Some results on Tchebycheffian spline functions

Kimeldorf, George; Wahba, Grace

doi:10.1016/0022-247x(71)90184-3

Cited by 1,107 publications

(748 citation statements)

References 7 publications

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“…Additional constraints are necessary to define the "optimal" hyperplane, typically in the form of maximal margin constraints maximizing the closest distance between the training points and the hyperplane. Under these assumptions (see references for details), the representer theorem 11,12 states that solving for the optimal hyperplane leads to a convex quadratic optimization problem such that the solution vector w is a linear combination of a subset of the training vectors, the support vectors, such that w ) Σ i)1 n R i x i , for some R i ∈ R, i ) 1, ..., n. Thus, f can thus be rewritten as As a side note, it is even possible to write the coefficients R i in the stronger form R i ) i y i with i g 0. If the problem is not exactly linearly separable, then there is a standard convex generalization of this approach using slack variables to allow for some of the classification constraints to be violated.…”

Section: Kernel Methodsmentioning

confidence: 99%

One- to Four-Dimensional Kernels for Virtual Screening and the Prediction of Physical, Chemical, and Biological Properties

Azencott

Ksikes

Swamidass

et al. 2007

J. Chem. Inf. Model.

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Many chemoinformatics applications, including high-throughput virtual screening, benefit from being able to rapidly predict the physical, chemical, and biological properties of small molecules to screen large repositories and identify suitable candidates. When training sets are available, machine learning methods provide an effective alternative to ab initio methods for these predictions. Here, we leverage rich molecular representations including 1D SMILES strings, 2D graphs of bonds, and 3D coordinates to derive efficient machine learning kernels to address regression problems. We further expand the library of available spectral kernels for small molecules developed for classification problems to include 2.5D surface and 3D kernels using Delaunay tetrahedrization and other techniques from computational geometry, 3D pharmacophore kernels, and 3.5D or 4D kernels capable of taking into account multiple molecular configurations, such as conformers. The kernels are comprehensively tested using cross-validation and redundancy-reduction methods on regression problems using several available data sets to predict boiling points, melting points, aqueous solubility, octanol/water partition coefficients, and biological activity with state-of-the art results. When sufficient training data are available, 2D spectral kernels in general tend to yield the best and most robust results, better than state-of-the art. On data sets containing thousands of molecules, the kernels achieve a squared correlation coefficient of 0.91 for aqueous solubility prediction and 0.94 for octanol/water partition coefficient prediction. Averaging over conformations improves the performance of kernels based on the three-dimensional structure of molecules, especially on challenging data sets. Kernel predictors for aqueous solubility (kSOL), LogP (kLOGP), and melting point (kMELT) are available over the Web through: http:// cdb.ics.uci.edu.

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Section: Kernel Methodsmentioning

confidence: 99%

One- to Four-Dimensional Kernels for Virtual Screening and the Prediction of Physical, Chemical, and Biological Properties

Azencott

Ksikes

Swamidass

et al. 2007

J. Chem. Inf. Model.

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“…However when using Boosting procedures on noisy realworld data, it turns out that regularization (e.g. [103,186,143,43]) is mandatory if overfitting is to be avoided (cf. Section 6).…”

Section: Learning From Data and The Pac Propertymentioning

confidence: 99%

“…, where we used the Representer Theorem [103,172] that shows that the maximum margin solution w can be written as a sum of the mapped examples, i.e. w = N n=1 β n Φ(x n ).…”

Section: Support Vector Machines (P = 2)mentioning

confidence: 99%

An Introduction to Boosting and Leveraging

Meir

Rätsch

2003

Advanced Lectures on Machine Learning

280

186

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Abstract. We provide an introduction to theoretical and practical aspects of Boosting and Ensemble learning, providing a useful reference for researchers in the field of Boosting as well as for those seeking to enter this fascinating area of research. We begin with a short background concerning the necessary learning theoretical foundations of weak learners and their linear combinations. We then point out the useful connection between Boosting and the Theory of Optimization, which facilitates the understanding of Boosting and later on enables us to move on to new Boosting algorithms, applicable to a broad spectrum of problems. In order to increase the relevance of the paper to practitioners, we have added remarks, pseudo code, "tricks of the trade", and algorithmic considerations where appropriate. Finally, we illustrate the usefulness of Boosting algorithms by giving an overview of some existing applications. The main ideas are illustrated on the problem of binary classification, although several extensions are discussed.

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“…Kernel logistic regression produces a nonlinear decision boundary, f (x), by forming a linear decision boundary in the space of the non-linearly transformed input vectors. By the representer theorem [19], the optimal f (x) has the form:…”

Section: Robust Kernel Logistic Regressionmentioning

confidence: 99%

Learning kernel logistic regression in the presence of class label noise

Bootkrajang

Kabán

2014

Pattern Recognition

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The classical machinery of supervised learning machines relies on a correct set of training labels. Unfortunately, there is no guarantee that all of the labels are correct. Labelling errors are increasingly noticeable in today's classification tasks, as the scale and difficulty of these tasks increases so much that perfect label assignment becomes nearly impossible. Several algorithms have been proposed to alleviate the problem, of which a robust Kernel Fisher Discriminant is a successful example. However, for classification, discriminative models are of primary interest, and rather curiously, the very few existing label-robust discriminative classifiers are limited to linear problems.In this paper, we build on the widely used and successful kernelising technique to introduce a label-noise robust Kernel Logistic Regression classifier. The main difficulty that we need to bypass is how to determine the model complexity parameters when no trusted validation set is available. We propose to adapt the Multiple Kernel Learning approach for this new purpose, together with a Bayesian regularisation scheme. Empirical results on 13 benchmark data sets and two real-world applications demonstrate the success of our approach.

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Some results on Tchebycheffian spline functions

Cited by 1,107 publications

References 7 publications

One- to Four-Dimensional Kernels for Virtual Screening and the Prediction of Physical, Chemical, and Biological Properties

One- to Four-Dimensional Kernels for Virtual Screening and the Prediction of Physical, Chemical, and Biological Properties

An Introduction to Boosting and Leveraging

Learning kernel logistic regression in the presence of class label noise

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