feat: documents SVM, SVR and SVC

src/svm/svr.rs

@@ -1,6 +1,24 @@
 //! # Epsilon-Support Vector Regression.
+//! 
+//! Support Vector Regression (SVR) is a popular algorithm used for regression that uses the same principle as SVM. 
+//! 
+//! Just like [SVC](../svc/index.html) SVR finds optimal decision boundary, \\(f(x)\\) that separates all training instances with the largest margin.
+//! Unlike SVC, in \\(\epsilon\\)-SVR regression the goal is to find a function \\(f(x)\\) that has at most \\(\epsilon\\) deviation from the 
+//! known targets \\(y_i\\) for all the training data. To find this function, we need to find solution to this optimization problem:
+//! 
+//! \\[\underset{w, \zeta}{minimize} \space \space \frac{1}{2} \lVert \vec{w} \rVert^2 + C\sum_{i=1}^m \zeta_i \\]
+//! 
+//! subject to:
+//! 
+//! \\[\lvert y_i - \langle\vec{w}, \vec{x}_i \rangle - b \rvert \leq \epsilon + \zeta_i \\]
+//! \\[\lvert \langle\vec{w}, \vec{x}_i \rangle + b - y_i \rvert \leq \epsilon + \zeta_i \\]
+//! \\[\zeta_i \geq 0 for \space any \space i = 1, ... , m\\]
+//! 
+//! Where \\( m \\) is a number of training samples, \\( y_i \\) is a target value and \\(\langle\vec{w}, \vec{x}_i \rangle + b\\) is a decision boundary. 
+//! 
+//! The parameter `C` > 0 determines the trade-off between the flatness of \\(f(x)\\) and the amount up to which deviations larger than \\(\epsilon\\) are tolerated
 //!
-//! Example
+//! Example:
 //!
 //! ```
 //! use smartcore::linalg::naive::dense_matrix::*;
@@ -44,10 +62,13 @@
 //!
 //! ## References:
 //!
-//! * ["Support Vector Machines" Kowalczyk A., 2017](https://www.svm-tutorial.com/2017/10/support-vector-machines-succinctly-released/)
+//! * ["Support Vector Machines", Kowalczyk A., 2017](https://www.svm-tutorial.com/2017/10/support-vector-machines-succinctly-released/)
 //! * ["A Fast Algorithm for Training Support Vector Machines", Platt J.C., 1998](https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/tr-98-14.pdf)
 //! * ["Working Set Selection Using Second Order Information for Training Support Vector Machines", Rong-En Fan et al., 2005](https://www.jmlr.org/papers/volume6/fan05a/fan05a.pdf)
-//! * ["A tutorial on support vector regression", SMOLA A.J., Scholkopf B., 2003](https://alex.smola.org/papers/2004/SmoSch04.pdf)
+//! * ["A tutorial on support vector regression", Smola A.J., Scholkopf B., 2003](https://alex.smola.org/papers/2004/SmoSch04.pdf)
+//! 
+//! <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
+//! <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
 
 use std::cell::{Ref, RefCell};
 use std::fmt::Debug;