Core AI and ML Topics Covered in Chapters 6-10 of the Maths-CS-AI Compendium

Chapters 6 through 10 of the HenryNdubuaku/maths-cs-ai-compendium form a comprehensive curriculum covering classical machine learning, computational linguistics, computer vision, audio processing, and multimodal learning architectures.

These five chapters progress from foundational algorithms to state-of-the-art deep learning systems. According to the source code in the repository, they provide a full-stack view of modern AI, starting with probabilistic models and optimization theory, advancing through domain-specific architectures for text and image data, and culminating in unified multimodal systems. The compendium emphasizes both mathematical rigor and practical implementation using frameworks like JAX.

Chapter 6: Machine Learning Foundations

Chapter 6 establishes the mathematical groundwork for all subsequent topics, divided into classical algorithms and gradient-based optimization methods.

Classical Algorithms and Decision Trees

The repository covers traditional supervised and unsupervised methods including Naïve Bayes, k-Nearest Neighbors, Decision Trees, Random Forests, SVM, k-means, and Gaussian Mixture Models (GMM). In chapter 06: machine learning/01. classical machine learning.md, the implementation details for decision tree splits using Gini impurity are provided:

import jax.numpy as jnp

def gini_impurity(y):
    _, counts = jnp.unique(y, return_counts=True)
    probs = counts / len(y)
    return 1.0 - jnp.sum(probs ** 2)

def information_gain(y, left_mask):
    parent = gini_impurity(y)
    left, right = y[left_mask], y[~left_mask]
    n = len(y)
    if len(left) == 0 or len(right) == 0:
        return 0.0
    child = (len(left)/n)*gini_impurity(left) + (len(right)/n)*gini_impurity(right)
    return float(parent - child)

This code demonstrates how the compendium implements fundamental splitting criteria for decision trees using JAX.

Gradient-Based Optimization and Modern Optimizers

The second part of Chapter 6 focuses on Linear and Logistic Regression, Softmax classifiers, and optimization algorithms. The source file chapter 06: machine learning/02. gradient machine learning.md covers loss functions (MSE, BCE, Hinge, MAE, Huber) and regularization techniques (L1/L2, Elastic Net), along with modern optimizers including SGD, Momentum, Adam, LION, and Muon.

Below is the Adam optimizer implementation on an elongated quadratic loss surface:

import jax, jax.numpy as jnp
import matplotlib.pyplot as plt

def loss(w):
    return 0.5*w[0]**2 + 10*w[1]**2       # elongated bowl

grad = jax.grad(loss)

def run_adam(w0, lr=0.05, steps=80):
    w, m, v = w0, jnp.zeros_like(w0), jnp.zeros_like(w0)
    path = [w]
    for t in range(1, steps+1):
        g = grad(w)
        m = 0.9*m + 0.1*g
        v = 0.999*v + 0.001*g**2
        m_hat = m/(1-0.9**t)
        v_hat = v/(1-0.999**t)
        w = w - lr*m_hat/(jnp.sqrt(v_hat)+1e-8)
        path.append(w)
    return jnp.stack(path)

traj = run_adam(jnp.array([8.0, 3.0]))
plt.plot(traj[:,0], traj[:,1], 'o-', color='#e74c3c')
plt.title('Adam on elongated quadratic')
plt.xlabel('w₁'); plt.ylabel('w₂')
plt.show()

This example illustrates the adaptive learning rate behavior that makes Adam effective for poorly conditioned optimization landscapes.

Chapter 7: Computational Linguistics and Modern NLP

Chapter 7 bridges linguistic theory and modern deep learning for natural language processing.

Linguistic Foundations and Text Preprocessing

The compendium covers phonetics, morphology, and syntax as prerequisites for computational modeling. In chapter 07: computational linguistics/01. linguistic foundations.md, the text preprocessing pipeline includes tokenisation, stemming, and stop-word removal, leading to classical representations like Bag-of-Words and TF-IDF.

Transformers and Large Language Models

The progression moves from static embeddings (Word2Vec, GloVe) to modern architectures. The file chapter 07: computational linguistics/04. transformers and language models.md contains the theoretical foundations of self-attention mechanisms, scaling laws, and the architecture patterns that underpin large language models.

Chapter 8: Computer Vision and Deep Learning Architectures

Chapter 8 addresses visual data processing, from pixel fundamentals to generative architectures.

Convolutional Neural Networks

The source file chapter 08: computer vision/02. convolutional networks.md explains image fundamentals (pixels, color spaces), receptive fields, and classic architectures including LeNet and ResNet. The compendium details pooling operations and the inductive biases that make CNNs effective for spatial data.

Vision Transformers and Advanced Architectures

Moving beyond convolutions, chapter 08: computer vision/04. vision transformers and generation.md covers Vision Transformers (ViT), object detection (YOLO, Mask R-CNN), segmentation, and video/3-D vision for action recognition and depth estimation.

The minimal ViT forward pass implementation demonstrates patch embedding and positional encoding:

import jax.numpy as jnp
from einops import rearrange

def vit_forward(patch_embeddings, cls_token, pos_embed, mlp):
    # patch_embeddings: (B, N, D)

    x = jnp.concatenate([cls_token, patch_embeddings], axis=1) + pos_embed
    # simple transformer block (omitted for brevity)

    # ...

    return mlp(x[:,0])   # classification token output

Chapter 9: Audio, Speech, and Signal Processing

Chapter 9 focuses on sequential signal data, covering both digital signal processing fundamentals and modern speech ML systems.

Digital Signal Processing Fundamentals

According to chapter 09: audio and speech/01. digital signal processing.md, the core topics include Fourier transforms, spectrograms, and the spectral analysis techniques that form the input representations for audio ML.

Speech Recognition and Synthesis

The compendium covers ASR pipelines and CTC loss for sequence-to-sequence speech recognition, text-to-speech (TTS) and voice synthesis, as well as speaker diarisation, emotion recognition, source separation, and noise reduction techniques.

Chapter 10: Multimodal Learning and Cross-Modal Integration

Chapter 10 synthesizes the preceding domains into systems that process heterogeneous data types simultaneously.

Multimodal Representations and Vision-Language Models

The file chapter 10: multimodal learning/01. multimodal representations.md introduces joint embeddings and contrastive learning. The compendium implements CLIP-style training objectives that align image and text representations:

import jax.numpy as jnp

def contrastive_loss(image_emb, text_emb, temperature=0.07):
    # Normalise

    image_emb = image_emb / jnp.linalg.norm(image_emb, axis=1, keepdims=True)
    text_emb  = text_emb  / jnp.linalg.norm(text_emb,  axis=1, keepdims=True)
    logits = image_emb @ text_emb.T / temperature
    labels = jnp.arange(logits.shape[0])
    loss_i = -jnp.mean(jnp.log(jnp.exp(jnp.diag(logits)) / jnp.sum(jnp.exp(logits), axis=1)))
    loss_t = -jnp.mean(jnp.log(jnp.exp(jnp.diag(logits)) / jnp.sum(jnp.exp(logits), axis=0)))
    return (loss_i + loss_t) / 2

This symmetric loss function trains models to maximize similarity between paired image-text embeddings while minimizing it for unpaired samples.

Unified Multimodal Architectures

The final section in chapter 10: multimodal learning/05. unified multimodal architectures.md covers cross-modal generation (image-to-text, text-to-image) and unified architectures like Perceiver and Gato that handle arbitrary modality mixtures through attention mechanisms.

Summary

• Chapters 6-10 provide a comprehensive AI/ML curriculum spanning from classical algorithms to modern multimodal systems according to the HenryNdubuaku/maths-cs-ai-compendium source code.
• Chapter 6 establishes foundations with decision trees, ensemble methods, and optimizers including Adam, LION, and Muon implemented in chapter 06: machine learning/.
• Chapter 7 transitions from linguistic theory to transformers and large language models in chapter 07: computational linguistics/.
• Chapter 8 covers computer vision through CNNs and Vision Transformers detailed in chapter 08: computer vision/.
• Chapter 9 addresses audio processing, speech recognition, and signal processing fundamentals in chapter 09: audio and speech/.
• Chapter 10 integrates these domains through multimodal representations and unified architectures like CLIP and Perceiver found in chapter 10: multimodal learning/.

Frequently Asked Questions

What optimization algorithms are covered in Chapter 6 beyond standard SGD?

Chapter 6 includes modern adaptive optimizers such as Adam, LION, and Muon alongside classical momentum and SGD. The run_adam() function in chapter 06: machine learning/02. gradient machine learning.md demonstrates the adaptive learning rate mechanism using bias-corrected first and second moment estimates.

How does the compendium explain the difference between CNNs and Vision Transformers?

The repository contrasts these architectures across two files: chapter 08: computer vision/02. convolutional networks.md covers CNNs with their inductive biases for spatial locality, while chapter 08: computer vision/04. vision transformers and generation.md explains how ViTs replace convolutional operations with self-attention over image patches, requiring less domain-specific engineering but more data.

What are the key components of the multimodal learning approach in Chapter 10?

Chapter 10 focuses on joint embedding spaces and contrastive learning objectives that align representations across modalities. The contrastive_loss() function implements the symmetric cross-entropy loss used in CLIP-style training, while the section on unified architectures covers models like Perceiver and Gato that process arbitrary modality combinations through attention mechanisms.

Does Chapter 9 include practical implementations for speech recognition?

Yes, Chapter 9 covers practical ASR pipelines including CTC loss for alignment-free sequence training and spectrogram-based feature extraction. The source file chapter 09: audio and speech/01. digital signal processing.md provides the signal processing foundations necessary for implementing modern speech recognition systems.

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    "instagit": {
      "command": "npx",
      "args": ["-y", "instagit@latest"]
    }
  }
}