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| The '''structured support vector machine''' is a [[machine learning]] algorithm that generalizes the [[Support vector machine|Support Vector Machine]] (SVM) classifier. Whereas the SVM classifier supports [[binary classification]], [[multiclass classification]] and [[Regression analysis|regression]], the structured SVM allows training of a classifier for general [[structured learning|structured output labels]].
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| As an example, a sample instance might be a natural language sentence, and the output label is an annotated [[parse tree]]. Training a classifier consists of showing pairs of correct sample and output label pairs. After training, the structured SVM model allows one to predict for new sample instances the corresponding output label; that is, given a natural language sentence, the classifier can produce the most likely parse tree.
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| ==Training==
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| For a set of <math>\ell</math> training instances <math>(\boldsymbol{x}_n,y_n) \in \mathcal{X}\times\mathcal{Y}</math>, <math>n=1,\dots,\ell</math> from a sample space <math>\mathcal{X}</math> and label space <math>\mathcal{Y}</math>, the structured SVM minimizes the following regularized risk function.
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| :<math>\underset{\boldsymbol{w}}{\min} \quad \|\boldsymbol{w}\|^2 + C \sum_{n=1}^{\ell} | |
| \underset{y\in\mathcal{Y}}{\max} \left(\Delta(y_n,y) + \boldsymbol{w}'\Psi(\boldsymbol{x}_n,y) - \boldsymbol{w}'\Psi(\boldsymbol{x}_n,y_n)\right)</math>
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| The function is convex in <math>\boldsymbol{w}</math> because the maximum of a set of affine functions is convex. The function <math>\Delta: \mathcal{Y} \times \mathcal{Y} \to \mathbb{R}_+</math> measures a distance in label space and is an arbitrary function (not necessarily a [[Metric (mathematics)|metric]]) satisfying <math>\Delta(y,z) \geq 0</math> and <math> \Delta(y,y)=0 \;\; \forall y,z \in \mathcal{Y}</math>. The function <math>\Psi: \mathcal{X} \times \mathcal{Y} \to \mathbb{R}^d</math> is a feature function, extracting some feature vector from a given sample and label. The design of this function depends very much on the application.
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| Because the regularized risk function above is non-differentiable, it is often reformulated in terms of a [[quadratic program]] by introducing one slack variable <math>\xi_n</math> for each sample, each representing the value of the maximum. The standard structured SVM primal formulation is given as follows.
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| :<math>\begin{array}{cl}
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| \underset{\boldsymbol{w},\boldsymbol{\xi}}{\min} & \|\boldsymbol{w}\|^2 + C \sum_{n=1}^{\ell} \xi_n\\
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| \textrm{s.t.} & \boldsymbol{w}' \Psi(\boldsymbol{x}_n,y_n) - \boldsymbol{w}' \Psi(\boldsymbol{x}_n,y) + \xi_n \geq \Delta(y_n,y),\qquad n=1,\dots,\ell,\quad \forall y \in \mathcal{Y}
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| \end{array}</math>
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| ==Inference==
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| At test time, only a sample <math>\boldsymbol{x} \in \mathcal{X}</math> is known, and a prediction function <math>f: \mathcal{X} \to \mathcal{Y}</math> maps it to a predicted label from the label space <math>\mathcal{Y}</math>. For structured SVMs, given the vector <math>\boldsymbol{w}</math> obtained from training, the prediction function is the following.
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| :<math>f(\boldsymbol{x}) = \underset{y \in \mathcal{Y}}{\textrm{argmax}} \quad \boldsymbol{w}'\Psi(\boldsymbol{x},y)</math>
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| Therefore, the maximizer over the label space is the predicted label. Solving for this maximizer is the so-called inference problem and similar to making a maximum a-posteriori (MAP) prediction in probabilistic models. Depending on the structure of the function <math>\Psi</math>, solving for the maximizer can be a hard problem.
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| == Separation ==
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| The above quadratic program involves a very large, possibly infinite number of linear inequality constraints. In general, the number of inequalities is too large to be optimized over explicitly. Instead the problem is solved by using [[delayed constraint generation]] where only a finite and small subset of the constraints is used. Optimizing over a subset of the constraints enlarges the [[feasible set]] and will yield a solution which provides a lower bound on the objective. To test whether the solution <math>\boldsymbol{w}</math> violates constraints of the complete set inequalities, a [[separation]]{{Disambiguation needed|date=June 2011}} problem needs to be solved. As the inequalities decompose over the samples, for each sample <math>(\boldsymbol{x}_n,y_n)</math> the following problem needs to be solved. | |
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| :<math>y_n^* = \underset{y \in \mathcal{Y}}{\textrm{argmax}} \left( | |
| \Delta(y_n,y) + \boldsymbol{w}'\Psi(\boldsymbol{x}_n,y) - \boldsymbol{w}'\Psi(\boldsymbol{x}_n,y_n) - \xi_n\right)</math>
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| The right hand side objective to be maximized is composed of the constant <math>-\boldsymbol{w}'\Psi(\boldsymbol{x}_n,y_n) - \xi_n</math> and a term dependent on the variables optimized over, namely <math>\Delta(y_n,y) + \boldsymbol{w}'\Psi(\boldsymbol{x}_n,y)</math>. If the achieved right hand side objective is smaller or equal to zero, no violated constraint for this sample exist. If it is strictly larger than zero, the most violated constraint with respect to this sample has been identified. The problem is enlarged by this constraint and resolved. The process continues until no violated inequalities can be identified.
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| If the constants are dropped from the above problem, we obtain the following problem to be solved.
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| :<math>y_n^* = \underset{y \in \mathcal{Y}}{\textrm{argmax}} \left(\Delta(y_n,y) + \boldsymbol{w}'\Psi(\boldsymbol{x}_n,y)\right)</math>
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| This problem looks very similar to the inference problem. The only difference is the addition of the term <math>\Delta(y_n,y)</math>. Most often, it is chosen such that it has a natural decomposition in label space. In that case, the influence of <math>\Delta</math> can be encoded into the inference problem and solving for the most violating constraint is equivalent to solving the inference problem.
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| == References ==
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| * Ioannis Tsochantaridis, Thorsten Joachims, Thomas Hofmann and Yasemin Altun (2005), [http://www.jmlr.org/papers/volume6/tsochantaridis05a/tsochantaridis05a.pdf Large Margin Methods for Structured and Interdependent Output Variables], JMLR, Vol. 6, pages 1453-1484.
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| * Thomas Finley and Thorsten Joachims (2008), [http://icml2008.cs.helsinki.fi/papers/279.pdf Training Structural SVMs when Exact Inference is Intractable], ICML 2008.
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| * Sunita Sarawagi and Rahul Gupta (2008), [http://icml2008.cs.helsinki.fi/papers/402.pdf Accurate Max-margin Training for Structured Output Spaces], ICML 2008.
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| * Gökhan BakIr, Ben Taskar, Thomas Hofmann, Bernhard Schölkopf, Alex Smola and SVN Vishwanathan (2007), [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=11332 Predicting Structured Data], MIT Press.
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| * Vojtěch Franc and Bogdan Savchynskyy [http://www.jmlr.org/papers/volume9/franc08a/franc08a.pdf Discriminative Learning of Max-Sum Classifiers], Journal of Machine Learning Research, 9(Jan):67—104, 2008, Microtome Publishing
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| * Kevin Murphy [http://www.cs.ubc.ca/~murphyk/MLbook/pml-print3-ch19.pdf] Machine Learning, Mit Press
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| [[Category:Structured prediction]]
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| [[Category:Support vector machines]]
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by Nas, is very fitting and the film agrees with it. The next step is to visit your Word - Press blog dashboard. These templates are professionally designed and are also Adsense ready. Word - Press also provides protection against spamming, as security is a measure issue. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed.
Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. Some of the Wordpress development services offered by us are:. If you have any issues pertaining to the place and how to use wordpress dropbox backup, you can speak to us at the webpage. This is the reason for the increased risk of Down Syndrome babies in women over age 35. So if you want to create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience.
Just ensure that you hire experienced Word - Press CMS developer who is experienced enough to perform the task of Word - Press customization to get optimum benefits of Word - Press CMS. The only problem with most is that they only offer a monthly plan, you never own the software and you can’t even install the software on your site, you must go to another website to manage your list and edit your autoresponder. Setting Up Your Business Online Using Free Wordpress Websites. You or your web designer can customize it as per your specific needs. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way.
There has been a huge increase in the number of developers releasing free premium Word - Press themes over the years. High Quality Services: These companies help you in creating high quality Word - Press websites. One of the great features of Wordpress is its ability to integrate SEO into your site. Fast Content Update - It's easy to edit or add posts with free Wordpress websites. OSDI, a Wordpress Development Company based on ahmedabad, India.
Internet is not only the source for information, it is also one of the source for passive income. As a website owner, you can easily manage CMS-based website in a pretty easy and convenient style. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. In addition, Word - Press design integration is also possible. The 2010 voting took place from July 7 through August 31, 2010.