The Palindrome

The Palindrome Substack delves into mathematics through the lens of real-world applications, optimization techniques, and machine learning models, aiming to clarify complex concepts for engineers, scientists, and the curious. It bridges theoretical fundamentals with practical insights, offering a thorough understanding of subjects like graph theory, linear algebra, statistical methods, and computational algorithms.

Mathematical Fundamentals Graph Theory Machine Learning Optimization Techniques Statistical Methods Linear Algebra Computational Algorithms Probability Theory Educational Theory Mathematics in Science

The hottest Substack posts of The Palindrome

And their main takeaways

How to implement a decision tree

3 implied HN points ā€¢ 08 Nov 24
šŸ•¹ Technology Machine Learning Data science Programming Artificial Intelligence Software Development
  1. A decision tree splits data based on features and thresholds, which helps in making predictions by creating branches. Each split leads to two outcomes based on whether the condition is met or not.
  2. Gini impurity is a key measure for evaluating how 'pure' the labels are in each leaf of the tree. A lower Gini impurity means better predictability for a leaf's classification.
  3. You can create both classification and regression trees by changing how you score the splits and define the predictions in the leaves. This flexibility allows for various applications in data analysis.
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The Computer Scientist's Guide to Graph Theory, ep. 02

3 implied HN points ā€¢ 13 Dec 23
šŸ”¬ Science Graph Theory Algorithms Mathematics
  1. Matching problems can be modeled using bipartite graphs where no edges go between vertices of the same type.
  2. In graph theory, a full matching of one partition of a bipartite graph implies that every vertex in that partition has at least as many neighbors in the other partition.
  3. Hall's theorem provides a necessary and sufficient condition for determining the existence of a full matching in a bipartite graph.

How large that number in the Law of Large Numbers is?

4 implied HN points ā€¢ 04 Sep 23
šŸ”¬ Science Probability
  1. The term 'large' is relative and depends on what you are comparing it to.
  2. The Law of Large Numbers states that sample averages converge to the true expected value as the number of samples increases.
  3. The speed of convergence in the Law of Large Numbers depends on the variance of the sample, with higher variance leading to slower convergence.

The mathematics of optimization for deep learning

2 implied HN points ā€¢ 12 Feb 24
šŸ•¹ Technology Mathematics Machine Learning Optimization
  1. The post discusses the mathematics of optimization for deep learning - essentially minimizing a function with many variables.
  2. The author reflects on their progression since 2019, highlighting growth and improvement in their writing.
  3. Readers can sign up for a 7-day free trial to access the full post archives on the topic of math and machine learning.

Building the machine learning interface

2 implied HN points ā€¢ 22 Jan 24
šŸ•¹ Technology Machine Learning Programming Python Object Oriented Programming Frameworks
  1. Building a modular interface is crucial as machine learning models complexity increases.
  2. Transitioning from procedural to object-oriented programming can greatly enhance understanding and performance in machine learning.
  3. Good design is essential in setting the framework for machine learning models, drawing inspiration from PyTorch and scikit-learn.

Why does gradient descent work?

5 implied HN points ā€¢ 06 Apr 23
šŸ”¬ Science Mathematics Machine Learning Optimization
  1. In machine learning, gradient descent is used to find local extrema by following the direction of steepest ascent or descent.
  2. Understanding derivatives helps us interpret the rate of change, such as speed in physics.
  3. Differential equations provide a mathematical framework to understand gradient descent and optimization, showing how systems flow towards equilibrium.

Is probability frequentist or Bayesian?

3 implied HN points ā€¢ 14 Aug 23
šŸ”¬ Science Probability Theory Mathematics Statistics
  1. Probability is a number that quantitatively measures the likelihood of events, always between 0 and 1.
  2. Probability is a well-defined mathematical concept, separate from how probabilities are assigned.
  3. The frequentist and Bayesian schools of thought differ in how they assign probabilities, but each has its own advantages in different situations.

A conversation with Alejandro Piad Morffis about education

2 implied HN points ā€¢ 23 Nov 23
šŸšŒ Education Teaching Curriculum Communication Skills Learning
  1. The role of talent in education is important, but having the right environment and support is crucial.
  2. The education system often focuses on problem-solving skills, but it's essential to also value teamwork, communication, and open-ended problem-solving.
  3. There is a gap in science communication where content needs to be accurate and rigorous without sacrificing accessibility to a wider audience.

The fascinating story of the exponential function

4 implied HN points ā€¢ 10 Feb 23
  1. The exponential function involves concepts like positive and negative exponents, as well as rational exponents defined in terms of roots.
  2. Wishful thinking and principles like the 'product of powers' and 'power of powers' are key in extending the definition of exponents to arbitrary powers.
  3. Matrix exponentials are derived using the powerful technique of Taylor series, allowing for complex mathematical operations with matrices.

Epsilons, no. 3: The LU decomposition

3 implied HN points ā€¢ 27 Mar 23
šŸ”¬ Science Mathematics Linear Algebra
  1. Matrix factorizations are a key part of linear algebra, used for inverting matrices and simplifying determinants.
  2. The LU decomposition method involves breaking a matrix into upper and lower triangular forms.
  3. Linear algebra helps in solving systems of linear equations by transforming them into echelon form using operations like multiplying by scalars and adding equations.

Epsilons, no. 1: The geometric series

3 implied HN points ā€¢ 08 Mar 23
šŸ”¬ Science Mathematics Applications
  1. The geometric series is a key concept in mathematics with many practical applications.
  2. Deriving the closed-form expression of the geometric series involves understanding its partial sums and limiting behavior.
  3. The geometric series is convergent for |q| < 1 and has a simple closed-form expression.

The science of science

3 implied HN points ā€¢ 01 Dec 22
  1. Our knowledge of the world is stored in propositions that are either true or false.
  2. Probability theory allows us to measure plausibility on a 0-1 scale, providing a more nuanced understanding than classical logic.
  3. Bayesian inference helps us update our beliefs based on new evidence, enabling more informed decision-making.

Why are neural networks so powerful?

1 implied HN point ā€¢ 11 Sep 23
šŸ”¬ Science Machine Learning Neural Networks Mathematics
  1. Neural networks are powerful due to their ability to closely approximate almost any function.
  2. Machine learning involves finding a function that approximates the relationship between data points and their ground truth.
  3. Approximation theory seeks to find a simple function close enough to a complex one by determining the right function family and precise approximation within that family.

How to measure the angle between two functions

1 implied HN point ā€¢ 08 Dec 22
  1. Defining angles and orthogonality for functions goes beyond traditional Euclidean spaces
  2. Generalizing concepts like vectors and inner products allows for broader applications in physics and mathematics
  3. Orthogonality in function spaces, like the LĀ² space, can be defined through the vanishing of the inner product

Where does the mean squared error come from?

0 implied HN points ā€¢ 21 Dec 23
šŸ”¬ Science Statistics Data Machine Learning
  1. Mean squared error is a common loss function for machine learning models due to its mathematical simplicity and alignment with statistical principles.
  2. Absolute value functions are not commonly chosen for loss function in machine learning due to issues with differentiability at zero.
  3. The linear model and mean squared error naturally arise when approaching machine learning with a statistical mindset.

Birthday Special Offer

0 implied HN points ā€¢ 05 Dec 23
šŸ’¼ Business Marketing Entrepreneurship Sales
  1. The Palindrome is offering a special birthday discount for annual subscriptions.
  2. You can get a 20% discount on your annual subscription and early access to a Mathematics of Machine Learning book.
  3. The offer is valid until December 31st. Email for early access to the book.

Linear regression in multiple variables

0 implied HN points ā€¢ 05 Mar 24
šŸ”¬ Science Mathematics Machine Learning Regression Data Analysis
  1. Real datasets often have multiple features, going beyond a single variable. Understanding how to handle multiple variables is crucial in machine learning.
  2. Linear regression can be generalized to handle multiple variables by using a regression coefficient vector and a bias term.
  3. The parameters of a multivariable linear regression model help define a d-dimensional plane, providing a way to map feature vectors to target values in a straightforward manner.

The taxonomy of machine learning paradigms

0 implied HN points ā€¢ 18 Sep 23
šŸ•¹ Technology Machine Learning Data science Algorithms Models AI
  1. Machine learning tasks involve three important parameters: the input, the output, and the training data.
  2. The basic machine learning setup consists of a dataset, a true relation function, and a parametric model as an estimation.
  3. Major paradigms of machine learning include supervised learning, unsupervised learning, semi-supervised learning, and reinforcement learning.

Linear regression and the least squares problem

0 implied HN points ā€¢ 12 Dec 23
šŸ”¬ Science Mathematics Machine Learning
  1. Linear regression can be optimized by hand, especially for single variable models where the loss function is simple.
  2. Gradient descent for linear regression can be like using a cannonball to shoot a sparrow, due to the simplicity of the loss function.
  3. Premium subscribers of The Palindrome can access exclusive content and chapters of 'Mathematics of Machine Learning' for an in-depth education.