Tensor Calculus: A Primer for Machine Learning & Optimization

Welcome to the Tensor Calculus Crash Course!

This mini-series is designed for learners who have a grounding in Linear Algebra, Multivariable Calculus, and the basics of Functional Analysis (as outlined in the preface and prerequisites section of the main “Mathematical Optimization Theory in ML” series). Tensors are fundamental mathematical objects for describing geometric and physical quantities, and they play an increasingly crucial role in modern machine learning. They provide a powerful framework for handling high-dimensional data, understanding complex transformations, and analyzing the geometric properties of model parameter spaces and loss landscapes.

In machine learning and optimization, we often encounter quantities like gradients of matrix-valued functions, Hessians in many dimensions, and concepts related to the curvature of optimization surfaces. Tensor calculus provides the natural language and tools to work with these objects rigorously and intuitively.

This crash course aims to:

• Demystify tensor notation (especially the Einstein summation convention).
• Explore fundamental tensor algebra (addition, products, contractions).
• Explain how tensor components change under coordinate transformations (covariance and contravariance).
• Introduce the metric tensor and its role in defining geometry.
• Provide a gentle introduction to tensor differentiation (the covariant derivative).

Our focus will be on building intuition and understanding the practical relevance of these concepts for subsequent topics in the main “Mathematical Optimization Theory in ML” series, particularly for modules on Differential Geometry and Information Geometry.

Course Outline

This crash course is structured into the following parts:

1. Part 1: From Vectors to Tensors – Definitions and Algebra
   • Motivation: Why do we need tensors in Machine Learning?
   • Revisiting Vectors and Covectors (Dual Vectors).
   • Introducing Tensors: Generalizing Scalars, Vectors, and Matrices.
   • The Language of Tensors: Einstein Summation Convention.
   • Fundamental Tensor Operations: Addition, Scalar Multiplication, Outer Product, Contraction.
   • Symmetry and Anti-symmetry in Tensors.
2. Part 2: Tensors in Motion – Coordinate Transformations and the Metric Tensor
   • The Importance of Transformation Rules.
   • Coordinate Transformations and Jacobian Matrices.
   • Contravariant Tensors (e.g., vectors).
   • Covariant Tensors (e.g., gradients, covectors).
   • Mixed Tensors and their Transformation Properties.
   • The Metric Tensor: Defining Inner Products, Distances, and Angles.
   • Raising and Lowering Indices using the Metric Tensor.
3. Part 3: Tensor Calculus – Differentiation and ML Applications
   • The Challenge: Why naive differentiation of tensor components fails.
   • Christoffel Symbols: Correcting for Changing Bases.
   • The Covariant Derivative: Differentiating Tensors Correctly.
   • Key Tensors in ML: Gradients, Hessians, and the Fisher Information Matrix.
   • A Glimpse into Curvature.

We encourage you to work through the examples and think about how these concepts might apply to problems you’ve encountered in machine learning. Let’s begin!