely used t-SNE and Kohonen’s map (self-organizing map, SOM).

I have also described the algorithm of Sammon Embedding, (see this) which attempts to capture the likeliness of pairwise Euclidean distances, and I implemented it using Theano. This blog entry is about its implementation in Tensorflow as a demonstration.

Let’s recall the formalism of Sammon Embedding, as outlined in the previous entry:

Assume there are high dimensional data described by d-dimensional vectors, X_i where i=1, 2, \ldots, N. And they will be mapped into vectors Y_i, with dimensions 2 or 3. Denote the distances to be d_{ij}^{*} = \sqrt{| X_i - X_j|^2} and d_{ij} = \sqrt{| Y_i - Y_j|^2}. In this problem, Y_i are the variables to be learned. The cost function to minimize is

E = \frac{1}{c} \sum_{i<j} \frac{(d_{ij}^{*} - d_{ij})^2}{d_{ij}^{*}},

where c = \sum_{i<j} d_{ij}^{*}.

Unlike in previous entry and original paper, I am going to optimize it using first-order gradient optimizer. If you are not familiar with Tensorflow, take a look at some online articles, for example, “Tensorflow demystified.” This demonstration can be found in this Jupyter Notebook in Github.

First of all, import all the libraries required:

import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf

Like previously, we want to use the points clustered around at the four nodes of a tetrahedron as an illustration, which is expected to give equidistant clusters. We sample points around them, as shown:

tetrahedron_points = [np.array([0., 0., 0.]), np.array([1., 0., 0.]), np.array([np.cos(np.pi/3), np.sin(np.pi/3), 0.]), np.array([0.5, 0.5/np.sqrt(3), np.sqrt(2./3.)])]

sampled_points = np.concatenate([np.random.multivariate_normal(point, np.eye(3)*0.0001, 10) for point in tetrahedron_points])

init_points = np.concatenate([np.random.multivariate_normal(point[:2], np.eye(2)*0.0001, 10) for point in tetrahedron_points])

Retrieve the number of points, N, and the resulting dimension, d:

N = sampled_points.shape[0]
d = sampled_points.shape[1]

One of the most challenging technical difficulties is to calculate the pairwise distance. Inspired by this StackOverflow thread and Travis Hoppe’s entry on Thomson’s problem, we know it can be computed. Assuming Einstein’s convention of summation over repeated indices, given vectors a_{ik}, the distance matrix is:

D_{ij} = (a_{ik}-a_{jk}) (a_{ik} - a_{jk})^T = a_{ik} a_{ik} + a_{jk} a_{jk} - 2 a_{ik} a_{jk},

where the first and last terms are simply the norms of the vectors. After computing the matrix, we will flatten it to vectors, for technical reasons omitted to avoid gradient overflow:

X = tf.placeholder('float')
Xshape = tf.shape(X)

sqX = tf.reduce_sum(X*X, 1)
sqX = tf.reshape(sqX, [-1, 1])
sqDX = sqX - 2*tf.matmul(X, tf.transpose(X)) + tf.transpose(sqX)
sqDXarray = tf.stack([sqDX[i, j] for i in range(N) for j in range(i+1, N)])
DXarray = tf.sqrt(sqDXarray)

Y = tf.Variable(init_points, dtype='float')
sqY = tf.reduce_sum(Y*Y, 1)
sqY = tf.reshape(sqY, [-1, 1])
sqDY = sqY - 2*tf.matmul(Y, tf.transpose(Y)) + tf.transpose(sqY)
sqDYarray = tf.stack([sqDY[i, j] for i in range(N) for j in range(i+1, N)])
DYarray = tf.sqrt(sqDYarray)

And DXarray and DYarray are the vectorized pairwise distances. Then we defined the cost function according to the definition:

Z = tf.reduce_sum(DXarray)*0.5
numerator = tf.reduce_sum(tf.divide(tf.square(DXarray-DYarray), DXarray))*0.5
cost = tf.divide(numerator, Z)

As we said, we used first-order gradient optimizers. For unknown reasons, the usually well-performing Adam optimizer gives overflow. I then picked Adagrad:

update_rule = tf.assign(Y, Y-0.01*grad_cost/lapl_cost)
train = tf.train.AdamOptimizer(0.01).minimize(cost)
init = tf.global_variables_initializer()

The last line initializes all variables in the Tensorflow session when it is run. Then start a Tensorflow session, and initialize all variables globally:

sess = tf.Session()
sess.run(init)

Then run the algorithm:

nbsteps = 1000
c = sess.run(cost, feed_dict={X: sampled_points})
print "epoch: ", -1, " cost = ", c
for i in range(nbsteps):
    sess.run(train, feed_dict={X: sampled_points})
    c = sess.run(cost, feed_dict={X: sampled_points})
    print "epoch: ", i, " cost =

Then extract the points and close the Tensorflow session:

calculated_Y = sess.run(Y, feed_dict={X: sampled_points})
sess.close()

Plot it using matplotlib:

embed1, embed2 = calculated_Y.transpose()
plt.plot(embed1, embed2, 'ro')

This gives, as expected,

download (5)

This code for Sammon Embedding has been incorporated into the Python package mogu, which is a collection of numerical routines. You can install it, and call:

from mogu.embed import sammon_embedding
calculated_Y = sammon_embedding(sampled_points, init_points)

Continue reading “Sammon Embedding with Tensorflow”

Release of shorttext 0.3.3

On November 21, 2016, the Python package `shorttext’ was published. Until today, more than seven versions have been published. There have been a drastic architecture change, but the overall purpose is still the same, as summarized in the first introduction entry:

This package `shorttext‘ was designed to tackle all these problems… It contains the following features:

  • example data provided (including subject keywords and NIH RePORT);
  • text preprocessing;
  • pre-trained word-embedding support;
  • gensim topic models (LDA, LSI, Random Projections) and autoencoder;
  • topic model representation supported for supervised learning using scikit-learn;
  • cosine distance classification; and
  • neural network classification (including ConvNet, and C-LSTM).

And since the first version, there have been updates, as summarized in the documention (News):

Version 0.3.3 (Apr 19, 2017)

  • Deleted CNNEmbedVecClassifier.
  • Added script ShortTextWord2VecSimilarity.

Version 0.3.2 (Mar 28, 2017)

  • Bug fixed for gensim model I/O;
  • Console scripts update;
  • Neural networks up to Keras 2 standard (refer to this).

Version 0.3.1 (Mar 14, 2017)

  • Compact model I/O: all models are in single files;
  • Implementation of stacked generalization using logistic regression.

Version 0.2.1 (Feb 23, 2017)

  • Removal attempts of loading GloVe model, as it can be run using gensim script;
  • Confirmed compatibility of the package with tensorflow;
  • Use of spacy for tokenization, instead of nltk;
  • Use of stemming for Porter stemmer, instead of nltk;
  • Removal of nltk dependencies;
  • Simplifying the directory and module structures;
  • Module packages updated.

Although there are still additions that I would love to add, but it would not change the overall architecture. I may add some more supervised learning algorithms, but under the same network. The upcoming big additions will be generative models or seq2seq models, but I do not see them coming in the short term. I will add corpuses.

I may add tutorials if I have time.

I am thankful that there is probably some external collaboration with other Python packages. Some people have already made some useful contributions. It will be updated if more things are confirmed.

Continue reading “Release of shorttext 0.3.3”

Python Package for Short Text Mining

There has been a lot of methods for natural language processing and text mining. However, in tweets, surveys, Facebook, or many online data, texts are short, lacking data to build enough information. Traditional bag-of-words (BOW) model gives sparse vector representation.

Semantic relations between words are important, because we usually do not have enough data to capture the similarity between words. We do not want “drive” and “drives,” or “driver” and “chauffeur” to be completely different.

The relation between or order of words become important as well. Or we want to capture the concepts that may be correlated in our training dataset.

We have to represent these texts in a special way and perform supervised learning with traditional machine learning algorithms or deep learning algorithms.

This package `shorttext‘ was designed to tackle all these problems. It is not a completely new invention, but putting everything known together. It contains the following features:

  • example data provided (including subject keywords and NIH RePORT);
  • text preprocessing;
  • pre-trained word-embedding support;
  • gensim topic models (LDA, LSI, Random Projections) and autoencoder;
  • topic model representation supported for supervised learning using scikit-learn;
  • cosine distance classification; and
  • neural network classification (including ConvNet, and C-LSTM).

Readers can refer this to the documentation.

Continue reading “Python Package for Short Text Mining”

Short Text Categorization using Deep Neural Networks and Word-Embedding Models

There are situations that we deal with short text, probably messy, without a lot of training data. In that case, we need external semantic information. Instead of using the conventional bag-of-words (BOW) model, we should employ word-embedding models, such as Word2Vec, GloVe etc.

Suppose we want to perform supervised learning, with three subjects, described by the following Python dictionary:

classdict={'mathematics': ['linear algebra',
           'topology',
           'algebra',
           'calculus',
           'variational calculus',
           'functional field',
           'real analysis',
           'complex analysis',
           'differential equation',
           'statistics',
           'statistical optimization',
           'probability',
           'stochastic calculus',
           'numerical analysis',
           'differential geometry'],
          'physics': ['renormalization',
           'classical mechanics',
           'quantum mechanics',
           'statistical mechanics',
           'functional field',
           'path integral',
           'quantum field theory',
           'electrodynamics',
           'condensed matter',
           'particle physics',
           'topological solitons',
           'astrophysics',
           'spontaneous symmetry breaking',
           'atomic molecular and optical physics',
           'quantum chaos'],
          'theology': ['divine providence',
           'soteriology',
           'anthropology',
           'pneumatology',
           'Christology',
           'Holy Trinity',
           'eschatology',
           'scripture',
           'ecclesiology',
           'predestination',
           'divine degree',
           'creedal confessionalism',
           'scholasticism',
           'prayer',
           'eucharist']}

And we implemented Word2Vec here. To add external information, we use a pre-trained Word2Vec model from Google, downloaded here. We can use it with Python package gensim. To load it, enter

from gensim.models import Word2Vec
wvmodel = Word2Vec.load_word2vec_format('<path-to>/GoogleNews-vectors-negative300.bin.gz', binary=True)

How do we represent a phrase in Word2Vec? How do we do the classification? Here I wrote two classes to do it.

Average

We can represent a sentence by summing the word-embedding representations of each word. The class, inside SumWord2VecClassification.py, is coded as follow:

from collections import defaultdict

import numpy as np
from nltk import word_tokenize
from scipy.spatial.distance import cosine

from utils import ModelNotTrainedException

class SumEmbeddedVecClassifier:
    def __init__(self, wvmodel, classdict, vecsize=300):
        self.wvmodel = wvmodel
        self.classdict = classdict
        self.vecsize = vecsize
        self.trained = False

    def train(self):
        self.addvec = defaultdict(lambda : np.zeros(self.vecsize))
        for classtype in self.classdict:
            for shorttext in self.classdict[classtype]:
                self.addvec[classtype] += self.shorttext_to_embedvec(shorttext)
            self.addvec[classtype] /= np.linalg.norm(self.addvec[classtype])
        self.addvec = dict(self.addvec)
        self.trained = True

    def shorttext_to_embedvec(self, shorttext):
        vec = np.zeros(self.vecsize)
        tokens = word_tokenize(shorttext)
        for token in tokens:
            if token in self.wvmodel:
                vec += self.wvmodel[token]
        norm = np.linalg.norm(vec)
        if norm!=0:
            vec /= np.linalg.norm(vec)
        return vec

    def score(self, shorttext):
        if not self.trained:
            raise ModelNotTrainedException()
        vec = self.shorttext_to_embedvec(shorttext)
        scoredict = {}
        for classtype in self.addvec:
            try:
                scoredict[classtype] = 1 - cosine(vec, self.addvec[classtype])
            except ValueError:
                scoredict[classtype] = np.nan
        return scoredict

Here the exception ModelNotTrainedException is just an exception raised if the model has not been trained yet, but scoring function was called by the user. (Codes listed in my Github repository.) The similarity will be calculated by cosine similarity.

Such an implementation is easy to understand and carry out. It is good enough for a lot of application. However, it has the problem that it does not take the relation between words or word order into account.

Convolutional Neural Network

To tackle the problem of word relations, we have to use deeper neural networks. Yoon Kim published a well cited paper regarding this in EMNLP in 2014, titled “Convolutional Neural Networks for Sentence Classification.” The model architecture is as follow: (taken from his paper)

cnn

Each word is represented by an embedded vector, but neighboring words are related through the convolutional matrix. And MaxPooling and a dense neural network were implemented afterwards. His paper involves multiple filters with variable window sizes / spatial extent, but for our cases of short phrases, I just use one window of size 2 (similar to dealing with bigram). While Kim implemented using Theano (see his Github repository), I implemented using keras with Theano backend. The codes, inside CNNEmbedVecClassification.py, are as follow:

import numpy as np
from keras.layers import Convolution1D, MaxPooling1D, Flatten, Dense
from keras.models import Sequential
from nltk import word_tokenize

from utils import ModelNotTrainedException

class CNNEmbeddedVecClassifier:
    def __init__(self,
                 wvmodel,
                 classdict,
                 n_gram,
                 vecsize=300,
                 nb_filters=1200,
                 maxlen=15):
        self.wvmodel = wvmodel
        self.classdict = classdict
        self.n_gram = n_gram
        self.vecsize = vecsize
        self.nb_filters = nb_filters
        self.maxlen = maxlen
        self.trained = False

    def convert_trainingdata_matrix(self):
        classlabels = self.classdict.keys()
        lblidx_dict = dict(zip(classlabels, range(len(classlabels))))

        # tokenize the words, and determine the word length
        phrases = []
        indices = []
        for label in classlabels:
            for shorttext in self.classdict[label]:
                category_bucket = [0]*len(classlabels)
                category_bucket[lblidx_dict[label]] = 1
                indices.append(category_bucket)
                phrases.append(word_tokenize(shorttext))

        # store embedded vectors
        train_embedvec = np.zeros(shape=(len(phrases), self.maxlen, self.vecsize))
        for i in range(len(phrases)):
            for j in range(min(self.maxlen, len(phrases[i]))):
                train_embedvec[i, j] = self.word_to_embedvec(phrases[i][j])
        indices = np.array(indices, dtype=np.int)

        return classlabels, train_embedvec, indices

    def train(self):
        # convert classdict to training input vectors
        self.classlabels, train_embedvec, indices = self.convert_trainingdata_matrix()

        # build the deep neural network model
        model = Sequential()
        model.add(Convolution1D(nb_filter=self.nb_filters,
                                filter_length=self.n_gram,
                                border_mode='valid',
                                activation='relu',
                                input_shape=(self.maxlen, self.vecsize)))
        model.add(MaxPooling1D(pool_length=self.maxlen-self.n_gram+1))
        model.add(Flatten())
        model.add(Dense(len(self.classlabels), activation='softmax'))
        model.compile(loss='categorical_crossentropy', optimizer='rmsprop')

        # train the model
        model.fit(train_embedvec, indices)

        # flag switch
        self.model = model
        self.trained = True

    def word_to_embedvec(self, word):
        return self.wvmodel[word] if word in self.wvmodel else np.zeros(self.vecsize)

    def shorttext_to_matrix(self, shorttext):
        tokens = word_tokenize(shorttext)
        matrix = np.zeros((self.maxlen, self.vecsize))
        for i in range(min(self.maxlen, len(tokens))):
            matrix[i] = self.word_to_embedvec(tokens[i])
        return matrix

    def score(self, shorttext):
        if not self.trained:
            raise ModelNotTrainedException()

        # retrieve vector
        matrix = np.array([self.shorttext_to_matrix(shorttext)])

        # classification using the neural network
        predictions = self.model.predict(matrix)

        # wrangle output result
        scoredict = {}
        for idx, classlabel in zip(range(len(self.classlabels)), self.classlabels):
            scoredict[classlabel] = predictions[0][idx]
        return scoredict

The output is a vector of length equal to the number of class labels, 3 in our example. The elements of the output vector add up to one, indicating its score, and a nature of probability.

Evaluation

A simple cross-validation to the example data set does not tell a difference between the two algorithms:

rplot_acc1

However, we can test the algorithm with a few examples:

Example 1: “renormalization”

  • Average: {‘mathematics’: 0.54135105096749336, ‘physics’: 0.63665460856632494, ‘theology’: 0.31014049736087901}
  • CNN: {‘mathematics’: 0.093827009201049805, ‘physics’: 0.85451591014862061, ‘theology’: 0.051657050848007202}

As renormalization was a strong word in the training data, it gives an easy result. CNN can distinguish much more clearly.

Example 2: “salvation”

  • Average: {‘mathematics’: 0.14939650156482298, ‘physics’: 0.21692765541184023, ‘theology’: 0.5698233329716329}
  • CNN: {‘mathematics’: 0.012395491823554039, ‘physics’: 0.022725773975253105, ‘theology’: 0.96487873792648315}

“Salvation” is not found in the training data, but it is closely related to “soteriology,” which means the doctrine of salvation. So it correctly identifies it with theology.

Example 3: “coffee”

  • Average: {‘mathematics’: 0.096820211601723272, ‘physics’: 0.081567332119268032, ‘theology’: 0.15962682945135631}
  • CNN: {‘mathematics’: 0.27321341633796692, ‘physics’: 0.1950736939907074, ‘theology’: 0.53171288967132568}

Coffee is not related to all subjects. The first architecture correctly indicates the fact, but CNN, with its probabilistic nature, has to roughly equally distribute it (but not so well.)

The code can be found in my Github repository: stephenhky/PyShortTextCategorization. (This repository has been updated since this article was published. The link shows the version of the code when this appeared online.)

Continue reading “Short Text Categorization using Deep Neural Networks and Word-Embedding Models”

Law Prediction

On August 1, my friends and I attended a meetup host by DC Data Science, titled “Predicting and Understanding Law with Machine Learning.” The speaker was John Nay, a Ph.D. candidate in Vanderbilt University. He presented his research which is at an application of natural language processing on legal enactment documents.

His talk was very interesting, from the similarity of presidents and the chambers, to the kind of topics each party focused on. He used a variety of techniques such as Word2Vec, STM (structural topic modeling), and some common textual and statistical analysis. It is quite a comprehensive study.

His work is demonstrated at predictgov.com. His work can be found in arXiv.

Continue reading “Law Prediction”

Probabilistic Theory of Word Embeddings: GloVe

The topic of word embedding algorithms has been one of the interests of this blog, as in this entry, with Word2Vec [Mikilov et. al. 2013] as one of the main examples. It is a great tool for text mining, (for example, see [Czerny 2015],) as it reduces the dimensions needed (compared to bag-of-words model). As an algorithm borrowed from computer vision, a lot of these algorithms use deep learning methods to train the model, while it was not exactly sure why it works. Despite that, there are many articles talking about how to train the model. [Goldberg & Levy 2014, Rong 2014 etc.] Addition and subtraction of the word vectors show amazing relationships that carry semantic values, as I have shown in my previous blog entry. [Ho 2015]

However, Tomas Mikolov is no longer working in Google, making the development of this algorithm discontinued. As a follow-up of their work, Stanford NLP group later proposed a model, called GloVe (Global Vectors), that embeds word vectors using probabilistic methods. [Pennington, Socher & Manning 2014] It can be implemented in the package glove-python in python, and text2vec in R (or see their CRAN post).  Their paper is neatly written, and a highly recommended read.

To explain the theory of GloVe, we start with some basic probabilistic picture in basic natural language processing (NLP). We suppose the relation between the words occur in certain text windows within a corpus, but the details are not important here. Assume that i, j, and k are three words, and the conditional probability P_{ik} is defined as

P_{ij} = P(j | i) = \frac{X_{ij}}{X_i},

where X‘s are the counts, and similarly for P_{jk}. And we are interested in the following ratio:

F(w_i, w_j, \tilde{w}_k) = \frac{P_{ik}}{P_{jk}}.

The tilde means “context,” but we will later assume it is also a word. Citing the example from their paper, take i as ice, and j as steam. if k is solid, then the ratio is expected to be large; or if k is gas, then it is expected to be low. But if k is water, which are related to both, or fashion, which is related to none, then the ratio is expected to be approximately 1.

And the addition and subtraction in Word2Vec is similar to this. We want the ratio to be like the subtraction as in Word2Vec (and multiplication as in addition), then we should modify the function F such that,

F(w_i - w_j, \tilde{w}_k) = \frac{P_{ik}}{P_{jk}}.

On the other hand, the input arguments of F are vectors, but the output is a scalar. We avoid the issue by making the input argument as a dot product,

F( (w_i - w_j)^T \tilde{w}_k) = \frac{P_{ik}}{P_{jk}}.

In NLP, the word-word co-occurrence matrices are symmetric, and our function F should also be invariant under switching the labeling. We first require F is be a homomorphism,

F((w_i - w_j)^T \tilde{w}_k) = \frac{F(w_i^T \tilde{w}_k) }{ F(w_j^T \tilde{w}_k)},

where we define,

F(w_i^T \tilde{w}_k) = P_{ik} = \frac{X_{ik}}{X_i}.

It is clear that F is an exponential function, but to ensure symmetry, we further define:

w_i^T \tilde{w}_k + b_i + \tilde{b}_k = \log X_{ik}.

As a result of this equation, the authors defined the following cost function to optimize for GloVe model:

J = \sum_{i, j=1}^V f(X_{ij}) \left( w_i^T \tilde{w}_j + b_i + \tilde{b}_j - \log X_{ik} \right)^2,

where w_j, \tilde{w}_j, b_i, and \tilde{b}_j are parameters to learn. f(x) is a weighting function. (Refer the details to the paper.) [Pennington, Socher & Manning 2014]

As Radim Řehůřek said in his blog entry, [Řehůřek 2014] it is a neat paper, but their evaluation is crappy.

This theory explained why certain similar relations can be achieved, such as Paris – France is roughly equal to Beijing – China, as both can be transformed to the ratio in the definition of F above.

It is a neat paper, as it employs optimization theory and probability theory, without any dark box deep learning.

Continue reading “Probabilistic Theory of Word Embeddings: GloVe”

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