# Total Internal Reflection

Technology and Art

## Machine Learning Theory Track

(Click the image below to see the full-size image)

• ## Functional Analysis Exercises 12 : Linear Operators

13 December 2021

This post lists solutions to the exercises in the Linear Operators section 2.6 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

• ## Functional Analysis Exercises 11 : Compactness and Finite Dimension

2 December 2021

This post lists solutions to the exercises in the Compactness and Finite Dimension section 2.5 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

• ## Functional Analysis Exercises 10 : Finite Dimensional Normed Spaces and Subspaces

22 November 2021

This post lists solutions to the exercises in the Finite Dimensional Normed Spaces and Subspaces section 2.4 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

• ## Functional Analysis Exercises 9 : Further Properties of Normed Spaces

19 November 2021

This post lists solutions to the exercises in the Further Properties of Normed Spaces section 2.3 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

• ## Two Phase Commit: Indistinguishable Commit Scenario

18 November 2021

We review the most interesting failure scenario for the Two Phase Commit (2PC) protocol. There are excellent explanations of 2PC out there, and I won’t bother too much with the basic explanation. The focus of this post is a walkthrough of the indistinguishable state scenario, where neither a global commit, nor a global abort command can be issued.

• ## Functional Analysis Exercises 8 : Normed and Banach Spaces

11 November 2021

This post lists solutions to the exercises in the Normed Space, Banach Space section 2.2 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

• ## Resilient Knowledge Bases : Fundamentals, not Buzzwords

6 November 2021

We start this tirade with a memorable quote from Alice Through the Looking-Glass:

• ## Functional Analysis Exercises 7 : Vector Spaces

3 November 2021

This post lists solutions to the exercises in the Vector Space section 2.1 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

• ## Functional Analysis Exercises 6 : Completion of Metric Spaces

25 October 2021

This post lists solutions to the exercises in the Completion of Metric Spaces section 1.6 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

• ## Functional Analysis Exercises 5 : Completeness Proofs

16 October 2021

This post lists solutions to the exercises in the Completeness Proofs section 1.5 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

• ## Functional Analysis Exercises 4 : Convergence, Cauchy Sequences, and Completeness

12 October 2021

This post lists solutions to the exercises in the Convergence, Cauchy Sequences, and Completeness section 1.4 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

• ## Assorted Analysis Proofs

11 October 2021

This post lists assorted proofs from Analysis, without any particular theme.

• ## Functional Analysis Exercises 3 : Sets, Continuous Mappings, and Separability

7 October 2021

This post lists solutions to many of the exercises in the Open Set, Closed Set, Neighbourhood section 1.3 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

• ## General Proof Tactics for Real and Functional Analysis

29 September 2021

This article represents a (very short) collection of my ongoing notes on proof tactics I’ve found useful when I’ve been stuck trying to solve proof exercises. I aim to continue documenting these in as much detail as possible. These are mostly aids while building intuition about how to prove something, and gradually should become part of one’s mental lexicon.

• ## Functional Analysis Exercises 2 : Distance Metrics

28 September 2021

This post lists solutions to many of the exercises in the Distance Metrics section 1.2 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

• ## Important Inequalities in Functional Analysis

27 September 2021

Continuing my self-study of Functional Analysis, this post describes proofs for the following important inequalities in the subject:

• Young’s Inequality
• Hölder’s Inequality
• Minkowski’s Inequality

• ## Functional Analysis Exercises 1 : Distance Metrics

21 September 2021

This post lists solutions to many of the exercises in the Distance Metrics section 1.1 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is definitely a work in progress, and proofs may be refined or added over time.

• ## Gaussian Processes: Theory

10 September 2021

In this article, we will build up our mathematical understanding of Gaussian Processes. We will understand the conditioning operation a bit more, since that is the backbone of inferring the posterior distribution. We will also look at how the covariance matrix evolves as training points are added.

• ## Gaussian Processes: Intuition

6 September 2021

In this article, we will build up our intuition of Gaussian Processes, and try to understand how it models uncertainty about data it has not encountered yet, while still being useful for regression. We will also see why the Covariance Matrix (and consequently, the Kernel) is a fundamental building block of our assumptions around the data we are trying to model.

• ## Geometry of the Multivariate Gaussian Distribution

30 August 2021

Continuing from the roadmap set out in Road to Gaussian Processes, we begin with the geometry of the central object which underlies this Machine Learning Technique, the Multivariate Gaussian Distribution. We will study its form to build up some geometric intuition around its interpretation.

• ## Statistics from Geometry and Linear Algebra

12 August 2021

This article covers some common statistical quantities/metrics which can be derived from Linear Algebra and corresponding intuitions from Geometry, without recourse to Probability or Calculus. Of course, those subjects add more rigour and insight into these concepts, but our aim is to provide a form of intuitive shorthand for the reader.

• ## Non-Linear Support Vector Machines: Radial Basis Function Kernel and the Kernel Trick

7 August 2021

This article builds upon the previous material on kernels and Support Vector Machines to introduce some simple examples of Reproducing Kernels, including a simplified version of the frequently-used Radial Basis Function kernel. Beyond that, we finally look at the actual application of kernels and the so-called Kernel Trick to avoid expensive computation of projections of data points into higher-dimensional space, when working with Support Vector Machines.

• ## Kernel Functions with Reproducing Kernel Hilbert Spaces

20 July 2021

This article uses the previous mathematical groundwork to discuss the construction of Reproducing Kernel Hilbert Spaces. We’ll make several assumptions that have been proved and discussed in those articles. There are multiple ways of discussing Kernel Functions, like the Moore–Aronszajn Theorem and Mercer’s Theorem. We may discuss some of those approaches in the future, but here we will focus on the constructive approach here to characterise Kernel Functions.

• ## Functional Analysis: Norms, Operators, and Some Theorems

19 July 2021

This article expands the groundwork laid in Kernel Functions: Functional Analysis and Linear Algebra Preliminaries to discuss some more properties and proofs for some of the properties of functions that we will use in future discussions on Kernel Methods in Machine Learning, including (but not restricted to) the construction of Reproducing Kernel Hilbert Spaces.

• ## Functional and Real Analysis Notes

18 July 2021

These are personal study notes, brief or expanded, complete or incomplete. Some concepts here will be alluded to in full-fledged Machine Learning posts.

• ## Kernel Functions: Functional Analysis and Linear Algebra Preliminaries

17 July 2021

This article lays the groundwork for an important construction called Reproducing Kernel Hilbert Spaces, which allows a certain class of functions (called Kernel Functions) to be a valid representation of an inner product in (potentially) higher-dimensional space. This construction will allow us to perform the necessary higher-dimensional computations, without projecting every point in our data set into higher dimensions, explicitly, in the case of Non-Linear Support Vector Machines, which will be discussed in the upcoming article.

• ## Real Analysis: Patterns for Proving Irrationality of Square Roots

9 July 2021

ld Continuing on my journey through Real Analysis, we will focus here on common proof patterns which apply to irrational square roots. These patterns apply to the following sort of proof exercises:

• ## The Cholesky and LDL* Factorisations

8 July 2021

This article discusses a set of two useful (and closely related) factorisations for positive-definite matrices: the Cholesky and the $$LDL^T$$ factorisations. Both of them find various uses: the Cholesky factorisation particularly is used when solving large systems of linear equations.

• ## The Gram-Schmidt Orthogonalisation

27 May 2021

We discuss an important factorisation of a matrix, which allows us to convert a linearly independent but non-orthogonal basis to a linearly independent orthonormal basis. This uses a procedure which iteratively extracts vectors which are orthonormal to the previously-extracted vectors, to ultimately define the orthonormal basis. This is called the Gram-Schmidt Orthogonalisation, and we will also show a proof for this.

• ## Real Analysis Proofs #1

18 May 2021

Since I’m currently self-studying Real Analysis, I’ll be listing down proofs I either initially had trouble understanding, or enjoyed proving, here. These are very mathematical posts, and are for personal documentation, mostly.

• ## Support Vector Machines from First Principles: Linear SVMs

10 May 2021

We have looked at how Lagrangian Multipliers and how they help build constraints as part of the function that we wish to optimise. Their relevance in Support Vector Machines is how the constraints about the classifier margin (i.e., the supporting hyperplanes) is incorporated in the search for the optimal hyperplane.

• ## Quadratic Optimisation: Lagrangian Dual, and the Karush-Kuhn-Tucker Conditions

10 May 2021

This article concludes the (very abbreviated) theoretical background required to understand Quadratic Optimisation. Here, we extend the Lagrangian Multipliers approach, which in its current form, admits only equality constraints. We will extend it to allow constraints which can be expressed as inequalities.

• ## Quadratic Optimisation: Mathematical Background

8 May 2021

This article continues the original discussion on Quadratic Optimisation, where we considered Principal Components Analysis as a motivation. Originally, this article was going to begin delving into the Lagrangian Dual and the Karush-Kuhn-Tucker Theorem, but the requisite mathematical machinery to understand some of the concepts necessitated breaking the preliminary setup into its own separate article (which you’re now reading).

• ## Common Ways of Looking at Matrix Multiplications

29 April 2021

We consider the more frequently utilised viewpoints of matrix multiplication, and relate it to one or more applications where using a certain viewpoint is more useful. These are the viewpoints we will consider.

• ## Intuitions about the Implicit Function Theorem

29 April 2021

We discussed the Implicit Function Theorem at the end of the article on Lagrange Multipliers, with some hand-waving to justify the linear behaviour on manifolds in arbitrary $$\mathbb{R}^N$$.

• ## Quadratic Optimisation using Principal Component Analysis as Motivation: Part Two

28 April 2021

We pick up from where we left off in Quadratic Optimisation using Principal Component Analysis as Motivation: Part One. We treated Principal Component Analysis as an optimisation, and took a detour to build our geometric intuition behind Lagrange Multipliers, wading through its proof to some level.

• ## Vector Calculus: Lagrange Multipliers, Manifolds, and the Implicit Function Theorem

24 April 2021

In this article, we finally put all our understanding of Vector Calculus to use by showing why and how Lagrange Multipliers work. We will be focusing on several important ideas, but the most important one is around the linearisation of spaces at a local level, which might not be smooth globally. The Implicit Function Theorem will provide a strong statement around the conditions necessary to satisfy this.

• ## Vector Calculus: Graphs, Level Sets, and Constraint Manifolds

20 April 2021

In this article, we take a detour to understand the mathematical intuition behind Constrained Optimisation, and more specifically the method of Lagrangian multipliers. We have been discussing Linear Algebra, specifically matrices, for quite a bit now. Optimisation theory, and Quadratic Optimisation as well, relies heavily on Vector Calculus for many of its results and proofs.

• ## Quadratic Optimisation using Principal Component Analysis as Motivation: Part One

19 April 2021

This series of articles presents the intuition behind the Quadratic Form of a Matrix, as well as its optimisation counterpart, Quadratic Optimisation, motivated by the example of Principal Components Analysis. PCA is presented here, not in its own right, but as an application of these two concepts. PCA proper will be presented in another article where we will discuss eigendecomposition, eigenvalues, and eigenvectors.

• ## Road to Gaussian Processes

17 April 2021

This article aims to start the road towards a theoretical intuition behind Gaussian Processes, another Machine Learning technique based on Bayes’ Rule. However, there is a raft of material that I needed to understand and relearn before fully appreciating some of the underpinnings of this technique.

• ## Support Vector Machines from First Principles: Part One

14 April 2021

We will derive the intuition behind Support Vector Machines from first principles. This will involve deriving some basic vector algebra proofs, including exploring some intuitions behind hyperplanes. Then we’ll continue adding to our understanding the concepts behind quadratic optimisation.

• ## Dot Product: Algebraic and Geometric Equivalence

11 April 2021

The dot product of two vectors is geometrically simple: the product of the magnitudes of these vectors multiplied by the cosine of the angle between them. What is not immediately obvious is the algebraic interpretation of the dot product.

• ## Linear Regression: Assumptions and Results using the Maximum Likelihood Estimator

5 April 2021

Let’s look at Linear Regression. The “linear” term refers to the fact that the output variable is a linear combination of the input variables.

• ## Matrix Rank and Some Results

4 April 2021

I’d like to introduce some basic results about the rank of a matrix. Simply put, the rank of a matrix is the number of independent vectors in a matrix. Note that I didn’t say whether these are column vectors or row vectors; that’s because of the following section which will narrow down the specific cases (we will also prove that these numbers are equal for any matrix).

• ## Assorted Intuitions about Matrices

3 April 2021

Some of these points about matrices are worth noting down, as aids to intuition. I might expand on some of these points into their own posts.

• ## Matrix Outer Product: Columns-into-Rows and the LU Factorisation

2 April 2021

We will discuss the Column-into-Rows computation technique for matrix outer products. This will lead us to one of the important factorisations (the LU Decomposition) that is used computationally when solving systems of equations, or computing matrix inverses.

• ## Intuitions about the Orthogonality of Matrix Subspaces

2 April 2021

This is the easiest way I’ve been able to explain to myself around the orthogonality of matrix spaces. The argument will essentially be based on the geometry of planes which extends naturally to hyperplanes.

• ## Matrix Outer Product: Value-wise computation and the Transposition Rule

1 April 2021

We will discuss the value-wise computation technique for matrix outer products. This will lead us to a simple sketch of the proof of reversal of order for transposed outer products.

• ## Matrix Outer Product: Linear Combinations of Vectors

30 March 2021

Matrix multiplication (outer product) is a fundamental operation in almost any Machine Learning proof, statement, or computation. Much insight may be gleaned by looking at different ways of looking matrix multiplication. In this post, we will look at one (and possibly the most important) interpretation: namely, the linear combination of vectors.

• ## Vectors, Normals, and Hyperplanes

29 March 2021

Linear Algebra deals with matrices. But that is missing the point, because the more fundamental component of a matrix is what will allow us to build our intuition on this subject. This component is the vector, and in this post, I will introduce vectors, along with common notations of expression.

• ## Machine Learning Theory Track

28 March 2021

I’ve always been fascinated by Machine Learning. This began in the seventh standard when I discovered a second-hand book on Neural Networks for my ZX Spectrum.

• ## First Post

28 March 2021

My previous Wordpress site went kaput. With it went most of the content that I’ve posted over the years. I have switched over to Jekyll, and will be adding new material, as well as reintroducing old material as needed, from a backup that I have. Stay posted!