|
Multiple Kernel Learning
Multiple kernel learning refers to a set of machine learning methods that use a predefined set of kernels and learn an optimal linear or non-linear combination of kernels as part of the algorithm. Reasons to use multiple kernel learning include a) the ability to select for an optimal kernel and parameters from a larger set of kernels, reducing bias due to kernel selection while allowing for more automated machine learning methods, and b) combining data from different sources (e.g. sound and images from a video) that have different notions of similarity and thus require different kernels. Instead of creating a new kernel, multiple kernel algorithms can be used to combine kernels already established for each individual data source. Multiple kernel learning approaches have been used in many applications, such as event recognition in video, object recognition in images, and biomedical data fusion. Algorithms Multiple kernel learning algorithms have been developed for supervised, semi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel Method
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). These methods involve using linear classifiers to solve nonlinear problems. The general task of pattern analysis is to find and study general types of relations (for example clusters, rankings, principal components, correlations, classifications) in datasets. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into feature vector representations via a user-specified ''feature map'': in contrast, kernel methods require only a user-specified ''kernel'', i.e., a similarity function over all pairs of data points computed using inner products. The feature map in kernel machines is infinite dimensional but only requires a finite dimensional matrix from user-input according to the representer theorem. Kernel machines are slow to compute for datasets larger than a couple of thousand exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reproducing Kernel Hilbert Spaces
In functional analysis, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H of functions from a set X (to \mathbb or \mathbb) is an RKHS if the point-evaluation functional L_x:H\to\mathbb, L_x(f)=f(x), is continuous for every x\in X. Equivalently, H is an RKHS if there exists a function K_x \in H such that, for all f \in H,\langle f, K_x \rangle = f(x).The function K_x is then called the ''reproducing kernel'', and it reproduces the value of f at x via the inner product. An immediate consequence of this property is that convergence in norm implies uniform convergence on any subset of X on which \, K_x\, is bounded. However, the converse does not necessarily hold. Often the set X carries a topology, and \, K_x\, depends continuously on x\in X, in which case: convergence in norm implies uniform convergence on compact subsets of X. It is not entirely straightforward ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tikhonov Regularization
Ridge regression (also known as Tikhonov regularization, named for Andrey Tikhonov) is a method of estimating the coefficients of multiple- regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. It is a method of regularization of ill-posed problems. It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters. In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias (see bias–variance tradeoff). The theory was first introduced by Hoerl and Kennard in 1970 in their ''Technometrics'' papers "Ridge regressions: biased estimation of nonorthogonal problems" and "Ridge regressions: applications in nonorthogonal problems". Ridge regression was developed as a possible solution to the imprecision of least s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support Vector Machine
In machine learning, support vector machines (SVMs, also support vector networks) are supervised max-margin models with associated learning algorithms that analyze data for classification and regression analysis. Developed at AT&T Bell Laboratories, SVMs are one of the most studied models, being based on statistical learning frameworks of VC theory proposed by Vapnik (1982, 1995) and Chervonenkis (1974). In addition to performing linear classification, SVMs can efficiently perform non-linear classification using the ''kernel trick'', representing the data only through a set of pairwise similarity comparisons between the original data points using a kernel function, which transforms them into coordinates in a higher-dimensional feature space. Thus, SVMs use the kernel trick to implicitly map their inputs into high-dimensional feature spaces, where linear classification can be performed. Being max-margin models, SVMs are resilient to noisy data (e.g., misclassified examples). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elastic Net Regularization
In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the ''L''1 and ''L''2 penalties of the lasso and ridge methods. Nevertheless, elastic net regularization is typically more accurate than both methods with regard to reconstruction. Specification The elastic net method overcomes the limitations of the LASSO (least absolute shrinkage and selection operator) method which uses a penalty function based on :\, \beta\, _1 = \textstyle \sum_^p , \beta_j, . Use of this penalty function has several limitations. For example, in the "large ''p'', small ''n''" case (high-dimensional data with few examples), the LASSO selects at most ''n'' variables before it saturates. Also if there is a group of highly correlated variables, then the LASSO tends to select one variable from a group and ignore the others. To overcome these limitations, the elastic net adds a quadratic part (\, \bet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structural Risk Minimization
Structural risk minimization (SRM) is an inductive principle of use in machine learning. Commonly in machine learning, a generalized model must be selected from a finite data set, with the consequent problem of overfitting – the model becoming too strongly tailored to the particularities of the training set and generalizing poorly to new data. The SRM principle addresses this problem by balancing the model's complexity against its success at fitting the training data. This principle was first set out in a 1974 book by Vladimir Vapnik and Alexey Chervonenkis and uses the VC dimension. In practical terms, Structural Risk Minimization is implemented by minimizing E_ + \beta H(W), where E_ is the train error, the function H(W) is called a regularization function, and \beta is a constant. H(W) is chosen such that it takes large values on parameters W that belong to high-capacity subsets of the parameter space. Minimizing H(W) in effect limits the capacity of the accessible subse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multinomial Probit
In statistics and econometrics, the multinomial probit model is a generalization of the probit model used when there are several possible categories that the dependent variable can fall into. As such, it is an alternative to the multinomial logit model as one method of multiclass classification. It is not to be confused with the ''multivariate'' probit model, which is used to model correlated binary outcomes for more than one independent variable. General specification It is assumed that we have a series of observations ''Y''''i'', for ''i'' = 1...''n'', of the outcomes of multi-way choices from a categorical distribution of size ''m'' (there are ''m'' possible choices). Along with each observation ''Y''''i'' is a set of ''k'' observed values ''x''''1,i'', ..., ''x''''k,i'' of explanatory variables (also known as independent variables, predictor variables, features, etc.). Some examples: *The observed outcomes might be "has disease A, has disease B, has disease C, has none ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gibbs Sampling
In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for sampling from a specified multivariate distribution, multivariate probability distribution when direct sampling from the joint distribution is difficult, but sampling from the Conditional probability distribution, conditional distribution is more practical. This sequence can be used to approximate the joint distribution (e.g., to generate a histogram of the distribution); to approximate the marginal distribution of one of the variables, or some subset of the variables (for example, the unknown parameters or latent variables); or to compute an integral (such as the expected value of one of the variables). Typically, some of the variables correspond to observations whose values are known, and hence do not need to be sampled. Gibbs sampling is commonly used as a means of statistical inference, especially Bayesian inference. It is a randomized algorithm (i.e. an algorithm that makes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Descent
Gradient descent is a method for unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a trajectory that maximizes that function; the procedure is then known as ''gradient ascent''. It is particularly useful in machine learning for minimizing the cost or loss function. Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisupervised Learning
Weak supervision (also known as semi-supervised learning) is a paradigm in machine learning, the relevance and notability of which increased with the advent of large language models due to large amount of data required to train them. It is characterized by using a combination of a small amount of human-labeled data (exclusively used in more expensive and time-consuming supervised learning paradigm), followed by a large amount of unlabeled data (used exclusively in unsupervised learning paradigm). In other words, the desired output values are provided only for a subset of the training data. The remaining data is unlabeled or imprecisely labeled. Intuitively, it can be seen as an exam and labeled data as sample problems that the teacher solves for the class as an aid in solving another set of problems. In the transductive setting, these unsolved problems act as exam questions. In the inductive setting, they become practice problems of the sort that will make up the exam. Problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proximal Gradient Methods For Learning
Proximal gradient (forward backward splitting) methods for learning is an area of research in optimization and statistical learning theory which studies algorithms for a general class of Convex function#Definition, convex Regularization (mathematics), regularization problems where the regularization penalty may not be Differentiable function, differentiable. One such example is \ell_1 regularization (also known as Lasso) of the form :\min_ \frac\sum_^n (y_i- \langle w,x_i\rangle)^2+ \lambda \, w\, _1, \quad \text x_i\in \mathbb^d\text y_i\in\mathbb. Proximal gradient methods offer a general framework for solving regularization problems from statistical learning theory with penalties that are tailored to a specific problem application. Such customized penalties can help to induce certain structure in problem solutions, such as ''sparsity'' (in the case of Lasso (statistics), lasso) or ''group structure'' (in the case of Lasso (statistics)#Group LASSO, group lasso). Relevant backgro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
