Linear Algebra: Vector Norms

Various ways to measure the distance between two points. Nice blog post: https://medium.com/@lakesare/vector-norms-fast-and-thorough-explanation-882745a79c47, https://udayablogs.wordpress.com/2018/07/25/vector-norms, https://ekamperi.github.io/machine%20learning/2019/10/19/norms-in-machine-learning.html.

9 flashcards · Mathematics · by lakesare

  1. What's vector norms?

    That's various ways to measure the magnitude of the vector. Vector norm is the magnitude (or length) of the vector.It’s the glorified distance between two points, measured in more traditional (2-norm) and less traditional (1-norm, 3-norm, any-norm) ways.

  2. Give the general formula for the p-norm.

    |aₓ - bₓ| here is the absolute value of the difference between the coordinate of point a along the x-axis, and the coordinate of point b along the x-axis.There can be as many dimensions as we’d like to (shown from x to z here).

  3. Give the name for: 0-norm 1-norm 2-norm ∞-norm

    0-norm - Hamming distance 1-norm - Manhattan/Taxicab distance 2-norm - Euclidean distance ∞-norm - Chebyshev distance/Infinity distance

  4. What's the formula for 0-norm?

    Hamming distance Simply the number of nonzero coordinates of the ab vector.

  5. What's the formula for 1-norm?

    Manhattan/Taxicab distance

  6. What's the formula for 2-norm?

    Euclidean distance

  7. What's the formula for ∞-norm?

    Chebyshev distance/Infinity distance The maximum distance along one of the dimensions.

  8. What is Euclidean distance VS Manhattan/taxicab distance graphically?

    Taxicab geometry versus Euclidean distance: In taxicab geometry, the red, yellow, and blue paths all have the same shortest path length of 12. In Euclidean geometry, the green line has length and is the unique shortest path.

  9. Find Manhattan distance (1-norm) Euclidean distance (2-norm) between these 2 points.