Compactly Supported RBF Kernels for Sparsifying the Gram Matrix in LS-SVM Regression Models

Hamers, Bart; Suykens, Johan A. K.; Moor, Bart De

doi:10.1007/3-540-46084-5_117

Cited by 13 publications

(10 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By compactifiying the RBF kernel in this way it is guaranteed to remain positive definite [30]. By using (10) and only data which is inside the compact region the results will be almost as accurate as when also including data outside of the compact region.…”

Section: ) Sparse Approximation Methodsmentioning

confidence: 99%

Indoor positioning using multi-frequency RSS with foot-mounted INS

Nilsson

Rantakokko

Skoglund

et al. 2014

2014 International Conference on Indoor Positioning and Indoor Navigation (IPIN)

View full text Add to dashboard Cite

Abstract-This paper presents a system which combines a zero-velocity-update-(ZUPT-)aided inertial navigation system (INS), using a foot-mounted inertial measurement unit (IMU), with opportunistic use of multi-frequency received signal strength (RSS) measurements. The system does not rely on maps or pre-collected data from surveys of the radio-frequency (RF) environment. Instead it builds its own database of collected RSS measurements during the course of the operation. New RSS measurements are continuously compared with the stored values in the database, and when the user returns to a previously visited area this can thus be detected. This enables loop-closures to be detected online and used for error drift correction. The system utilises a distributed particle simultaneous localization and mapping (DP-SLAM) algorithm which provides a flexible 2D navigation platform that can be extended with more sensors. The experimental results presented in this paper indicates that the developed RSS SLAM algorithm can, in many cases, significantly improve the positioning performance of a foot-mounted INS.

show abstract

Section: ) Sparse Approximation Methodsmentioning

confidence: 99%

Indoor positioning using multi-frequency RSS with foot-mounted INS

Nilsson

Rantakokko

Skoglund

et al. 2014

2014 International Conference on Indoor Positioning and Indoor Navigation (IPIN)

View full text Add to dashboard Cite

show abstract

“…For example, the RBF kernel can be compactified as (3) where and are the RBF kernel parameters, defines the compact region over which the kernel has support, and is an odd number set to either or where is the dimensionality of the input. This modified kernel can be shown to be positive definite [19].…”

Section: Online Sparse Matrix Gaussian Processesmentioning

confidence: 99%

“…To ensure the sparsity of the covariance matrix, we use compactified kernel functions [19]. This is because although kernels such as the RBF may produce a covariance matrix with many small entries, the entries in the matrix should be exactly zero for sparse matrix algorithms to be applicable.…”

Section: Online Sparse Matrix Gaussian Processesmentioning

confidence: 99%

“…We are now required to remove the row and column from this matrix and update its Cholesky factor (16) The kernel matrix after the row-removal is (17) and its updated Cholesky factor can be written as (18) where the only component that changes is . This is given by the equation of [11] ( 19) It is clear that (19) involves nothing but a one rank update to the submatrix again. Hence, deleting an arbitrary row can be accomplished by a sparse rank update to the Cholesky factor.…”

Section: A Removing An Arbitrary Row and Column From The Matrixmentioning

confidence: 99%

See 1 more Smart Citation

Online Sparse Gaussian Process Regression and Its Applications

Ranganathan

Yang

2011

IEEE Trans. on Image Process.

102

View full text Add to dashboard Cite

Abstract-We present a new Gaussian process (GP) inference algorithm, called online sparse matrix Gaussian processes (OSMGP), and demonstrate its merits by applying it to the problems of head pose estimation and visual tracking. The OSMGP is based upon the observation that for kernels with local support, the Gram matrix is typically sparse. Maintaining and updating the sparse Cholesky factor of the Gram matrix can be done efficiently using Givens rotations. This leads to an exact, online algorithm whose update time scales linearly with the size of the Gram matrix. Further, we provide a method for constant time operation of the OSMGP using matrix downdates. The downdates maintain the Cholesky factor at a constant size by removing certain rows and columns corresponding to discarded training examples. We demonstrate that, using these matrix downdates, online hyperparameter estimation can be included at cost linear in the number of total training examples. We describe a robust appearance-based head pose estimation system based upon the OSMGP. Numerous experiments and comparisons with existing methods using a large dataset system demonstrate the efficiency and accuracy of our system. Further, to showcase the applicability of OSMGP to a wide variety of problems, we also describe a regression-based visual tracking method. Experiments show that our OSMGP algorithm generalizes well using online learning.Index Terms-Gaussian process, matrix algebra, online algorithm, pose estimation, visual tracking.

show abstract

“…To ensure the sparsity of the covariance matrix, we use "compactified" kernel functions [5]. This is because although kernels such as the RBF may produce a covariance matrix with many small entries, the entries in the matrix should be exactly zero for sparse matrix algorithms to be applicable.…”

Section: Online Sparse Matrix Gaussian Processesmentioning

confidence: 99%

Online Sparse Matrix Gaussian Process Regression and Vision Applications

Ranganathan

Yang

2008

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. We present a new Gaussian Process inference algorithm, called Online Sparse Matrix Gaussian Processes (OSMGP), and demonstrate its merits with a few vision applications. The OSMGP is based on the observation that for kernels with local support, the Gram matrix is typically sparse. Maintaining and updating the sparse Cholesky factor of the Gram matrix can be done efficiently using Givens rotations. This leads to an exact, online algorithm whose update time scales linearly with the size of the Gram matrix. Further, if approximate updates are permissible, the Cholesky factor can be maintained at a constant size using hyperbolic rotations to remove certain rows and columns corresponding to discarded training examples. We demonstrate that, using these matrix downdates, online hyperparameter estimation can be included without affecting the linear runtime complexity of the algorithm. The OSMGP algorithm is applied to headpose estimation and visual tracking problems. Experimental results demonstrate that the proposed method is accurate, efficient and generalizes well using online learning.

show abstract

Compactly Supported RBF Kernels for Sparsifying the Gram Matrix in LS-SVM Regression Models

Cited by 13 publications

References 5 publications

Indoor positioning using multi-frequency RSS with foot-mounted INS

Indoor positioning using multi-frequency RSS with foot-mounted INS

Online Sparse Gaussian Process Regression and Its Applications

Online Sparse Matrix Gaussian Process Regression and Vision Applications

Contact Info

Product

Resources

About