Math for Machine Learning for Signal Processing (WIP)

This section is dedicated to foundation of signal processing and machine learning. We will examine machine learning methods from the lens of signal processing using vector space methods, vectors and matrices representation, and linear algebras.

A Short Technical Primer

Signal processing and machine learning have become foundational pillars in the technological landscape, powering advancements from voice recognition to medical imaging. At their core lies a rich tapestry of mathematical concepts. This primer offers a concise overview of these mathematical underpinnings, fostering a deeper understanding and appreciation of their roles in these disciplines.

Vector Spaces & Dimensionality

In both signal processing and machine learning, data often reside in high-dimensional vector spaces. Understanding these spaces is crucial, especially when considering dimensionality reduction techniques or feature spaces.

Example: In machine learning, an image with 100×100 pixels can be considered a point in a 10,000-dimensional space.

Fourier and Wavelet Transforms

Signal processing frequently operates in the frequency domain. Fourier transforms allow signals to be represented in terms of their frequency components, essential for tasks like filtering or compression. Wavelet transforms offer a joint time-frequency analysis.

Example: Noise reduction in audio signals often involves removing specific frequency components.

Optimization Techniques

Training machine learning models often boils down to optimization problems. Techniques like gradient descent seek to minimize a loss function, guiding the model towards better performance.

Example: Neural networks adjust their weights using backpropagation, a form of gradient descent, to reduce prediction errors.

Probability and Statistics

A cornerstone in both signal processing and machine learning, understanding data distributions, variability, and uncertainty is pivotal. From Bayesian inference to hypothesis testing, these concepts help in making data-driven decisions and model uncertainties.

Example: In Bayesian machine learning, prior beliefs about model parameters are updated with data to get posterior distributions, aiding in model training.

Linear Algebra in ML

Matrix operations form the backbone of numerous algorithms. Eigen-decomposition, singular value decomposition, and other matrix factorizations offer insights and tools for data compression, feature extraction, and more.

Example: Principal Component Analysis (PCA) uses singular value decomposition for dimensionality reduction.

Neural Networks and Deep Learning

These computational models, inspired by biological neural networks, have revolutionized machine learning. They consist of layers of interconnected nodes or "neurons" and are particularly adept at handling large-scale and complex data.

Example: Convolutional Neural Networks (CNNs) are a class of deep learning models exceptionally well-suited for image recognition tasks.

Kernel Methods

Kernels allow data to be implicitly mapped into higher-dimensional spaces, facilitating nonlinear classifications or regressions without explicitly transforming the data.

Example: Support Vector Machines (SVM) with radial basis function (RBF) kernels can classify non-linearly separable data by implicitly mapping it to a higher-dimensional space.

Information Theory

This field provides tools to quantify information, helping in data compression, error correction, and understanding data distributions in machine learning.

Example: The Kullback-Leibler divergence measures the difference between two probability distributions, aiding in algorithms like t-SNE for dimensionality reduction.

Learning Theory

Understanding how models learn and generalize is crucial. Concepts like the bias-variance tradeoff offer insights into a model's performance and its potential pitfalls.

Example: Overfitting occurs when a model is too complex, capturing noise in the data. Regularization techniques counteract this by adding constraints to the model.