Groups
Category
Level
A Banach space is a vector space with a norm where every Cauchy sequence actually converges within the space.
A Hilbert space is an inner product space that is complete, meaning Cauchy sequences converge to points inside the space.
Lebesgue integration measures how much time a function spends near each value and adds up value ร size of the set where it occurs.
A ฯ-algebra is a collection of subsets that is closed under complements and countable unions, giving us a stable universe of sets where measure makes sense.
Natural gradient scales the ordinary gradient by the inverse Fisher information matrix to account for the geometry of probability distributions.
Lie groups model continuous symmetries like rotations and rigid-body motions, combining algebra (group law) and calculus (smooth manifolds).
Curvature measures how a geometric object bends, and it comes in several flavors: Gaussian, sectional, and Ricci curvature.
Geodesics are the โstraightest possibleโ paths on curved spaces (manifolds) and locally minimize distance.
A Riemannian metric assigns an inner product to each tangent space, giving you a way to measure lengths and angles on curved spaces (manifolds).
A smooth manifold is a space that looks like ordinary Euclidean space when you zoom in, glued together using charts that transition smoothly.
Betti numbers count independent k-dimensional holes: ฮฒโ counts connected components, ฮฒโ counts independent loops/tunnels, and ฮฒโ counts voids.
Topological Data Analysis (TDA) studies the shape of data using tools from algebraic topology, producing summaries like Betti numbers, barcodes, and persistence diagrams.