Mastering Linear Algebra, Probability, and Statistics for Machine Learning and NLP

Natural language processing (NLP) and machine learning (ML) are two fields that have significantly benefited from mathematical concepts, particularly linear algebra and probability theory. These fundamental tools enable the analysis of the relationships between variables, forming the basis of many NLP and ML models. This chapter provides a comprehensive introduction to linear algebra and probability theory, including their practical applications in NLP and ML. The chapter commences with an overview of vectors and matrices and covers essential operations. Additionally, the basics of statistics, required for understanding the concepts and models in subsequent chapters, will be explained. Finally, the chapter introduces the fundamentals of optimization, which are critical for solving NLP problems and understanding the relationships between variables. By the end of this chapter, you will have a solid foundation...

Introduction to linear algebra

Let’s start by first understanding scalars, vectors, and matrices:

• Scalars: A scalar is a single numerical value that usually comes from the real domain in most ML applications. Examples of scalars in NLP include the frequency of a word in a text corpus.
• Vectors: A vector is a collection of numerical elements. Each of these elements can be termed as an entry, component, or dimension, and the count of these components defines the vector’s dimensionality. Within NLP, a vector could hold components related to elements such as word frequency, sentiment ranking, and more. NLP and ML are two domains that have reaped substantial benefits from mathematical disciplines, particularly linear algebra and probability theory. These foundational tools aid in evaluating the correlation between variables and are at the heart of numerous NLP and ML models. This segment presents a detailed primer on linear algebra and probability theory, along...

Eigenvalues and eigenvectors

A vector x, belonging to a d × d matrix A, is an eigenvector if it satisfies the equation Ax = λx, where λ represents the eigenvalue associated with the matrix. This relationship delineates the link between matrix A and its corresponding eigenvector x, which can be perceived as the “stretching direction” of the matrix. In the case where A is a matrix that can be diagonalized, it can be deconstructed into a d × d invertible matrix, V, and a diagonal d × d matrix, Δ, such that

The columns of V encompass d eigenvectors, while the diagonal entries of Δ house the corresponding eigenvalues. The linear transformation Ax can be visually understood through a sequence of three operations. Initially, the multiplication of x by calculates x’s co-ordinates in a non-orthogonal basis associated with V’s columns. Subsequently, the multiplication of x by Δ scales these co-ordinates using...

Basic probability for machine learning

Probability provides information about the likelihood of an event occurring. In this field, there are several key terms that are important to understand:

• Trial or experiment: An action that results in a certain outcome with a certain likelihood
• Sample space: This encompasses all potential outcomes of a given experiment
• Event: This denotes a non-empty portion of the sample space

Therefore, in technical terms, probability is a measure of the likelihood of an event occurring when an experiment is conducted.

In this very simple case, the probability of event A with one outcome is equal to the chance of event A divided by the chance of all possible events. For example, in flipping a fair coin, there are two outcomes with the same chance: heads and tails. The chance of having heads will be 1/(1+1) = ½.

In order to calculate the probability, given an event, A, with n outcomes and a sample space, S, the probability of...

Summary

This chapter was about linear algebra and probability for ML, and it covers the fundamental mathematical concepts that are essential to understanding many machine learning algorithms. The chapter began with a review of linear algebra, covering topics such as matrix multiplication, determinants, eigenvectors, and eigenvalues. It then moved on to discuss probability theory, introducing the basic concepts of random variables and probability distributions. We also covered key concepts in statistical inference, such as maximum likelihood estimation and Bayesian inference.

In the next chapter, we will cover the fundamentals of machine learning for NLP, including topics such as data exploration, feature engineering, selection methods, and model training and validation.

Further reading

Please find the additional reading content as follows:

• Householder reflection matrix: A Householder reflection matrix, or Householder matrix, is a type of linear transformation utilized in numerical linear algebra due to its computational effectiveness and numerical stability. This matrix is used to perform reflections of a given vector about a plane or hyperplane, transforming the vector so that it only has non-0 components in one specific dimension. The Householder matrix (H) is defined by

Here, I is the identity matrix, and u is a unit vector defining the reflection plane.

The main purpose of Householder transformations is to perform QR factorization and to reduce matrices to a tridiagonal or Hessenberg form. The properties of being symmetric and orthogonal make the Householder matrix computationally efficient and numerically stable.

• Diagonalizable: A matrix is said to be diagonalizable if it can be written in the form , where A is the...

References

• Alter O, Brown PO, Botstein D. (2000) Singular value decomposition for genome-wide expression data processing and modeling. Proc Natl Acad Sci U S A, 97, 10101-6.
• Golub, G.H., and Van Loan, C.F. (1989) Matrix Computations, 2nd ed. (Baltimore: Johns Hopkins University Press).
• Greenberg, M. (2001) Differential equations & Linear algebra (Upper Saddle River, N.J. : Prentice Hall).
• Strang, G. (1998) Introduction to linear algebra (Wellesley, MA : Wellesley-Cambridge Press).
• Lax, Peter D. Linear algebra and its applications. Vol. 78. John Wiley & Sons, 2007.
• Dangeti, Pratap. Statistics for machine learning. Packt Publishing Ltd, 2017.
• DasGupta, Anirban. Probability for statistics and machine learning: fundamentals and advanced topics. New York: Springer, 2011.