Abstract We show that when KP (Kadomtsev–Petviashvili) τ functions allow special symmetries, the discrete BKP equation can be expressed as a linear combination of AKP and its reflected symmetric forms. Thus equations share same with these symmetries. Such connection is extended to 4 dimensional (i.e. higher order) in corresponding hierarchies. Various explicit forms such functions, including Hi...

For a Banach space $\mathcal{U}$ and Hilbert $\mathcal{H}, B(\mathcal{U}, \mathcal{H})$-valued measures are studied. $B(\mathcal{U}, \mathcal{H})$ is right $B(\mathcal{U})$-module has $B\left(\mathcal{U}, \mathcal{U}^*\right)$-valued gramian. Gramian orthogonally scattered defined some necessary sufficient conditions obtained for measure to have gramian dilation. In particular, if Schauder basi...

The objective of this paper is to make, in a simple and rigorous way, some contributions the notion output controllability. We first examine, for general linear time-invariant systems, state controllability, introduced 60s by Bertram Sarachik. More precisely, we extend Hautus test, well-known case propose controllability Gramian matrix, allowing us build continuous control achieving transfer wi...

We investigate the problem of approximate Bayesian inference for a general class of observation models by means of the expectation propagation (EP) framework for large systems under some statistical assumptions. Our approach tries to overcome the numerical bottleneck of EP caused by the inversion of large matrices. Assuming that the measurement matrices are realizations of specific types of ens...

Category-based statistic language model is an important method to solve the problem of sparse data. But there are two bottlenecks about this model: (1) the problem of word clustering, it is hard to find a suitable clustering method that has good performance and not large amount of computation. (2) class-based method always loses some prediction ability to adapt the text of different domain. The...

We study the problem of evaluating the goodness of a kernel matrix for a classification task. As kernel matrix evaluation is usually used in other expensive procedures like feature and model selections, the goodness measure must be calculated efficiently. Most previous approaches are not efficient, except for Kernel Target Alignment (KTA) that can be calculated in O(n) time complexity. Although...

We consider the problem of clustering incomplete data drawn from a union of subspaces. Classical subspace clustering methods are not applicable to this problem because the data are incomplete, while classical low-rank matrix completion methods may not be applicable because data in multiple subspaces may not be low rank. This paper proposes and evaluates two new approaches for subspace clusterin...

The kinematics of a robot manipulator are described in terms of the mapping connecting its joint space and the 6-dimensional Euclidean group of motions SE(3). The associated Jacobian matrices map into its Lie algebra se(3), the space of twists describing infinitesimal motion of a rigid body. Control methods generally require knowledge of an inverse for the Jacobian. However for an arm with fewe...

This chapter serves as a brief introduction to the supervised learning setting and kernel methods. Moreover, several results from linear algebra, probability theory, and functional analysis are reviewed which will be used throughout the thesis. 2.1 Some notational conventions We begin by introducing some basic notational conventions. The sets N, Z, R, C denote the natural, integer, real, and co...

We propose hashing to facilitate efficient kernels. This generalizes previous work using sampling and we show a principled way to compute the kernel matrix for data streams and sparse feature spaces. Moreover, we give deviation bounds from the exact kernel matrix. This has applications to estimation on strings and graphs.