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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.
In machine learning, kernel methods are a class of algorithms for pattern analysis, whose best known member is the support vector machine (SVM). The general task of pattern analysis is to find and study general types of relations 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 pairs of data points in raw representation.
Multiple kernel learning approaches have been used in many applications, such as event recognition in video, [1] object recognition in images, [2] and biomedical data fusion. [3]
Multiple kernel learning algorithms have been developed for supervised, semi-supervised, as well as unsupervised learning. Most work has been done on the supervised learning case with linear combinations of kernels, however, many algorithms have been developed. The basic idea behind multiple kernel learning algorithms is to add an extra parameter to the minimization problem of the learning algorithm. As an example, consider the case of supervised learning of a linear combination of a set of kernels . We introduce a new kernel , where is a vector of coefficients for each kernel. Because the kernels are additive (due to properties of reproducing kernel Hilbert spaces), this new function is still a kernel. For a set of data with labels , the minimization problem can then be written as
where is an error function and is a regularization term. is typically the square loss function (Tikhonov regularization) or the hinge loss function (for SVM algorithms), and is usually an norm or some combination of the norms (i.e. elastic net regularization). This optimization problem can then be solved by standard optimization methods. Adaptations of existing techniques such as the Sequential Minimal Optimization have also been developed for multiple kernel SVM-based methods. [4]
Tikhonov regularization, named for Andrey Tikhonov, is a method of regularization of ill-posed problems. Also known as ridge regression, 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.
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 L1 and L2 penalties of the lasso and ridge methods.
For supervised learning, there are many other algorithms that use different methods to learn the form of the kernel. The following categorization has been proposed by Gonen and Alpaydın (2011) [5]
Fixed rules approaches such as the linear combination algorithm described above use rules to set the combination of the kernels. These do not require parameterization and use rules like summation and multiplication to combine the kernels. The weighting is learned in the algorithm. Other examples of fixed rules include pairwise kernels, which are of the form
These pairwise approaches have been used in predicting protein-protein interactions. [6]
These algorithms use a combination function that is parameterized. The parameters are generally defined for each individual kernel based on single-kernel performance or some computation from the kernel matrix. Examples of these include the kernel from Tenabe et al. (2008). [7] Letting be the accuracy obtained using only , and letting be a threshold less than the minimum of the single-kernel accuracies, we can define
Other approaches use a definition of kernel similarity, such as
Using this measure, Qui and Lane (2009) [8] used the following heuristic to define
These approaches solve an optimization problem to determine parameters for the kernel combination function. This has been done with similarity measures and structural risk minimization approaches. For similarity measures such as the one defined above, the problem can be formulated as follows: [9]
where is the kernel of the training set.
Structural risk minimization approaches that have been used include linear approaches, such as that used by Lanckriet et al. (2002). [10] We can define the implausibility of a kernel to be the value of the objective function after solving a canonical SVM problem. We can then solve the following minimization problem:
where is a positive constant. Many other variations exist on the same idea, with different methods of refining and solving the problem, e.g. with nonnegative weights for individual kernels and using non-linear combinations of kernels.
Bayesian approaches put priors on the kernel parameters and learn the parameter values from the priors and the base algorithm. For example, the decision function can be written as
can be modeled with a Dirichlet prior and can be modeled with a zero-mean Gaussian and an inverse gamma variance prior. This model is then optimized using a customized multinomial probit approach with a Gibbs sampler.
[11] These methods have been used successfully in applications such as protein fold recognition and protein homology problems [12] [13]
Boosting approaches add new kernels iteratively until some stopping criteria that is a function of performance is reached. An example of this is the MARK model developed by Bennett et al. (2002) [14]
The parameters and are learned by gradient descent on a coordinate basis. In this way, each iteration of the descent algorithm identifies the best kernel column to choose at each particular iteration and adds that to the combined kernel. The model is then rerun to generate the optimal weights and .
Semisupervised learning approaches to multiple kernel learning are similar to other extensions of supervised learning approaches. An inductive procedure has been developed that uses a log-likelihood empirical loss and group LASSO regularization with conditional expectation consensus on unlabeled data for image categorization. We can define the problem as follows. Let be the labeled data, and let be the set of unlabeled data. Then, we can write the decision function as follows.
The problem can be written as
where is the loss function (weighted negative log-likelihood in this case), is the regularization parameter (Group LASSO in this case), and is the conditional expectation consensus (CEC) penalty on unlabeled data. The CEC penalty is defined as follows. Let the marginal kernel density for all the data be
where (the kernel distance between the labeled data and all of the labeled and unlabeled data) and is a non-negative random vector with a 2-norm of 1. The value of is the number of times each kernel is projected. Expectation regularization is then performed on the MKD, resulting in a reference expectation and model expectation . Then, we define
where is the Kullback-Leibler divergence. The combined minimization problem is optimized using a modified block gradient descent algorithm. For more information, see Wang et al. [15]
Unsupervised multiple kernel learning algorithms have also been proposed by Zhuang et al. The problem is defined as follows. Let be a set of unlabeled data. The kernel definition is the linear combined kernel . In this problem, the data needs to be "clustered" into groups based on the kernel distances. Let be a group or cluster of which is a member. We define the loss function as . Furthermore, we minimize the distortion by minimizing . Finally, we add a regularization term to avoid overfitting. Combining these terms, we can write the minimization problem as follows.
where . One formulation of this is defined as follows. Let be a matrix such that means that and are neighbors. Then, . Note that these groups must be learned as well. Zhuang et al. solve this problem by an alternating minimization method for and the groups . For more information, see Zhuang et al. [16]
Available MKL libraries include
Supervised learning is the machine learning task of learning a function that maps an input to an output based on example input-output pairs. It infers a function from labeled training data consisting of a set of training examples. In supervised learning, each example is a pair consisting of an input object and a desired output value. A supervised learning algorithm analyzes the training data and produces an inferred function, which can be used for mapping new examples. An optimal scenario will allow for the algorithm to correctly determine the class labels for unseen instances. This requires the learning algorithm to generalize from the training data to unseen situations in a "reasonable" way.
In machine learning, support-vector machines are supervised learning models with associated learning algorithms that analyze data used for classification and regression analysis. Given a set of training examples, each marked as belonging to one or the other of two categories, an SVM training algorithm builds a model that assigns new examples to one category or the other, making it a non-probabilistic binary linear classifier. An SVM model is a representation of the examples as points in space, mapped so that the examples of the separate categories are divided by a clear gap that is as wide as possible. New examples are then mapped into that same space and predicted to belong to a category based on the side of the gap on which they fall.
Multi-task learning (MTL) is a subfield of machine learning in which multiple learning tasks are solved at the same time, while exploiting commonalities and differences across tasks. This can result in improved learning efficiency and prediction accuracy for the task-specific models, when compared to training the models separately. Early versions of MTL were called "hints".
In machine learning and statistics, feature selection, also known as variable selection, attribute selection or variable subset selection, is the process of selecting a subset of relevant features for use in model construction. Feature selection techniques are used for several reasons:
In mathematics, statistics, and computer science, particularly in machine learning and inverse problems, regularization is the process of adding information in order to solve an ill-posed problem or to prevent overfitting.
Semi-supervised learning is a class of machine learning tasks and techniques that also make use of unlabeled data for training – typically a small amount of labeled data with a large amount of unlabeled data. Semi-supervised learning falls between unsupervised learning and supervised learning. Many machine-learning researchers have found that unlabeled data, when used in conjunction with a small amount of labeled data, can produce considerable improvement in learning accuracy. The acquisition of labeled data for a learning problem often requires a skilled human agent or a physical experiment. The cost associated with the labeling process thus may render a fully labeled training set infeasible, whereas acquisition of unlabeled data is relatively inexpensive. In such situations, semi-supervised learning can be of great practical value. Semi-supervised learning is also of theoretical interest in machine learning and as a model for human learning.
In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses, including function approximation, time series prediction, classification, and system control. They were first formulated in a 1988 paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment.
In computer science, online machine learning is a method of machine learning in which data becomes available in a sequential order and is used to update our best predictor for future data at each step, as opposed to batch learning techniques which generate the best predictor by learning on the entire training data set at once. Online learning is a common technique used in areas of machine learning where it is computationally infeasible to train over the entire dataset, requiring the need of out-of-core algorithms. It is also used in situations where it is necessary for the algorithm to dynamically adapt to new patterns in the data, or when the data itself is generated as a function of time, e.g., stock price prediction. Online learning algorithms may be prone to catastrophic interference, a problem that can be addressed by incremental learning approaches.
Least-squares support-vector machines (LS-SVM) are least-squares versions of support-vector machines (SVM), which are a set of related supervised learning methods that analyze data and recognize patterns, and which are used for classification and regression analysis. In this version one finds the solution by solving a set of linear equations instead of a convex quadratic programming (QP) problem for classical SVMs. Least-squares SVM classifiers were proposed by Suykens and Vandewalle. LS-SVMs are a class of kernel-based learning methods.
Augmented Lagrangian methods are a certain class of algorithms for solving constrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem by a series of unconstrained problems and add a penalty term to the objective; the difference is that the augmented Lagrangian method adds yet another term, designed to mimic a Lagrange multiplier. The augmented Lagrangian is not the same as the method of Lagrange multipliers.
Regularization perspectives on support-vector machines provide a way of interpreting support-vector machines (SVMs) in the context of other machine-learning algorithms. SVM algorithms categorize multidimensional data, with the goal of fitting the training set data well, but also avoiding overfitting, so that the solution generalizes to new data points. Regularization algorithms also aim to fit training set data and avoid overfitting. They do this by choosing a fitting function that has low error on the training set, but also is not too complicated, where complicated functions are functions with high norms in some function space. Specifically, Tikhonov regularization algorithms choose a function that minimizes the sum of training-set error plus the function's norm. The training-set error can be calculated with different loss functions. For example, regularized least squares is a special case of Tikhonov regularization using the squared error loss as the loss function.
In machine learning, kernel methods arise from the assumption of an inner product space or similarity structure on inputs. For some such methods, such as support vector machines (SVMs), the original formulation and its regularization were not Bayesian in nature. It is helpful to understand them from a Bayesian perspective. Because the kernels are not necessarily positive semidefinite, the underlying structure may not be inner product spaces, but instead more general reproducing kernel Hilbert spaces. In Bayesian probability kernel methods are a key component of Gaussian processes, where the kernel function is known as the covariance function. Kernel methods have traditionally been used in supervised learning problems where the input space is usually a space of vectors while the output space is a space of scalars. More recently these methods have been extended to problems that deal with multiple outputs such as in multi-task learning.
In statistical learning theory, a representer theorem is any of several related results stating that a minimizer of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data.
T-distributed Stochastic Neighbor Embedding (t-SNE) is a machine learning algorithm for visualization developed by Laurens van der Maaten and Geoffrey Hinton. It is a nonlinear dimensionality reduction technique well-suited for embedding high-dimensional data for visualization in a low-dimensional space of two or three dimensions. Specifically, it models each high-dimensional object by a two- or three-dimensional point in such a way that similar objects are modeled by nearby points and dissimilar objects are modeled by distant points with high probability.
Spectral regularization is any of a class of regularization techniques used in machine learning to control the impact of noise and prevent overfitting. Spectral regularization can be used in a broad range of applications, from deblurring images to classifying emails into a spam folder and a non-spam folder. For instance, in the email classification example, spectral regularization can be used to reduce the impact of noise and prevent overfitting when a machine learning system is being trained on a labeled set of emails to learn how to tell a spam and a non-spam email apart.
In machine learning, the kernel embedding of distributions comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS). A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis. This learning framework is very general and can be applied to distributions over any space on which a sensible kernel function may be defined. For example, various kernels have been proposed for learning from data which are: vectors in , discrete classes/categories, strings, graphs/networks, images, time series, manifolds, dynamical systems, and other structured objects. The theory behind kernel embeddings of distributions has been primarily developed by Alex Smola, Le Song , Arthur Gretton, and Bernhard Schölkopf. A review of recent works on kernel embedding of distributions can be found in.
Kernel methods are a well-established tool to analyze the relationship between input data and the corresponding output of a function. Kernels encapsulate the properties of functions in a computationally efficient way and allow algorithms to easily swap functions of varying complexity.
In machine learning, Manifold regularization is a technique for using the shape of a dataset to constrain the functions that should be learned on that dataset. In many machine learning problems, the data to be learned do not cover the entire input space. For example, a facial recognition system may not need to classify any possible image, but only the subset of images that contain faces. The technique of manifold learning assumes that the relevant subset of data comes from a manifold, a mathematical structure with useful properties. The technique also assumes that the function to be learned is smooth: data with different labels are not likely to be close together, and so the labeling function should not change quickly in areas where there are likely to be many data points. Because of this assumption, a manifold regularization algorithm can use unlabeled data to inform where the learned function is allowed to change quickly and where it is not, using an extension of the technique of Tikhonov regularization. Manifold regularization algorithms can extend supervised learning algorithms in semi-supervised learning and transductive learning settings, where unlabeled data are available. The technique has been used for applications including medical imaging, geographical imaging, and object recognition.
Regularized least squares (RLS) is a family of methods for solving the least-squares problem while using regularization to further constrain the resulting solution.
Structured sparsity regularization is a class of methods, and an area of research in statistical learning theory, that extend and generalize sparsity regularization learning methods. Both sparsity and structured sparsity regularization methods seek to exploit the assumption that the output variable to be learned can be described by a reduced number of variables in the input space . Sparsity regularization methods focus on selecting the input variables that best describe the output. Structured sparsity regularization methods generalize and extend sparsity regularization methods, by allowing for optimal selection over structures like groups or networks of input variables in .