Early adopter version of my book - mathematical foundations of data science - key maths ideas you should know for mathematical foundations of AI

Background

I am creating a small community for my book - mathematical foundations of data science. You get pdf when released but you also chapters as they are released and you get to engage and ask questions.?

The price is a one off 40 USD. If you are interested please DM me. I am trying to keep spaces limited since I want to learn from feedback so that's an important criterion as well.???

The idea of the book is simple: In an age when a majority of the code could be LLM generated, its very useful to approach AI from first principles i.e. from the maths. The good news is .. there are only four things to know: linear algebra, statistics, optimization and probability theory. The bad news is: its not easy to tie these four ideas to every machine learning and deep learning algorithm considering that the field itself is rapidly evolving. In this sense, the book helps by creating a concise structure. Since these ideas are known to many people at A levels (around age 18) - the book creates a foundation to know AI based on ideas that you already know - even if you have studied them years ago!

Some notes

1. I want to keep the book concise. Hence, there is no point in explaining the basics of matrix algebra etc - when you can easily get that information from chatGPT! Its more important to understand how these ideas fit together.
2. The traditional statistical approach depends on understanding of the underlying probability distribution of a phenomenon. As the number of parameters to be determined increases (some cases to millions and billions)? - there is a decoupling from core statistical ideas - the implications of which are important to understand.

Key mathematical ideas

Here is a list of key maths ideas you should know for mathematical foundations of AI that I cover. ? This is in no particular order.?

Core

• ?Implications of ‘Deep learning works’ and what it means for AGI
• Learning from data
• Deep learning as automatic feature detection
• Statistical inference is not the same as machine learning inference
• Linearity vs non linearity
• Linear regression and its evolution can explain many ideas of machine learning.?
• Algorithms can be expressed as discerning a hidden function
• Implications of highly parameterised models?
• The model evaluation metric is the ‘north star’ in the sense that the goal of a machine learning process is to optimise the metric and that each step of the machine learning pipeline ties back to that goal??
• decision making under uncertainty?
• Linear Algebra: Understanding vectors, matrices, and operations like matrix multiplication, determinants, and eigenvalues is crucial for data representation and transformations in ML.
• Probability Theory: Key concepts include probability distributions, conditional probability, Bayes' theorem, and understanding random variables, which underpin many ML models.
• Statistics: Includes understanding descriptive statistics, hypothesis testing, p-values, confidence intervals, and various distributions like normal and Poisson distributions.
• Optimization: Knowledge of optimization techniques such as gradient descent, Lagrange multipliers, and convex optimization is vital for training models.
• Calculus: Specifically, differential calculus (gradients, Jacobians, Hessians) is used to optimize ML models, while integral calculus helps in understanding probability distributions.
• Loss Functions: Critical for model training, including Mean Squared Error, Cross-Entropy Loss, and Hinge Loss, which quantify the error between predictions and actual values.
• Overfitting and Underfitting: Recognizing how models can perform too well on training data (overfit) or fail to capture the data's complexity (underfit) is key to model generalization.
• Regularization: Techniques like L1, L2, and dropout are essential to prevent overfitting by introducing constraints into the model.
• Bias-Variance Tradeoff: Understanding the balance between a model's complexity and its ability to generalize and its implication for model performance.
• Sparsity: The principle of sparsity is crucial in feature selection, regularization, and reducing model complexity.
• Sampling Methods: Techniques like Monte Carlo methods and Gibbs sampling are important for probabilistic and Bayesian approaches.
• Expressing machine learning as a search problem

Machine learning algorithms

• Kernel Methods: Understanding kernel functions is vital for support vector machines and non-linear feature transformations.
• Linear Models: Concepts such as linear regression, logistic regression, and generalized linear models are foundational to many ML algorithms.
• Bayesian Methods: Bayesian inference, probabilistic models, and understanding prior and posterior distributions are important for probabilistic ML.
• Principal Component Analysis (PCA): PCA is key for dimensionality reduction, feature extraction, and data compression.
• Information Theory: Concepts like entropy, KL divergence, and mutual information are important for understanding data compression and model efficiency.
• Clustering: Concepts like k-means, hierarchical clustering, and DBSCAN are essential for unsupervised learning tasks.
• Support Vector Machines (SVM): Understanding the geometry of hyperplanes and margin maximization is essential for classification tasks.
• Decision Trees and Ensembles: Principles behind decision trees, random forests, and boosting (e.g., AdaBoost, XGBoost) are vital for classification and regression tasks.
• Eigenvalues and Eigenvectors: Important in understanding dimensionality reduction techniques like Principal Component Analysis (PCA) and spectral clustering.
• ?Bayesian Networks: A graphical model representing probabilistic relationships among a set of variables, widely used in AI for decision-making under uncertainty.

Neural networks and deep learning

• The basic structure of artificial neurons, activation functions, and how backpropagation works is fundamental for deep learning.
• Explaining deep neural networks as nested functions and the use of the chain rule. understanding backpropagation and how gradients are computed across layers in a deep network.
• The convolutional operation?
• The convolutional and the recurrent neural network operation as exceptions to the multilayer perceptron
• The basic maths of deep learning?
• Gradient Descent and its Variants: Techniques like stochastic gradient descent, batch gradient descent, and Adam optimization are key to model training.
• Convexity: Recognizing convex functions and optimization problems is essential for ensuring that optimization algorithms converge.
• Backpropagation: The algorithm used to compute gradients for each layer in the neural network, enabling efficient parameter updates.
• Activation Functions: Understanding functions like ReLU, sigmoid, tanh, and softmax, which introduce non-linearity into deep neural networks.
• Convolution Operations:? understanding convolutional neural networks (CNNs), including the use of filters and feature maps.
• Pooling Layers:? in CNNs, such as max pooling and average pooling, that reduce the dimensionality of feature maps.
• Vanishing and Exploding Gradients: Issues that arise in deep networks where gradients become too small or too large, preventing effective learning.
• Attention Mechanisms: understanding transformer architectures. how models can "pay attention" to different parts of an input sequence when making predictions.

What have I missed?

About the book community

As I mentioned before, I am creating a small community for my book - mathematical foundations of data science. You get pdf when released but you also receive chapters as they are released and you get to engage and ask questions.?

The price is a one off 40 USD. If you are interested please DM me. I am trying to keep spaces limited since I want to learn from feedback so that's an important criterion as well. ?