text: This paper presents a kernel-based discriminative learning framework on probability measures. Rather than relying on large collections of vectorial training examples, our framework learns using a collection of probability distributions that have been constructed to meaningfully represent training data. By representing these probability distributions as mean embeddings in the reproducing kernel Hilbert space (RKHS), we are able to apply many standard kernel-based learning techniques in straightforward fashion. To accomplish this, we construct a generalization of the support vector machine (SVM) called a support measure machine (SMM). Our analyses of SMMs provides several insights into their relationship to traditional SVMs. Based on such insights, we propose a flexible SVM (Flex-SVM) that places different kernel functions on each training example. Experimental results on both synthetic and real-world data demonstrate the effectiveness of our proposed framework.
  @cite_3: The expected kernel for missing features is introduced and applied to training a support vector machine. The expected kernel is a measure of the mean similarity with respect to the distribution of the missing features. We compare the expected kernel SVM with the robust second-order cone program (SOCP) SVM, which accounts for missing kernel values by estimating the mean and covariance of missing similarities. Further, we extend the SOCP SVM to utilize the expected kernel by deriving the expected kernel variance. Results show that the expected kernel-used with a traditional SVM solver-shows competitive performance on benchmark datasets to the SOCP SVM at a far-reduced computational burden.
  @cite_20: We propose a novel second order cone programming formulation for designing robust classifiers which can handle uncertainty in observations. Similar formulations are also derived for designing regression functions which are robust to uncertainties in the regression setting. The proposed formulations are independent of the underlying distribution, requiring only the existence of second order moments. These formulations are then specialized to the case of missing values in observations for both classification and regression problems. Experiments show that the proposed formulations outperform imputation.
summary: The use of expected kernels in dealing with the uncertainty in the input data has a connection to robust SVMs. For instance, a generalized form of the SVM in @cite_20 incorporates the probabilistic uncertainty into the maximization of the margin. This results in a second-order cone programming (SOCP) that generalizes the standard SVM. In SOCP, one needs to specify the parameter @math that reflects the probability of correctly classifying the @math th training example. The parameter @math is therefore closely related to the parameter @math , which specifies the variance of the distribution centered at the @math th example. @cite_3 showed the equivalence between SVMs using expected kernels and SOCP when @math . When @math , the mean and covariance of missing kernel entries have to be estimated explicitly, making the SOCP more involved for nonlinear kernels. Although achieving comparable performance to the standard SVM with expected kernels, the SOCP requires a more computationally extensive SOCP solver, as opposed to simple quadratic programming (QP).