19:28 "Since this holds for any alpha1 and alpha2 that are non-negative it also holds for a1 and a2 that sum to 1 and therefore a cone has to be convex" Geometrically this makes sense if we fill the interior of the cone but how can we guarantee convexity for all cones. For example, the graph of y=|x| is a cone that is not convex; however, the locus of points (x,y) with y≥|x| is a convex cone. There's a whole wiki on convex cones so I assume you made this statement solely based on the definition above and not as a general truth? I'm a bit confused.
Hello, professor. I may have misunderstood something, but at 15:22, shouldn't vectors (1, -1) be vectors heading right down from the origin? BTW, thank you so much for your lectures!
Convex set definition 0:00
Subspace, range(span), null space, affine space 4:54
Line, hyperplane 9:52
Half space 15:58
Cones and polar cones 18:38
Tangent cones 21:54
Normal cones 24:46
Your lectures are SO AMAZING. Thanks so much for your sharing.
Great and interesting lecture series!! Helpful for ML researchers.
Yes. It's an amazing playlist for someone like me, who is researching at the intersection of optimization and machine learning.
19:28 "Since this holds for any alpha1 and alpha2 that are non-negative it also holds for a1 and a2 that sum to 1 and therefore a cone has to be convex"
Geometrically this makes sense if we fill the interior of the cone but how can we guarantee convexity for all cones. For example, the graph of y=|x| is a cone that is not convex; however, the locus of points (x,y) with y≥|x| is a convex cone. There's a whole wiki on convex cones so I assume you made this statement solely based on the definition above and not as a general truth? I'm a bit confused.
I just found this example too drive.google.com/file/d/1lCPb48aW2kfd-yaOUmxKWILsCQ9UnvBh/view
Thank you for this amazing lecture. At 15:27, is the normal vector (1, -1) not supposed to be in the fourth quadrant, pointing away from the origin?
Yes, you're right, thanks! (Since in this case we have = 0, though, we could have multiplied through by -1, to get (-1,1).)
Hello, professor. I may have misunderstood something, but at 15:22, shouldn't vectors (1, -1) be vectors heading right down from the origin? BTW, thank you so much for your lectures!
Yes, you're right, thanks! (Since in this case we have = 0, though, we could have multiplied through by -1, to get (-1,1).)