fix: formatting

Author: Volodymyr Orlov

src/svm/mod.rs

@@ -1,20 +1,20 @@
 //! # Support Vector Machines
-//! 
-//! Support Vector Machines (SVM) is one of the most performant off-the-shelf machine learning algorithms. 
+//!
+//! Support Vector Machines (SVM) is one of the most performant off-the-shelf machine learning algorithms.
 //! SVM is based on the [Vapnik–Chervonenkiy theory](https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_theory) that was developed during 1960–1990 by Vladimir Vapnik and Alexey Chervonenkiy.
-//! 
-//! SVM splits data into two sets using a maximal-margin decision boundary, \\(f(x)\\). For regression, the algorithm uses a value of the function \\(f(x)\\) to predict a target value. 
+//!
+//! SVM splits data into two sets using a maximal-margin decision boundary, \\(f(x)\\). For regression, the algorithm uses a value of the function \\(f(x)\\) to predict a target value.
 //! To classify a new point, algorithm calculates a sign of the decision function to see where the new point is relative to the boundary.
-//! 
+//!
 //! SVM is memory efficient since it uses only a subset of training data to find a decision boundary. This subset is called support vectors.
-//! 
-//! In SVM distance between a data point and the support vectors is defined by the kernel function. 
-//! SmartCore supports multiple kernel functions but you can always define a new kernel function by implementing the `Kernel` trait. Not all functions can be a kernel. 
-//! Building a new kernel requires a good mathematical understanding of the [Mercer theorem](https://en.wikipedia.org/wiki/Mercer%27s_theorem) 
+//!
+//! In SVM distance between a data point and the support vectors is defined by the kernel function.
+//! SmartCore supports multiple kernel functions but you can always define a new kernel function by implementing the `Kernel` trait. Not all functions can be a kernel.
+//! Building a new kernel requires a good mathematical understanding of the [Mercer theorem](https://en.wikipedia.org/wiki/Mercer%27s_theorem)
 //! that gives necessary and sufficient condition for a function to be a kernel function.
-//! 
+//!
 //! Pre-defined kernel functions:
-//! 
+//!
 //! * *Linear*, \\( K(x, x') = \langle x, x' \rangle\\)
 //! * *Polynomial*, \\( K(x, x') = (\gamma\langle x, x' \rangle + r)^d\\), where \\(d\\) is polynomial degree, \\(\gamma\\) is a kernel coefficient and \\(r\\) is an independent term in the kernel function.
 //! * *RBF (Gaussian)*, \\( K(x, x') = e^{-\gamma \lVert x - x' \rVert ^2} \\), where \\(\gamma\\) is kernel coefficient

src/svm/svc.rs

@@ -1,27 +1,27 @@
 //! # Support Vector Classifier.
-//! 
+//!
 //! Support Vector Classifier (SVC) is a binary classifier that uses an optimal hyperplane to separate the points in the input variable space by their class.
-//!  
-//! During training, SVC chooses a Maximal-Margin hyperplane that can separate all training instances with the largest margin. 
-//! The margin is calculated as the perpendicular distance from the boundary to only the closest points. Hence, only these points are relevant in defining 
+//!
+//! During training, SVC chooses a Maximal-Margin hyperplane that can separate all training instances with the largest margin.
+//! The margin is calculated as the perpendicular distance from the boundary to only the closest points. Hence, only these points are relevant in defining
 //! the hyperplane and in the construction of the classifier. These points are called the support vectors.
-//! 
-//! While SVC selects a hyperplane with the largest margin it allows some points in the training data to violate the separating boundary. 
-//! The parameter `C` > 0 gives you control over how SVC will handle violating points. The bigger the value of this parameter the more we penalize the algorithm 
-//! for incorrectly classified points. In other words, setting this parameter to a small value will result in a classifier that allows for a big number 
-//! of misclassified samples. Mathematically, SVC optimization problem can be defined as: 
-//! 
+//!
+//! While SVC selects a hyperplane with the largest margin it allows some points in the training data to violate the separating boundary.
+//! The parameter `C` > 0 gives you control over how SVC will handle violating points. The bigger the value of this parameter the more we penalize the algorithm
+//! for incorrectly classified points. In other words, setting this parameter to a small value will result in a classifier that allows for a big number
+//! of misclassified samples. Mathematically, SVC optimization problem can be defined as:
+//!
 //! \\[\underset{w, \zeta}{minimize} \space \space \frac{1}{2} \lVert \vec{w} \rVert^2 + C\sum_{i=1}^m \zeta_i \\]
-//! 
+//!
 //! subject to:
-//! 
+//!
 //! \\[y_i(\langle\vec{w}, \vec{x}_i \rangle + b) \geq 1 - \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 label value (either 1 or -1) and \\(\langle\vec{w}, \vec{x}_i \rangle + b\\) is a decision boundary. 
-//! 
-//! To solve this optimization problem, SmartCore uses an [approximate SVM solver](https://leon.bottou.org/projects/lasvm). 
-//! The optimizer reaches accuracies similar to that of a real SVM after performing two passes through the training examples. You can choose the number of passes 
+//!
+//! Where \\( m \\) is a number of training samples, \\( y_i \\) is a label value (either 1 or -1) and \\(\langle\vec{w}, \vec{x}_i \rangle + b\\) is a decision boundary.
+//!
+//! To solve this optimization problem, SmartCore uses an [approximate SVM solver](https://leon.bottou.org/projects/lasvm).
+//! The optimizer reaches accuracies similar to that of a real SVM after performing two passes through the training examples. You can choose the number of passes
 //! through the data that the algorithm takes by changing the `epoch` parameter of the classifier.
 //!
 //! Example:
@@ -73,7 +73,7 @@
 //!
 //! * ["Support Vector Machines", Kowalczyk A., 2017](https://www.svm-tutorial.com/2017/10/support-vector-machines-succinctly-released/)
 //! * ["Fast Kernel Classifiers with Online and Active Learning", Bordes A., Ertekin S., Weston J., Bottou L., 2005](https://www.jmlr.org/papers/volume6/bordes05a/bordes05a.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>
 

src/svm/svr.rs

@@ -1,21 +1,21 @@
 //! # Epsilon-Support Vector Regression.
-//! 
-//! Support Vector Regression (SVR) is a popular algorithm used for regression that uses the same principle as SVM. 
-//! 
+//!
+//! 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 
+//! 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. 
-//! 
+//!
+//! 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:
@@ -66,7 +66,7 @@
 //! * ["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)
-//! 
+//!
 //! <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>