Vector Norms - Explained
Key Takeaways
This video explains vector norms, including L1, L2, and L-infinity norms, for measuring vector size in linear algebra
Full Transcript
How do you measure the size of a vector? You might think there's one answer, but there are infinitely many. Take the vector 3 4. How big is it? That's where norms come in. The most familiar way is the Euclidean norm, just the Pythagorean theorem. The horizontal leg is three, the vertical leg is four, and the hypotenuse is the square root of 3 squared plus 4 squared, giving us five. In general, the L2 norm is the square root of the sum of each component squared. But what if you couldn't walk in a straight line? Imagine a city grid. You go three blocks over and four blocks up. Total distance, seven. That's the L1 norm, the Manhattan norm. Just add up the absolute values. So, the same vector measures five by L2, but seven by L1. Size depends on how you measure. Now, what's the largest component of our vector? It's four. That's the L infinity norm, just the maximum absolute value. And here's the beautiful part. All three are special cases of one general formula, the P norm. Raise each component's absolute value to the power P, sum them up, take the path root. Set P to one, you get seven. Set P to two, five. As P goes to infinity, four. So, what does the set of all vectors with norm equal to one look like? For L2, it's a circle. For L1, a diamond. And for L infinity, a square. Now, watch as we smoothly vary P. The shape morphs from a star to a diamond, then a circle, and finally into a square. One formula, infinitely many geometries. In machine learning, norms drive regularization. An L1 penalty has a diamond-shaped constraint, so the optimum lands on a corner, or they might where a coordinate is zero. That's sparsity. An L2 penalty's smooth circle keeps all weights small, but nonzero. L1 pushes to zero. L2 keeps things small. And that's basically it. If you found this helpful, hit that like button, subscribe for more, and drop a comment if you have any questions. See you in the next one. Bye-bye.
Original Description
Understand vector norms in linear algebra with this clear explanation of L1 norm (Manhattan distance), L2 norm (Euclidean distance), and L-infinity norm. Learn how to measure vector size, explore the p-norm formula, and see how different norms shape geometry and impact machine learning regularization (L1 vs L2). Perfect for beginners in math, data science, and machine learning.
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