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## Tensor Libraries

Data in deep neural networks is typically represented as multi-dimensional arrays, also called tensors. Tensors are nowadays the basic data structures running under the hood of most machine learning libraries and are used to learn different data representations through various operations and layers. The most popular library is probably Tensorflow by Google. However, thanks to its ease of use the Keras library (which is based on Tensorflow) even overtook vanilla Tensorflow most recently. It can also be seen that Pytorch gained significant traction of the last years, too.

## Tensors in R

Tensors can be seen as generalizations of vectors and matrices to an arbitrary number of dimensions.

In R we have the analogy of vector and matrix objects for 1- or 2-dimensional tensors and array for higher dimensions. Training deep neural networks means that we need to find data representations from the input layer through the (hidden) layers to the output which fit into tensors.

Tensors can be described through the following properties:

• Rank: The rank describes the number of axes or dimensions, e.g a matrix has rank 2 (2 dimensions), a 3D tensor has rank 3.
• Shape: Describing the dimension size along each axis.
• Data type: The type of the Tensor which typically falls in the numeric category (integer, double).

## Exercise: Tensor Attributes

Let’s examine the attributes of our x_train data from the MNIST dataset.

library(keras)
mnist <- dataset_mnist()
x_train <- mnist$train$x
y_train <- mnist$train$y
x_test <- mnist$test$x
y_test <- mnist$test$y

Given our loaded MNIST dataset, we can get the shape using dim():

The number of axes (rank) is simply the length of the shape vector:

The type of the tensor can be retrieved through typeof() as

Example based on Deep Learning with R, page 31

## Practical Examples

For practical applications the data will almost always fall into one of the following categories:

• Vector data: 2D tensors of shape (samples, features).
• Time series data or sequence data: 3D tensors of shape (samples, timesteps, features)
• Images: 4D tensors of shape (samples, height, width, channels) or (samples, channels, height, width)
• Video: 5D tensors of shape (samples, frames, height, width, channels) or (samples, frames, channels, height, width)

How could each of the following real-world examples be encoded as data-tensors?

## Tensor Operations

A deep neural network consists of a set a layers which can be stacked on top of each other using the pipe %>% operator. Thanks to the intuitive interface we do not have to think of how the layers are exactly wired together.

We use the function layer_dense, define the number of hidden units and the popular "relu" activation function which stands for rectified linear unit :

layer_dense(units = 512, activation = "relu")

Activations functions can be used through layer_activation(), or using the activation argument supported by all forward layers. See also the RStudio keras documentation for other available activation functions and further documentation.

The actual operation has a 2D tensor as input and outputs another 2D tensor as follows:

$relu( W \cdot input + b)$

$$input \dots$$ input tensor

$$W \dots$$ weight tensor

$$b \dots$$ bias tensor

We could also write the operations performed using the code below:

output = relu(dot(W, input) + b)

## Element-wise Operations

Basic arithmetic (addition, subtraction, multiplication) and relu are element-wise operations. These operations are performed independently on each element of the tensor and can thus be easily parallelized on multiple CPUs/GPUs. The relu operation is is defined as $relu(x) = max(x, 0)$

## Different Dimensions

The R function sweep() allows us to calculate operations on tensors of different dimensionality. Try to guess the shape and result of the output z first before evaluating the result.

### Tensor Dot

The tensor dot operation is comparable to a matrix multiplication %*% but extending to higher dimensions, e.g.

(a, b, c, d) . (d) -> (a, b, c)

(a, b, c, d) . (d, e) -> (a, b, c, e)

## Tensor Reshaping

Another important tensor operation is reshaping using array_reshape for data-preprocessing as

library(keras)
mnist <- dataset_mnist()
x_train <- mnist$train$x
dim(x_train)
## [1] 60000    28    28
x_train <- array_reshape(x_train, c(60000, 28*28))
dim(x_train)
## [1] 60000   784

Given the following matrix as

x <- matrix(c(0, 1,
2, 3,
4, 5),
nrow = 3, ncol = 2, byrow = TRUE)
x

let’s reshape x to have 6 rows and 1 column:

Or 2 rows and 3 columns:

We can also reshape tensor by transposition or simply exchanging rows and columns of a matrix.

The t() function can be used as

x <- matrix(0, nrow = 300, ncol = 20)
dim(x)
## [1] 300  20
x <- t(x)
dim(x)
## [1]  20 300