Almost three years ago, I wrote a blog entry titled Useful Python Packages, which listed the essential packages that I deemed important. How has the list been changed over the past three years?

First of all, three years ago, most people were still writing Python 2.7. But now there is a trend to switch to Python 3. I admitted that I still have not started the switch yet, but in the short term, I will have no choice and I will.

What are some of the essential packages?
Numerical Packages

• numpy: numerical Python, containing most basic numerical routines such as matrix manipulation, linear algebra, random sampling, numerical integration etc. There is a built-in wrapper for Fortran as well. Actually, numpy is so important that some Linux system includes it with Python.
• scipy: scientific Python, containing some functions useful for scientific computing, such as sparse matrices, numerical differential equations, advanced linear algebra, special functions etc.
• networkx: package that handles various types of networks
• PuLP: linear programming
• cvxopt: convex optimization

Data Visualization

• matplotlib: basic plotting.
• ggplot2: the ggplot2 counterpart in Python for producing quality publication plots.

Data Manipulation

• pandas: data manipulation, working with data frames in Python, and save/load of various formats such as CSV and Excel

Machine Learning

• scikit-learn: machine-learning library in Python, containing classes and functions for supervised and unsupervised learning

Probabilistic Programming

• PyMC: Metropolis-Hasting algorithm
• Edward: deep probabilistic programing

Deep Learning Frameworks

• TensorFlow: because of Google’s marketing effort, TensorFlow is now the industrial standard for building deep learning networks, with rich source of mathematical functions, esp. for neural network cells, with GPU capability
• Keras: containing routines of high-level layers for deep learning neural networks, with TensorFlow, Theano, or CNTK as the backbone
• PyTorch: a rivalry against TensorFlow

Natural Language Processing

• nltk: natural language processing toolkit for Python, containing bag-of-words model, tokenizer, stemmers, chunker, lemmatizers, part-of-speech taggers etc.
• gensim: a useful natural language processing package useful for topic modeling, word-embedding, latent semantic indexing etc., running in a fast fashion
• shorttext: text mining package good for handling short sentences, that provide high-level routines for training neural network classifiers, or generating feature represented by topic models or autoencodings.
• spacy: industrial standard for natural language processing common tools

GUI

I can probably list more, but I think I covered most of them. If you do not find something useful, it is probably time for you to write a brand new package.

In text mining, it is important to create the document-term matrix (DTM) of the corpus we are interested in. A DTM is basically a matrix, with documents designated by rows and words by columns, that the elements are the counts or the weights (usually by tf-idf). Subsequent analysis is usually based creatively on DTM.

Exploring with DTM therefore becomes an important issues with a good text-mining tool. How do we perform exploratory data analysis on DTM using R and Python? We will demonstrate it using the data set of U. S. Presidents’ Inaugural Address, preprocessed, and can be downloaded here.

# R: textmineR

In R, we can use the package textmineR, which has been in introduced in a previous post. Together with other packages such as dplyr (for tidy data analysis) and snowBall (for stemming), load all of them at the beginning:

library(dplyr)
library(textmineR)
library(SnowballC)


usprez.df<- read.csv('inaugural.csv', stringsAsFactors = FALSE)


Then we create the DTM, while we remove all digits and punctuations, make all letters lowercase, and stem all words using Porter stemmer.

dtm<- CreateDtm(usprez.df$speech, doc_names = usprez.df$yrprez,
ngram_window = c(1, 1),
lower = TRUE,
remove_punctuation = TRUE,
remove_numbers = TRUE,
stem_lemma_function = wordStem)


Then defining a set of functions:

get.doc.tokens<- function(dtm, docid)
dtm[docid, ] %>% as.data.frame() %>% rename(count=".") %>%
mutate(token=row.names(.)) %>% arrange(-count)

get.token.occurrences<- function(dtm, token)
dtm[, token] %>% as.data.frame() %>% rename(count=".") %>%
mutate(token=row.names(.)) %>% arrange(-count)

get.total.freq<- function(dtm, token) dtm[, token] %>% sum

get.doc.freq<- function(dtm, token)
dtm[, token] %>% as.data.frame() %>% rename(count=".") %>%
filter(count>0) %>% pull(count) %>% length


Then we can happily extract information. For example, if we want to get the top-most common words in 2009’s Obama’s speech, enter:

dtm %>% get.doc.tokens('2009-Obama') %>% head(10)


Or which speeches have the word “change”: (but need to stem the word before extraction)

dtm %>% get.token.occurrences(wordStem('change')) %>% head(10)


You can also get the total number of occurrence of the words by:

dtm %>% get.doc.freq(wordStem('change'))   # gives 28


# Python: shorttext

In Python, similar things can be done using the package shorttext, described in a previous post. It uses other packages such as pandas and stemming. Load all packages first:

import shorttext
import numpy as np
import pandas as pd
from stemming.porter import stem

import re


And define the preprocessing pipelines:

pipeline = [lambda s: re.sub('[^\w\s]', '', s),
lambda s: re.sub('[\d]', '', s),
lambda s: s.lower(),
lambda s: ' '.join(map(stem, shorttext.utils.tokenize(s)))
]
txtpreproceesor = shorttext.utils.text_preprocessor(pipeline)


The function <code>txtpreprocessor</code> above perform the functions we talked about in R.

usprezdf = pd.read_csv('inaugural.csv')


The corpus needs to be preprocessed before putting into the DTM:

docids = list(usprezdf['yrprez'])    # defining document IDs
corpus = [txtpreproceesor(speech).split(' ') for speech in usprezdf['speech']]


Then create the DTM:

dtm = shorttext.utils.DocumentTermMatrix(corpus, docids=docids, tfidf=False)


Then we do the same thing as we have done above. To get the top-most common words in 2009’s Obama’s speech, enter:

dtm.get_doc_tokens('2009-Obama')


Or we look up which speeches have the word “change”:

dtm.get_token_occurences(stem('change'))


Or to get the document frequency of the word:

dtm.get_doc_frequency(stem('change'))


They Python and R codes give different document frequencies probably because the two stemmers work slightly differently.

When it was proposed that GAN uses Wasserstein distance as the training metric, GAN is usually seen as a transportation problem. Previously, it was mentioned in a previous post that GAN can be seen as a transportation problem, and because of that, some computation can be simplified by relating a kernel in the discriminator and the generator.

GAN can be used in word translation problem too. In a recent preprint in arXiv (refer to arXiv:1710.04087), Wasserstein GAN has been used to train a machine translation machine, given that there are no parallel data between the word embeddings between two languages. The translation mapping is seen as a generator, and the mapping is described using Wasserstein distance. The training objective is cross-domain similarity local scaling (CSLS). Their work has been performed in English-Russian and English-Chinese mappings.

It seems to work. Given GAN sometimes does not work for unknown reasons, it is an excitement that it works.

Recently, Geoffrey Hinton and his colleagues made the article about capsules available. He has been known to heavily criticize the use of pooling and back propagation.

“A capsule is a group of neurons whose activity vector represents the instantiation parameters of a specific type of entity such as an object or object part.” The nodes of inputs and outputs are vectors, instead of scalars as in neural networks. A cheat sheet comparing the traditional neurons and capsules is as follow:

Based on the capsule, the authors suggested a new type of layer called CapsNet.

Huadong Liao implemented CapsNet with TensorFlow according to the paper. (Refer to his repository.)

The theory and the interpretability of deep neural networks have always been called into questions. In the recent few years, there have been several ideas uncovering the theory of neural networks.

# Renormalization Group (RG)

Mehta and Schwab analytically connected renormalization group (RG) with one particular type of deep learning networks, the restricted Boltzmann machines (RBM). (See their paper and a previous post.) RBM is similar to Heisenberg model in statistical physics. This weakness of this work is that it can only explain only one type of deep learning algorithms.

However, this insight gives rise to subsequent work, with the use of density matrix renormalization group (DMRG), entanglement renormalization (in quantum information), and tensor networks, a new supervised learning algorithm was invented. (See their paper and a previous post.)

# Neural Networks as Polynomial Approximation

Lin and Tegmark were not satisfied with the RG intuition, and pointed out a special case that RG does not explain. However, they argue that neural networks are good approximation to several polynomial and asymptotic behaviors of the physical universe, making neural networks work so well in predictive analytics. (See their paper, Lin’s reply on Quora, and a previous post.)

# Information Bottleneck (IB)

Tishby and his colleagues have been promoting information bottleneck as a backing theory of deep learning. (See previous post.) In recent papers such as arXiv:1612.00410, on top of his information bottleneck, they devised an algorithm using variation inference.

# Generalization

Recently, Kawaguchi, Kaelbling, and Bengio suggested that “deep model classes have an exponential advantage to represent certain natural target functions when compared to shallow model classes.” (See their paper and a previous post.) They provided their proof using generalization theory. With this, they introduced a new family of regularization methods.

# Geometric View on Generative Adversarial Networks (GAN)

Recently, Lei, Su, Cui, Yau, and Gu tried to offer a geometric view of generative adversarial networks (GAN), and provided a simpler method of training the discriminator and generator with a large class of transportation problems. However, I am still yet to understand their work, and their experimental results were done on low-dimensional feature spaces. (See their paper.) Their work is very mathematical.

In their paper, Kawaguchi, Kaelbling, and Bengio explored the theory of why generalization in deep learning is so good. Based on their theoretical insights, they proposed a new regularization method, called Directly Approximately Regularizing Complexity (DARC), in addition to commonly used Lp-regularization and dropout methods.

This paper explains why deep learning can generalize well, despite large capacity and possible algorithmic instability, nonrobustness, and sharp minima, effectively addressing an open problem in the literature. Based on our theoretical insight, this paper also proposes a family of new regularization methods. Its simplest member was empirically shown to improve base models and achieve state-of-the-art performance on MNIST and CIFAR-10 benchmarks. Moreover, this paper presents both data-dependent and data-independent generalization guarantees with improved convergence rates. Our results suggest several new open areas of research.

If you have been taking Andrew Ng’s deeplearning.ai course on Coursera, you must have learned in Course 1 about the graph operations, and the method of back propagation using derivatives in terms of graph. In fact, it is the basis of TensorFlow, a Python package commonly used in deep learning. Because it is based on the graph model of computation, we can see it as a “programming language.”

Google published a paper about the big picture of computational model in TensorFlow:

TensorFlow is a powerful, programmable system for machine learning. This paper aims to provide the basics of a conceptual framework for understanding the behavior of TensorFlow models during training and inference: it describes an operational semantics, of the kind common in the literature on programming languages. More broadly, the paper suggests that a programming-language perspective is fruitful in designing and in explaining systems such as TensorFlow.

Beware that this model is not limited to deep learning.

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.

(diagram adapted from Gao and Duan’s article)

Quantum computing has been picking up the momentum, and there are many startups and scholars discussing quantum machine learning. A basic knowledge of quantum two-level computation ought to be acquired.

Recently, Rigetti, a startup for quantum computing service in Bay Area, published that they opened to public their cloud server for users to simulate the use of quantum instruction language, as described in their blog and their White Paper. It is free.

Go to their homepage, http://rigetti.com/, click on “Get Started,” and fill in your information and e-mail. Then you will be e-mailed keys of your cloud account. Copy the information to a file .pyquil_config, and in your .bash_profile, add a line

export PYQUIL_CONFIG="\$HOME/.pyquil_config"

More information can be found in their Installation tutorial. Then install the Python package pyquil, by typing in the command line:

pip install -U pyquil

Some of you may need to root (adding sudo in front).

Then we can go ahead to open Python, or iPython, or Jupyter notebook, to play with it. For the time being, let me play with creating an entangled singlet state, $\frac{1}{\sqrt{2}} (|01\rangle - |10\rangle )$. The corresponding quantum circuit is like this:

First of all, import all necessary libraries:

import numpy as np

from pyquil.quil import Program
import pyquil.api as api
from pyquil.gates import H, X, Z, CNOT


You can see that the package includes a lot of quantum gates. First, we need to instantiate a quantum simulator:

# starting the quantum simulator
quantum_simulator = api.SyncConnection()


Then we implement the quantum circuit with a “program” as follow:

# generating singlet state
# 2. Pauli-Z
# 3. CNOT
# 4. NOT
p = Program(H(0), Z(0), CNOT(0, 1), X(1))
wavefunc, _ = quantum_simulator.wavefunction(p)


The last line gives the final wavefunction after running the quantum circuit, or “program.” For the ket, the rightmost qubit is qubit 0, and the left of it is qubit 1, and so on. Therefore, in the first line of the program, H, the Hadamard gate, acts on qubit 0, i.e., the rightmost qubit. Running a simple print statement:

print wavefunc


gives

(-0.7071067812+0j)|01> + (0.7071067812+0j)|10>

The coefficients are complex, and the imaginary part is described by j. You can extract it as a numpy array:

wavefunc.amplitudes


If we want to calculate the metric of entanglement, we can use the Python package pyqentangle, which can be installed by running on the console:

pip install -U pyqentangle

Import them:

from pyqentangle import schmidt_decomposition
from pyqentangle.schmidt import bipartitepurestate_reduceddensitymatrix
from pyqentangle.metrics import entanglement_entropy, negativity


Because pyqentangle does not recognize the coefficients in the same way as pyquil, but see each element as the coefficients of $|j i \rangle$, we need to reshape the final state first, by:

tensorcomp = wavefunc.amplitudes.reshape((2, 2))


Then perform Schmidt decomposition (which the Schmidt modes are actually trivial in this example):

# Schmidt decomposition
schmidt_modes = schmidt_decomposition(tensorcomp)
for prob, modeA, modeB in schmidt_modes:
print prob, ' : ', modeA, ' ', modeB


This outputs:

0.5  :  [ 0.+0.j  1.+0.j]   [ 1.+0.j  0.+0.j]
0.5  :  [-1.+0.j  0.+0.j]   [ 0.+0.j  1.+0.j]

Calculate the entanglement entropy and negativity from its reduced density matrix:

print 'Entanglement entropy = ', entanglement_entropy(bipartitepurestate_reduceddensitymatrix(tensorcomp, 0))
print 'Negativity = ', negativity(bipartitepurestate_reduceddensitymatrix(tensorcomp, 0))


which prints:

Entanglement entropy =  0.69314718056
Negativity =  -1.11022302463e-16

The calculation can be found in this thesis.

P.S.: The circuit was drawn by using the tool in this website, introduced by the Marco Cezero’s blog post. The corresponding json for the circuit is:

{"gate":[],{"gate":[], "circuit": [{"type":"h", "time":0, "targets":[0], "controls":[]},     {"type":"z", "time":1, "targets":[0], "controls":[]},     {"type":"x", "time":2, "targets":[1], "controls":[0]},     {"type":"x", "time":3, "targets":[1], "controls":[]}], "qubits":2,"input":[0,0]}
Continue reading "A First Glimpse of Rigetti’s Quantum Computing Cloud" →