We consider the two-group classification problem and propose a kernel classifier based on the optimal scoring framework. Unlike previous approaches, we provide theoretical guarantees on the expected risk consistency of the method. We also allow for feature selection by imposing structured sparsity using weighted kernels. We propose fully-automated methods for selection of all tuning parameters, and in particular adapt kernel shrinkage ideas for ridge parameter selection. Numerical studies demonstrate the superior classification performance of the proposed approach compared to existing nonparametric classifiers.
To our knowledge, the kernelized version of the optimal scoring problem has not been considered in the literature except for @cite_29 . Unlike @cite_29 , we fix the scores and provide theoretical guarantees for the method. Another major distinction of our method is the feature selection which is achieved by weighting the kernel and adding a sparsity penalty to the weights.
@cite_29
2185932763 2134373213 2104978738 1990470142
One approach to improving the running time of kernel-based methods is to build a small sketch of the kernel matrix and use it in lieu of the full matrix in the machine learning task of interest. Here, we describe a version of this approach that comes with running time guarantees as well as improved guarantees on its statistical performance. By extending the notion of statistical leverage scores to the setting of kernel ridge regression, we are able to identify a sampling distribution that reduces the size of the sketch (i.e., the required number of columns to be sampled) to the effective dimensionality of the problem. This latter quantity is often much smaller than previous bounds that depend on the maximal degrees of freedom. We give an empirical evidence supporting this fact. Our second contribution is to present a fast algorithm to quickly compute coarse approximations to these scores in time linear in the number of samples. More precisely, the running time of the algorithm is O(np2) with p only depending on the trace of the kernel matrix and the regularization parameter. This is obtained via a variant of squared length sampling that we adapt to the kernel setting. Lastly, we discuss how this new notion of the leverage of a data point captures a fine notion of the difficulty of the learning problem. We present a geometric formulation of the Multiple Kernel Learning (MKL) problem. To do so, we reinterpret the problem of learning kernel weights as searching for a kernel that maximizes the minimum (kernel) distance between two convex polytopes. This interpretation combined with novel structural insights from our geometric formulation allows us to reduce the MKL problem to a simple optimization routine that yields provable convergence as well as quality guarantees. As a result our method scales efficiently to much larger data sets than most prior methods can handle. Empirical evaluation on eleven datasets shows that we are significantly faster and even compare favorably with a uniform unweighted combination of kernels. Discriminative learning is challenging when examples are sets of features, and the sets vary in cardinality and lack any sort of meaningful ordering. Kernel-based classification methods can learn complex decision boundaries, but a kernel over unordered set inputs must somehow solve for correspondences epsivnerally a computationally expensive task that becomes impractical for large set sizes. We present a new fast kernel function which maps unordered feature sets to multi-resolution histograms and computes a weighted histogram intersection in this space. This "pyramid match" computation is linear in the number of features, and it implicitly finds correspondences based on the finest resolution histogram cell where a matched pair first appears. Since the kernel does not penalize the presence of extra features, it is robust to clutter. We show the kernel function is positive-definite, making it valid for use in learning algorithms whose optimal solutions are guaranteed only for Mercer kernels. We demonstrate our algorithm on object recognition tasks and show it to be accurate and dramatically faster than current approaches Despite their theoretical appeal and grounding in tractable convex optimization techniques, kernel methods are often not the first choice for large-scale speech applications due to their significant memory requirements and computational expense. In recent years, randomized approximate feature maps have emerged as an elegant mechanism to scale-up kernel methods. Still, in practice, a large number of random features is required to obtain acceptable accuracy in predictive tasks. In this paper, we develop two algorithmic schemes to address this computational bottleneck in the context of kernel ridge regression. The first scheme is a specialized distributed block coordinate descent procedure that avoids the explicit materialization of the feature space data matrix, while the second scheme gains efficiency by combining multiple weak random feature models in an ensemble learning framework. We demonstrate that these schemes enable kernel methods to match the performance of state of the art Deep Neural Networks on TIMIT for speech recognition and classification tasks. In particular, we obtain the best classification error rates reported on TIMIT using kernel methods.