Contents
What is inner product in Hilbert space?
product spaces and complete inner product spaces, called Hilbert spaces. Inner product spaces are special normed spaces, as we shall see. Historically they are older than general normed spaces. Their theory is richer and retains many features of Euclidean space, a central concept being orthogonality.
What is complete inner product space?
An inner product space is a vector space together with an inner product on it. If the inner product defines a complete metric, then the inner product space is called a Hilbert space. Historically, inner product spaces are sometimes referred to as pre-Hilbert spaces.
What is the function of inner product space?
Inner product space, In mathematics, a vector space or function space in which an operation for combining two vectors or functions (whose result is called an inner product) is defined and has certain properties.
How to calculate inner product in machine learning?
As long as we can calculate the inner product in the feature space, we do not need the mapping explicitly Many common geometric operations (angles, distances) can be expressed by inner products Define the kernel function Kby m i j j T i j i j i m i i yy 121 1 , max
How to calculate the inner product in SVM?
Recall the SVM optimization problem The data points only appear asinner product As long as we can calculate the inner product in the feature space, we do not need the mapping explicitly Many common geometric operations (angles, distances) can be expressed by inner products Define the kernel function Kby
What is the kernel trick in machine learning?
The Kernel Trick, Reproducing Kernel Hilbert Space, and the Representer Theorem Eric Xing Lecture 6, September 24, 2014 Reading: © Eric Xing @ CMU, 20141 Recap: the SVM problem We solve the following constrained opt problem: This is a quadratic programmingproblem.
Which is an example of a polynomial kernel?
© Eric Xing @ CMU, 20149 More examples of kernel functions Linear kernel (we’ve seen it) Polynomial kernel (we just saw an example) where p= 2, 3, … To get the feature vectors we concatenate all pth order polynomial terms of the components of x (weighted appropriately)