# What is the support vector machine?

What IS the support vector machine? Can someone clarify my confusion?

1. The SVM is the problem: given data $$(xn,yn),n=1,…,N(x_n, y_n), n = 1, \ldots, N$$

$$min\min_{w, b}\frac{1}{2}||w||^2$$
$$\text{ subject to: } y_n(w \cdot x_n + b) \geq 1, n=1,…,N\text{ subject to: } y_n(w \cdot x_n + b) \geq 1, n=1,...,N$$

See Solving Support Vector Machine
with Many Examples

Paweł Białoń, “…Various methods of dealing with linear support vector machine (SVM) problems with a large number of examples
are presented and compared…”

See http://www.cs.tau.ac.il/~mansour/ml-course-10/scribe9.pdf “We begin with building the intuition behind SVMs, continue to define
SVM as an optimization problem

1. SVM is the algorithm that solves the problem,

given data $$(x_n, y_n), n = 1, \ldots, N(x_n, y_n), n = 1, \ldots, N$$

$$\min_{w, b}\frac{1}{2}||w||^2\min_{w, b}\frac{1}{2}||w||^2$$
$$\text{ subject to: } y_n(w \cdot x_n + b) \geq 1, n=1,…,N\text{ subject to: } y_n(w \cdot x_n + b) \geq 1, n=1,...,N$$

“SVMs are among the best (and many believe are indeed the best)
“off-the-shelf” supervised learning algorithms.” – Andrew Ng

See reference, “SVM is a supervised learning algorithm”

But this means that QP solvers are SVMs…

1. SVM is the solution to the following problem:

given data $$(x_n, y_n), n = 1, \ldots, N(x_n, y_n), n = 1, \ldots, N$$

$$\min_{w, b}\frac{1}{2}||w||^2\min_{w, b}\frac{1}{2}||w||^2$$
$$\text{ subject to: } y_n(w \cdot x_n + b) \geq 1, n=1,…,N\text{ subject to: } y_n(w \cdot x_n + b) \geq 1, n=1,...,N$$

See, https://cel.archives-ouvertes.fr/cel-01003007/file/Lecture1_Linear_SVM_Primal.pdf (Page 22) “SVM is the solution to the problem….” (but then the author immediately contradicts himself by calling the SVM a quadratic programming problem – so is it the solution or the problem??)

See, reference “SVM is a discriminative classifier”