Positive Semi-definite Kernel Research Articles

Polynomial summaries of positive semidefinite kernels

Polynomials have proven to be useful tools to tailor generic kernels to specific applications. Nevertheless, we had only restricted knowledge for selecting fertile polynomials which consistently produce positive semidefinite kernels. For example, the well-known polynomial kernel can only take advantage of a very narrow range of polynomials, that is, the univariate polynomials with positive coefficients. This restriction not only hinders intensive exploitation of the flexibility of the kernel method, but also causes misuse of indefinite kernels. Our main theorem significantly relaxes the restriction by asserting that a polynomial consistently produces positive semidefinite kernels, if it has a positive semidefinite coefficient matrix. This sufficient condition is quite natural, and hence, it can be a good characterization of the fertile polynomials. In fact, we prove that the converse of the assertion of the theorem also holds true in the case of degree 1. We also prove the effectiveness of our main theorem by showing three corollaries relating to certain applications known in the literature: the first and second corollaries, respectively, give generalizations of the polynomial kernel and the principal-angle (determinant) kernel. The third corollary shows extended and corrected sufficient conditions for the codon-improved kernel and the weighted-degree kernel with shifts to be positive semidefinite.

Ｈａｕｓｓｌｅｒ畳み込みカーネルの一般化と応用 マッピングカーネル

Haussler's convolution kernel provides a successful framework for engineering new positive semidefinite kernels, and has been applied to a wide range of data types and applications. In the framework, each data object represents a finite set of finer grained components. Then, Haussler's convolution kernel takes a pair of data objects as input, and returns the sum of the return values of the predetermined primitive positive semidefinite kernel calculated for all the possible pairs of the components of the input data objects. On the other hand, the mapping kernel that we introduce in this paper is a natural generalization of Haussler's convolution kernel, in that the input to the primitive kernel moves over a predetermined subset rather than the entire cross product. Although we have plural instances of the mapping kernel in the literature, their positive semidefiniteness was investigated in case-by-case manners, and worse yet, was sometimes incorrectly concluded. In fact, there exists a simple and easily checkable necessary and sufficient condition, which is generic in the sense that it enables us to investigate the positive semidefiniteness of an arbitrary instance of the mapping kernel. This paper presents and proves the validity of the condition. In addition, we introduce two important instances of the mapping kernel, which we refer to as the size-of-index-structure-distribution kernel and the edit-cost-distribution kernel. Both of them are naturally derived from well known (dis)similarity measurements in the literature (the maximum agreement tree, the edit distance), and are reasonably expected to improve the performance of the existing measures by evaluating their distributional features rather than their peak (maximum/minimum) features.

Engineering Positive Semidefinite Kernels for Trees A Framework and a Survey

This paper proposes two frameworks to be used in engineering tree kernels. One is to ensure that the resulting tree kernels be positive semidefinite, while the other is for efficient algorithms to compute the kernels based on the dynamic programming. The first framework provides a method to construct tree kernels using primitive kernels for simpler structures (e.g. for labels, strings) as building blocks and a easy-to-check sufficient condition for the resultant tree kernels to be positive semidefinite. The second framework provides a set of templates of algorithms to calculate a wide range of tree kernels in O(|X|3) or O(|X|2)-time, where |X| denotes the number of vertices of trees.

Agler interpolation families of kernels

An abstract Pick interpolation theorem for a family of positive semi-definite kernels on as etX is formulated. The result complements those in (Ag) and (AM02) and will subsequently be applied to Pick interpolation on distinguished varieties (JKM).

10.1162/15324430152748218

An implicit Lagrangian for the dual of a simple reformulation of the standard quadratic program of a linear support vector machine proposed. This leads to the minimization of an unconstrained differentiable convex function in a space of dimensionality equal to the number of classified points. This problem solvable by an extremely simple linearly convergent Lagrangian support vector machine (LSVM) algorithm. LSVM requires the inversion at the outset of a single matrix of the order of the much smaller dimensionality of the original input space plus one. The full algorithm given in this paper in 11 lines of MATLAB code without any special optimization tools such linear or quadratic programming solvers. This LSVM code can be used as is to solve classification problems with millions of points. For example, 2 million points in 10 dimensional input space were classified by a linear surface in 82 minutes on a Pentium III 500 MHz notebook with 384 megabytes of memory (and additional swap space), and in 7 minutes on a 250 MHz UltraSPARC II processor with 2 gigabytes of memory. Other standard classification test problems were also solved. Nonlinear kernel classification can also be solved by LSVM. Although it does not scale up to very large problems, it can handle any positive semidefinite kernel and guaranteed to converge. A short MATLAB code also given for nonlinear kernels and tested on a number of problems.

Position and quantum theory

It is customarily maintained that the concept of unmeasured (i.e., unobserved) position is without meaning in quantum theory. However, the vast literature on the subject, as shown here through the typical example of the double-slit experiment, really does not provide an iron-clad proof that this is indeed so. The question of the possible meaning of unobserved position is thus open. The problem is then resolved by showing quite rigorously that, after all, unobserved position is meaningless in quantum theory. The criterion for unobserved position to be meaningful in any theory (probabilistic or not) is taken to be that the probability density for position must satisfy the Einstein-Chapman-Kolmogorov equation with a positive-semidefinite kernel which is also properly normalized.