*“Today you are You, that is truer than true. There is no one alive who is Youer than You.”*

Dr. Seuss

*“Explanations exist; they have existed for all time; there is always a well-known solution to every human problem — neat, plausible, and wrong.”*

H L Mencken

# Introduction

In this 6th instalment of ‘Deep Learning from first principles in Python, R and Octave-Part6’, I look at a couple of different initialization techniques used in Deep Learning, L2 regularization and the ‘dropout’ method. Specifically, I implement “He initialization” & “Xavier Initialization”. My earlier posts in this series of Deep Learning included

1. Part 1 – In the 1st part, I implemented logistic regression as a simple 2 layer Neural Network

2. Part 2 – In part 2, implemented the most basic of Neural Networks, with just 1 hidden layer, and any number of activation units in that hidden layer. The implementation was in vectorized Python, R and Octave

3. Part 3 -In part 3, I derive the equations and also implement a L-Layer Deep Learning network with either the relu, tanh or sigmoid activation function in Python, R and Octave. The output activation unit was a sigmoid function for logistic classification

4. Part 4 – This part looks at multi-class classification, and I derive the Jacobian of a Softmax function and implement a simple problem to perform multi-class classification.

5. Part 5 – In the 5th part, I extend the L-Layer Deep Learning network implemented in Part 3, to include the Softmax classification. I also use this L-layer implementation to classify MNIST handwritten digits with Python, R and Octave.

The code in Python, R and Octave are identical, and just take into account some of the minor idiosyncrasies of the individual language. In this post, I implement different initialization techniques (random, He, Xavier), L2 regularization and finally dropout. Hence my generic L-Layer Deep Learning network includes these additional enhancements for enabling/disabling initialization methods, regularization or dropout in the algorithm. It already included sigmoid & softmax output activation for binary and multi-class classification, besides allowing relu, tanh and sigmoid activation for hidden units.

A video presentation of regularization and initialization techniques can be also be viewed in Neural Networks 6

This R Markdown file and the code for Python, R and Octave can be cloned/downloaded from **Github** at DeepLearning-Part6

Checkout my book ‘Deep Learning from first principles: Second Edition – In vectorized Python, R and Octave’. My book starts with the implementation of a simple 2-layer Neural Network and works its way to a generic L-Layer Deep Learning Network, with all the bells and whistles. The derivations have been discussed in detail. The code has been extensively commented and included in its entirety in the Appendix sections. My book is available on Amazon as paperback ($18.99) and in kindle version($9.99/Rs449).

You may also like my companion book “Practical Machine Learning with R and Python:Second Edition- Machine Learning in stereo” available in Amazon in paperback($10.99) and Kindle($7.99/Rs449) versions. This book is ideal for a quick reference of the various ML functions and associated measurements in both R and Python which are essential to delve deep into Deep Learning.

## 1. Initialization techniques

The usual initialization technique is to generate Gaussian or uniform random numbers and multiply it by a small value like 0.01. Two techniques which are used to speed up convergence is the He initialization or Xavier. These initialization techniques enable gradient descent to converge faster.

## 1.1 a Default initialization – Python

This technique just initializes the weights to small random values based on Gaussian or uniform distribution

```
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
exec(open("DLfunctions61.py").read())
train_X, train_Y, test_X, test_Y = load_dataset()
layersDimensions = [2,7,1]
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid",learningRate = 0.6, num_iterations = 9000, initType="default", print_cost = True,figure="fig1.png")
plt.clf()
plt.close()
plot_decision_boundary(lambda x: predict(parameters, x.T), train_X, train_Y,str(0.6),figure1="fig2.png")
```

## 1.1 b He initialization – Python

‘He’ initialization attributed to He et al, multiplies the random weights by

```
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
exec(open("DLfunctions61.py").read())
train_X, train_Y, test_X, test_Y = load_dataset()
layersDimensions = [2,7,1]
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid", learningRate =0.6, num_iterations = 10000,initType="He",print_cost = True, figure="fig3.png")
plt.clf()
plt.close()
plot_decision_boundary(lambda x: predict(parameters, x.T), train_X, train_Y,str(0.6),figure1="fig4.png")
```

## 1.1 c Xavier initialization – Python

Xavier initialization multiply the random weights by

```
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
exec(open("DLfunctions61.py").read())
train_X, train_Y, test_X, test_Y = load_dataset()
layersDimensions = [2,7,1]
# Train a L layer Deep Learning network
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid",
learningRate = 0.6,num_iterations = 10000, initType="Xavier",print_cost = True,
figure="fig5.png")
plot_decision_boundary(lambda x: predict(parameters, x.T), train_X, train_Y,str(0.6),figure1="fig6.png")
```

## 1.2a Default initialization – R

```
source("DLfunctions61.R")
#Load the data
z <- as.matrix(read.csv("circles.csv",header=FALSE))
x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
#Set the layer dimensions
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
numIterations = 8000,
initType="default",
print_cost = True)
```

```
iterations <- seq(0,8000,1000)
costs=retvals$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
ggtitle("Costs vs iterations") + xlab("No of iterations") + ylab("Cost")
```

```
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",lr=0.5)
```

## 1.2b He initialization – R

The code for ‘He’ initilaization in R is included below

# He Initialization model for L layers
# Input : List of units in each layer
# Returns: Initial weights and biases matrices for all layers
# He initilization multiplies the random numbers with sqrt(2/layerDimensions[previouslayer])
HeInitializeDeepModel <- function(layerDimensions){
set.seed(2)
# Initialize empty list
layerParams <- list()
# Note the Weight matrix at layer 'l' is a matrix of size (l,l-1)
# The Bias is a vectors of size (l,1)
# Loop through the layer dimension from 1.. L
# Indices in R start from 1
for(l in 2:length(layersDimensions)){
# Initialize a matrix of small random numbers of size l x l-1
# Create random numbers of size l x l-1
w=rnorm(layersDimensions[l]*layersDimensions[l-1])
# Create a weight matrix of size l x l-1 with this initial weights and
# Add to list W1,W2... WL
# He initialization - Divide by sqrt(2/layerDimensions[previous layer])
layerParams[[paste('W',l-1,sep="")]] = matrix(w,nrow=layersDimensions[l],
ncol=layersDimensions[l-1])*sqrt(2/layersDimensions[l-1])
layerParams[[paste('b',l-1,sep="")]] = matrix(rep(0,layersDimensions[l]),
nrow=layersDimensions[l],ncol=1)
}
return(layerParams)
}

The code in R below uses He initialization to learn the data

```
source("DLfunctions61.R")
# Load the data
z <- as.matrix(read.csv("circles.csv",header=FALSE))
x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
# Set the layer dimensions
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
numIterations = 9000,
initType="He",
print_cost = True)
```

```
iterations <- seq(0,9000,1000)
costs=retvals$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
ggtitle("Costs vs iterations") + xlab("No of iterations") + ylab("Cost")
```

```
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",0.5,lr=0.5)
```

## 1.2c Xavier initialization – R

```
# Set the layer dimensions
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
numIterations = 9000,
initType="Xav",
print_cost = True)
```

```
iterations <- seq(0,9000,1000)
costs=retvals$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
ggtitle("Costs vs iterations") + xlab("No of iterations") + ylab("Cost")
```

```
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",0.5)
```

## 1.3a Default initialization – Octave

source("DL61functions.m")
# Read the data
data=csvread("circles.csv");
X=data(:,1:2);
Y=data(:,3);
# Set the layer dimensions
layersDimensions = [2 11 1]; #tanh=-0.5(ok), #relu=0.1 best!
# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
lambd=0,
keep_prob=1,
numIterations = 10000,
initType="default");
# Plot cost vs iterations
plotCostVsIterations(10000,costs)
#Plot decision boundary
plotDecisionBoundary(data,weights, biases,keep_prob=1, hiddenActivationFunc="relu")

##

## 1.3b He initialization – Octave

source("DL61functions.m")
#Load data
data=csvread("circles.csv");
X=data(:,1:2);
Y=data(:,3);
# Set the layer dimensions
layersDimensions = [2 11 1]; #tanh=-0.5(ok), #relu=0.1 best!
# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
lambd=0,
keep_prob=1,
numIterations = 8000,
initType="He");
plotCostVsIterations(8000,costs)
#Plot decision boundary
plotDecisionBoundary(data,weights, biases,keep_prob=1,hiddenActivationFunc="relu")

## 1.3c Xavier initialization – Octave

The code snippet for Xavier initialization in Octave is shown below

source("DL61functions.m")
# Xavier Initialization for L layers
# Input : List of units in each layer
# Returns: Initial weights and biases matrices for all layers
function [W b] = XavInitializeDeepModel(layerDimensions)
rand ("seed", 3);
# note the Weight matrix at layer 'l' is a matrix of size (l,l-1)
# The Bias is a vectors of size (l,1)
# Loop through the layer dimension from 1.. L
# Create cell arrays for Weights and biases
for l =2:size(layerDimensions)(2)
W{l-1} = rand(layerDimensions(l),layerDimensions(l-1))* sqrt(1/layerDimensions(l-1)); # Multiply by .01
b{l-1} = zeros(layerDimensions(l),1);
endfor
end

The Octave code below uses Xavier initialization

source("DL61functions.m")
#Load data
data=csvread("circles.csv");
X=data(:,1:2);
Y=data(:,3);
#Set layer dimensions
layersDimensions = [2 11 1]; #tanh=-0.5(ok), #relu=0.1 best!
# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
lambd=0,
keep_prob=1,
numIterations = 8000,
initType="Xav");
plotCostVsIterations(8000,costs)
plotDecisionBoundary(data,weights, biases,keep_prob=1,hiddenActivationFunc="relu")

## 2.1a Regularization : Circles data – Python

The cross entropy cost for Logistic classification is given as The regularized L2 cost is given by

```
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
exec(open("DLfunctions61.py").read())
train_X, train_Y, test_X, test_Y = load_dataset()
layersDimensions = [2,7,1]
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",learningRate = 0.6, lambd=0.1, num_iterations = 9000,
initType="default", print_cost = True,figure="fig7.png")
plt.clf()
plt.close()
plot_decision_boundary(lambda x: predict(parameters, x.T), train_X, train_Y,str(0.6),figure1="fig8.png")
plt.clf()
plt.close()
#Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T,keep_prob=0.9), train_X, train_Y,str(2.2),"fig8.png",)
```

## 2.1 b Regularization: Spiral data – Python

```
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
exec(open("DLfunctions61.py").read())
N = 100
D = 2
K = 3
X = np.zeros((N*K,D))
y = np.zeros(N*K, dtype='uint8')
for j in range(K):
ix = range(N*j,N*(j+1))
r = np.linspace(0.0,1,N)
t = np.linspace(j*4,(j+1)*4,N) + np.random.randn(N)*0.2
X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]
y[ix] = j
plt.scatter(X[:, 0], X[:, 1], c=y, s=40, cmap=plt.cm.Spectral)
plt.clf()
plt.close()
#Set layer dimensions
layersDimensions = [2,100,3]
y1=y.reshape(-1,1).T
# Train a deep learning network
parameters = L_Layer_DeepModel(X.T, y1, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax",
learningRate = 1,lambd=1e-3, num_iterations = 5000, print_cost = True,figure="fig9.png")
plt.clf()
plt.close()
W1=parameters['W1']
b1=parameters['b1']
W2=parameters['W2']
b2=parameters['b2']
plot_decision_boundary1(X, y1,W1,b1,W2,b2,figure2="fig10.png")
```

## 2.2a Regularization: Circles data – R

```
source("DLfunctions61.R")
#Load data
df=read.csv("circles.csv",header=FALSE)
z <- as.matrix(read.csv("circles.csv",header=FALSE))
x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
#Set layer dimensions
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
lambd=0.1,
numIterations = 9000,
initType="default",
print_cost = True)
```

```
```

```
iterations <- seq(0,9000,1000)
costs=retvals$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
ggtitle("Costs vs iterations") + xlab("No of iterations") + ylab("Cost")
```

```
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",0.5)
```

## 2.2b Regularization:Spiral data – R

```
#Load the data
source("DLfunctions61.R")
Z <- as.matrix(read.csv("spiral.csv",header=FALSE))
X <- Z[,1:2]
y <- Z[,3]
X <- t(X)
Y <- t(y)
layersDimensions = c(2, 100, 3)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="softmax",
learningRate = 0.5,
lambd=0.01,
numIterations = 9000,
print_cost = True)
```

`print_cost = True)`

```
parameters<-retvals$parameters
plotDecisionBoundary1(Z,parameters)
```

2.3a Regularization: Circles data – Octave

source("DL61functions.m")
#Load data
data=csvread("circles.csv");
X=data(:,1:2);
Y=data(:,3);
layersDimensions = [2 11 1]; #tanh=-0.5(ok), #relu=0.1 best!
# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
lambd=0.2,
keep_prob=1,
numIterations = 8000,
initType="default");
plotCostVsIterations(8000,costs)
#Plot decision boundary
plotDecisionBoundary(data,weights, biases,keep_prob=1,hiddenActivationFunc="relu")

## 2.3b Regularization:Spiral data 2 – Octave

source("DL61functions.m")
data=csvread("spiral.csv");
# Setup the data
X=data(:,1:2);
Y=data(:,3);
layersDimensions = [2 100 3]
# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="softmax",
learningRate = 0.6,
lambd=0.2,
keep_prob=1,
numIterations = 10000);
plotCostVsIterations(10000,costs)
#Plot decision boundary
plotDecisionBoundary1(data,weights, biases,keep_prob=1,hiddenActivationFunc="relu")

## 3.1 a Dropout: Circles data – Python

The ‘dropout’ regularization technique was used with great effectiveness, to prevent overfitting by Alex Krizhevsky, Ilya Sutskever and Prof Geoffrey E. Hinton in the Imagenet classification with Deep Convolutional Neural Networks

The technique of dropout works by dropping a random set of activation units in each hidden layer, based on a ‘keep_prob’ criteria in the forward propagation cycle. Here is the code for Octave. A ‘dropoutMat’ is created for each layer which specifies which units to drop **Note**: The same ‘dropoutMat has to be used which computing the gradients in the backward propagation cycle. Hence the dropout matrices are stored in a cell array.

for l =1:L-1
...
D=rand(size(A)(1),size(A)(2));
D = (D < keep_prob) ;
# Zero out some hidden units
A= A .* D;
# Divide by keep_prob to keep the expected value of A the same
A = A ./ keep_prob;
# Store D in a dropoutMat cell array
dropoutMat{l}=D;
...
endfor

In the backward propagation cycle we have

for l =(L-1):-1:1
...
D = dropoutMat{l};
# Zero out the dAl based on same dropout matrix
dAl= dAl .* D;
# Divide by keep_prob to maintain the expected value
dAl = dAl ./ keep_prob;
...
endfor

```
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
exec(open("DLfunctions61.py").read())
train_X, train_Y, test_X, test_Y = load_dataset()
layersDimensions = [2,7,1]
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",learningRate = 0.6, keep_prob=0.7, num_iterations = 9000,
initType="default", print_cost = True,figure="fig11.png")
plt.clf()
plt.close()
plot_decision_boundary(lambda x: predict(parameters, x.T,keep_prob=0.7), train_X, train_Y,str(0.6),figure1="fig12.png")
```

### 3.1b Dropout: Spiral data – Python

```
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
exec(open("DLfunctions61.py").read())
N = 100
D = 2
K = 3
X = np.zeros((N*K,D))
y = np.zeros(N*K, dtype='uint8')
for j in range(K):
ix = range(N*j,N*(j+1))
r = np.linspace(0.0,1,N)
t = np.linspace(j*4,(j+1)*4,N) + np.random.randn(N)*0.2
X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]
y[ix] = j
plt.scatter(X[:, 0], X[:, 1], c=y, s=40, cmap=plt.cm.Spectral)
plt.clf()
plt.close()
layersDimensions = [2,100,3]
y1=y.reshape(-1,1).T
# Train a deep learning network
parameters = L_Layer_DeepModel(X.T, y1, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax",
learningRate = 1,keep_prob=0.9, num_iterations = 5000, print_cost = True,figure="fig13.png")
plt.clf()
plt.close()
W1=parameters['W1']
b1=parameters['b1']
W2=parameters['W2']
b2=parameters['b2']
#Plot decision boundary
plot_decision_boundary1(X, y1,W1,b1,W2,b2,figure2="fig14.png")
```

## 3.2a Dropout: Circles data – R

```
source("DLfunctions61.R")
#Load data
df=read.csv("circles.csv",header=FALSE)
z <- as.matrix(read.csv("circles.csv",header=FALSE))
x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
keep_prob=0.8,
numIterations = 9000,
initType="default",
print_cost = True)
```

```
plotDecisionBoundary(z,retvals,keep_prob=0.6, hiddenActivationFunc="relu",0.5)
```

## 3.2b Dropout: Spiral data – R

```
source("DLfunctions61.R")
# Load data
Z <- as.matrix(read.csv("spiral.csv",header=FALSE))
X <- Z[,1:2]
y <- Z[,3]
X <- t(X)
Y <- t(y)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="softmax",
learningRate = 0.1,
keep_prob=0.90,
numIterations = 9000,
print_cost = True)
```

```
parameters<-retvals$parameters
#Plot decision boundary
plotDecisionBoundary1(Z,parameters)
```

## 3.3a Dropout: Circles data – Octave

data=csvread("circles.csv");
X=data(:,1:2);
Y=data(:,3);
layersDimensions = [2 11 1]; #tanh=-0.5(ok), #relu=0.1 best!
# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
lambd=0,
keep_prob=0.8,
numIterations = 10000,
initType="default");
plotCostVsIterations(10000,costs)
#Plot decision boundary
plotDecisionBoundary1(data,weights, biases,keep_prob=1, hiddenActivationFunc="relu")

## 3.3b Dropout Spiral data – Octave

source("DL61functions.m")
data=csvread("spiral.csv");
# Setup the data
X=data(:,1:2);
Y=data(:,3);
layersDimensions = [numFeats numHidden numOutput];
# Train a deep learning network
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="softmax",
learningRate = 0.1,
lambd=0,
keep_prob=0.8,
numIterations = 10000);
plotCostVsIterations(10000,costs)
#Plot decision boundary
plotDecisionBoundary1(data,weights, biases,keep_prob=1, hiddenActivationFunc="relu")

**Note**: The Python, R and Octave code can be cloned/downloaded from Github at DeepLearning-Part6

**Conclusion**

This post further enhances my earlier L-Layer generic implementation of a Deep Learning network to include options for initialization techniques, L2 regularization or dropout regularization

**References**

1. Deep Learning Specialization

2. Neural Networks for Machine Learning

Also see

1. Architecting a cloud based IP Multimedia System (IMS)

2. Using Linear Programming (LP) for optimizing bowling change or batting lineup in T20 cricket

3. My book ‘Practical Machine Learning with R and Python’ on Amazon

4. Simulating a Web Joint in Android

5. Inswinger: yorkr swings into International T20s

6. Introducing QCSimulator: A 5-qubit quantum computing simulator in R

7. Computer Vision: Ramblings on derivatives, histograms and contours

8. Bend it like Bluemix, MongoDB using Auto-scale – Part 1!

9. The 3rd paperback & kindle editions of my books on Cricket, now on Amazon

To see all posts click Index of posts

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