Math for Machine Learning - Chapter 5: Norms and Distance Metrics

This quiz provides an in-depth assessment of your understanding of norms and distance metrics, fundamental concepts in linear algebra that are critical for machine learning. You will be challenged on the theoretical underpinnings and practical implications of various vector norms, including their definitions, properties, and geometric interpretations. The quiz covers the L1 (Manhattan), L2 (Euclidean), L-infinity (Chebyshev), and the general L-p norms.

Key formulas you should be familiar with:

• L-p Norm: For a vector $$\mathbf{x} \in \mathbb{R}^n$$, the L-p norm is defined as:

  $$||\mathbf{x}||_p = \left(\sum_{i=1}^{n} |x_i|^p\right)^{1/p}$$

• L1 Norm (p=1):

  $$||\mathbf{x}||_1 = \sum_{i=1}^{n} |x_i|$$

• L2 Norm (p=2):

  $$||\mathbf{x}||_2 = \sqrt{\sum_{i=1}^{n} x_i^2}$$

• L-infinity Norm (p -> infinity):

  $$||\mathbf{x}||_\infty = \max_{i} |x_i|$$

• Distance Metric: The distance between two vectors $$\mathbf{x}$$ and $$\mathbf{y}$$ induced by a norm is:

  $$d_p(\mathbf{x}, \mathbf{y}) = ||\mathbf{x} - \mathbf{y}||_p$$

Questions will require you to go beyond simple calculations and apply these concepts to solve complex problems related to optimization (regularization), geometry in high-dimensional spaces, and the behavior of machine learning algorithms. Prepare to analyze the properties of norms, compare their effects, and understand their role in shaping model characteristics like sparsity and robustness to outliers.