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-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 n {\displaystyle n} kernels K {\displaystyle K} . We introduce a new kernel K ′ = ∑ i = 1 n β i K i {\displaystyle K'=\sum _{i=1}^{n}\beta _{i}K_{i}} , where β {\displaystyle \beta } 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 X {\displaystyle X} with labels Y {\displaystyle Y} , the minimization problem can then be written as
min β , c E ( Y , K ′ c ) + R ( K , c ) {\displaystyle \min _{\beta ,c}\mathrm {E} (Y,K'c)+R(K,c)}
where E {\displaystyle \mathrm {E} } is an error function and R {\displaystyle R} is a regularization term. E {\displaystyle \mathrm {E} } is typically the square loss function (Tikhonov regularization) or the hinge loss function (for SVM algorithms), and R {\displaystyle R} is usually an ℓ n {\displaystyle \ell _{n}} 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.
Supervised learning 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)
Fixed rules approaches 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
k ( ( x 1 i , x 1 j ) , ( x 2 i , x 2 j ) ) = k ( x 1 i , x 2 i ) k ( x 1 j , x 2 j ) + k ( x 1 i , x 2 j ) k ( x 1 j , x 2 i ) {\displaystyle k((x_{1i},x_{1j}),(x_{2i},x_{2j}))=k(x_{1i},x_{2i})k(x_{1j},x_{2j})+k(x_{1i},x_{2j})k(x_{1j},x_{2i})} . These pairwise approaches have been used in predicting protein-protein interactions.
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