diff --git a/src/svm/mod.rs b/src/svm/mod.rs index c9aea15..84a405e 100644 --- a/src/svm/mod.rs +++ b/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 diff --git a/src/svm/svc.rs b/src/svm/svc.rs index 5cce80a..04b3b7b 100644 --- a/src/svm/svc.rs +++ b/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) -//! +//! //! //! diff --git a/src/svm/svr.rs b/src/svm/svr.rs index be5d7b9..0fcaa30 100644 --- a/src/svm/svr.rs +++ b/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) -//! +//! //! //!