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Appendix A. Mathematical Foundations and Advanced TensorFlow

Here we will discuss some of the concepts that will be useful to understand details provided in the chapters. First we will discuss several mathematical data structures found throughout the book, followed by a description about various operations performed on those data structures. Next, we will discuss the concept of probabilities. Probabilities play a vital role in machine learning, as they usually give insights to how uncertain a model is about its prediction. Thereafter, we discuss a high-level library known as Keras in TensorFlow, as well as how to implement a neural machine translator with the seq2seq sublibrary in TensorFlow. Finally we conclude this section with a guide on how to use the TensorBoard as a visualization tool for word embeddings.

Basic data structures

Scalar

A scalar is a single number unlike a matrix or a vector. For example, 1.3 is a scalar. A scalar can be mathematically denoted as follows:

Here, R is the real number space.

Vectors

A vector is an array of numbers. Unlike a set, where there is no order to elements, a vector has a certain order to the elements. An example vector is [1.0, 2.0, 1.4, 2.3]. Mathematically, it can be denoted as follows:

Alternatively, we can write this as:

Here, R is the real number space and n is the number of elements in the vector.

Matrices

A matrix can be thought of as a two-dimensional arrangement of a collection of scalars. In other words, a matrix can be thought of as a vector of vectors. An example matrix would be as shown here:

A more general matrix of size can be mathematically defined like this:

And:

Here, m is the number of rows of the matrix, n is the number of columns in the matrix, and R is the real number space.

Indexing of a matrix

We will be using zero-indexed notation (that is, indexes start with 0).

To index a single element from a matrix at (i, j)th position, we use the following notation:

Referring to the previously defined matrix, we get the following:

We index an element from A like this:

We denote a single row of any matrix A as shown here:

For our example matrix, we can denote the second row (indexed as 1) of the matrix as shown here:

We denote the slice starting from the (i, k)th index to the (j, l)th index of any matrix A as shown here:

In our example matrix, we can denote the slice from first row third column to second row fourth column as shown here:

Special types of matrices

Identity matrix

An identity matrix is where it is equal to 1 on the diagonal of the matrix and 0 everywhere else. Mathematically, it can be shown as follows:

This would look like the following:

Here, .

The identity matrix gives the following nice property when multiplied with another matrix A :

Diagonal matrix

A diagonal matrix is a more general case of the identity matrix, where the values along the diagonal can take any value and the off-diagonal values are zeros:

Tensors

An n -dimensional matrix is called a tensor . In other words, a matrix with an arbitrary number of dimensions is called a tensor. For example, a four-dimensional tensor can be denoted as shown here:

Here, R is the real number space.

Tensor/matrix operations

Transpose

Transpose is an important operation defined for matrices or tensors. For a matrix, the transpose is defined as follows:

Here, AT denotes the transpose of

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