Sparse kernel density construction using orthogonal forward regression with leave-one-out test score and local regularization

This paper presents an efficient construction algorithm for obtaining sparse kernel density estimates based on a regression approach that directly optimizes model generalization capability. Computational efficiency of the density construction is ensured using an orthogonal forward regression, and the algorithm incrementally minimizes the leave-one-out test score. A local regularization method is incorporated naturally into the density construction process to further enforce sparsity. An additional advantage of the proposed algorithm is that it is fully automatic and the user is not required to specify any criterion to terminate the density construction procedure. This is in contrast to an existing state-of-art kernel density estimation method using the support vector machine (SVM), where the user is required to specify some critical algorithm parameter. Several examples are included to demonstrate the ability of the proposed algorithm to effectively construct a very sparse kernel density estimate with comparable accuracy to that of the full sample optimized Parzen window density estimate. Our experimental results also demonstrate that the proposed algorithm compares favorably with the SVM method, in terms of both test accuracy and sparsity, for constructing kernel density estimates.

An orthogonal forward regression technique for sparse kernel density estimation

Using the classical Parzen window (PW) estimate as the desired response, the kernel density estimation is formulated as a regression problem and the orthogonal forward regression technique is adopted to construct sparse kernel density (SKD) estimates. The proposed algorithm incrementally minimises a leave-one-out test score to select a sparse kernel model, and a local regularisation method is incorporated into the density construction process to further enforce sparsity. The kernel weights of the selected sparse model are finally updated using the multiplicative nonnegative quadratic programming algorithm, which ensures the nonnegative and unity constraints for the kernel weights and has the desired ability to reduce the model size further. Except for the kernel width, the proposed method has no other parameters that need tuning, and the user is not required to specify any additional criterion to terminate the density construction procedure. Several examples demonstrate the ability of this simple regression-based approach to effectively construct a SKID estimate with comparable accuracy to that of the full-sample optimised PW density estimate. (c) 2007 Elsevier B.V. All rights reserved.

Sparse modeling using orthogonal forward regression with PRESS statistic and regularization

The paper introduces an efficient construction algorithm for obtaining sparse linear-in-the-weights regression models based on an approach of directly optimizing model generalization capability. This is achieved by utilizing the delete-1 cross validation concept and the associated leave-one-out test error also known as the predicted residual sums of squares (PRESS) statistic, without resorting to any other validation data set for model evaluation in the model construction process. Computational efficiency is ensured using an orthogonal forward regression, but the algorithm incrementally minimizes the PRESS statistic instead of the usual sum of the squared training errors. A local regularization method can naturally be incorporated into the model selection procedure to further enforce model sparsity. The proposed algorithm is fully automatic, and the user is not required to specify any criterion to terminate the model construction procedure. Comparisons with some of the existing state-of-art modeling methods are given, and several examples are included to demonstrate the ability of the proposed algorithm to effectively construct sparse models that generalize well.

Sparse model identification using orthogonal forward regression with basis pursuit and D-optimality

An efficient model identification algorithm for a large class of linear-in-the-parameters models is introduced that simultaneously optimises the model approximation ability, sparsity and robustness. The derived model parameters in each forward regression step are initially estimated via the orthogonal least squares (OLS), followed by being tuned with a new gradient-descent learning algorithm based on the basis pursuit that minimises the l(1) norm of the parameter estimate vector. The model subset selection cost function includes a D-optimality design criterion that maximises the determinant of the design matrix of the subset to ensure model robustness and to enable the model selection procedure to automatically terminate at a sparse model. The proposed approach is based on the forward OLS algorithm using the modified Gram-Schmidt procedure. Both the parameter tuning procedure, based on basis pursuit, and the model selection criterion, based on the D-optimality that is effective in ensuring model robustness, are integrated with the forward regression. As a consequence the inherent computational efficiency associated with the conventional forward OLS approach is maintained in the proposed algorithm. Examples demonstrate the effectiveness of the new approach.

A forward-constrained regression algorithm for sparse kernel density estimation

Using the classical Parzen window (PW) estimate as the target function, the sparse kernel density estimator is constructed in a forward-constrained regression (FCR) manner. The proposed algorithm selects significant kernels one at a time, while the leave-one-out (LOO) test score is minimized subject to a simple positivity constraint in each forward stage. The model parameter estimation in each forward stage is simply the solution of jackknife parameter estimator for a single parameter, subject to the same positivity constraint check. For each selected kernels, the associated kernel width is updated via the Gauss-Newton method with the model parameter estimate fixed. The proposed approach is simple to implement and the associated computational cost is very low. Numerical examples are employed to demonstrate the efficacy of the proposed approach.

On the near periodicity of eigenvalues of Toeplitz matrices

Let $A$ be an infinite Toeplitz matrix with a real symbol $f$ defined on $[-\pi, \pi]$. It is well known that the sequence of spectra of finite truncations $A_N$ of $A$ converges to the convex hull of the range of $f$. Recently, Levitin and Shargorodsky, on the basis of some numerical experiments, conjectured, for symbols $f$ with two discontinuities located at rational multiples of $\pi$, that the eigenvalues of $A_N$ located in the gap of $f$ asymptotically exhibit periodicity in $N$, and suggested a formula for the period as a function of the position of discontinuities. In this paper, we quantify and prove the analog of this conjecture for the matrix $A^2$ in a particular case when $f$ is a piecewise constant function taking values $-1$ and $1$.

Molecular t-matrices for low-energy electron diffraction (TMOL v1.1)*1

We describe a FORTRAN-90 program that computes scattering t-matrices for a molecule. These can be used in a Low-Energy Electron Diffraction program to solve the molecular structural problem very efficiently. The intramolecular multiple scattering is computed within a Dyson-like approach, using free space Green propagators in a basis of spherical waves. The advantage of this approach is related to exploiting the chemical identity of the molecule, and to the simplicity to translate and rotate these t-matrices without performing a new multiple-scattering calculation for each configuration. FORTRAN-90 routines for rotating the resulting t-matrices using Wigner matrices are also provided.

Sparse kernel density estimation technique based on zero-norm constraint

A sparse kernel density estimator is derived based on the zero-norm constraint, in which the zero-norm of the kernel weights is incorporated to enhance model sparsity. The classical Parzen window estimate is adopted as the desired response for density estimation, and an approximate function of the zero-norm is used for achieving mathemtical tractability and algorithmic efficiency. Under the mild condition of the positive definite design matrix, the kernel weights of the proposed density estimator based on the zero-norm approximation can be obtained using the multiplicative nonnegative quadratic programming algorithm. Using the -optimality based selection algorithm as the preprocessing to select a small significant subset design matrix, the proposed zero-norm based approach offers an effective means for constructing very sparse kernel density estimates with excellent generalisation performance.

Regression based D-optimality experimental design for sparse kernel density estimation

This paper derives an efficient algorithm for constructing sparse kernel density (SKD) estimates. The algorithm first selects a very small subset of significant kernels using an orthogonal forward regression (OFR) procedure based on the D-optimality experimental design criterion. The weights of the resulting sparse kernel model are then calculated using a modified multiplicative nonnegative quadratic programming algorithm. Unlike most of the SKD estimators, the proposed D-optimality regression approach is an unsupervised construction algorithm and it does not require an empirical desired response for the kernel selection task. The strength of the D-optimality OFR is owing to the fact that the algorithm automatically selects a small subset of the most significant kernels related to the largest eigenvalues of the kernel design matrix, which counts for the most energy of the kernel training data, and this also guarantees the most accurate kernel weight estimate. The proposed method is also computationally attractive, in comparison with many existing SKD construction algorithms. Extensive numerical investigation demonstrates the ability of this regression-based approach to efficiently construct a very sparse kernel density estimate with excellent test accuracy, and our results show that the proposed method compares favourably with other existing sparse methods, in terms of test accuracy, model sparsity and complexity, for constructing kernel density estimates.

Using the Cayley-Hamilton theorem with N-partitioned matrices

A definition is given for the characteristic equation of anN-partitioned matrix. It is then proved that this matrix satisfies its own characteristic equation. This can then be regarded as a version of the Cayley-Hamilton theorem, of use withN-dimensional systems.

Using zero-norm constraint for sparse probability density function estimation

A new sparse kernel probability density function (pdf) estimator based on zero-norm constraint is constructed using the classical Parzen window (PW) estimate as the target function. The so-called zero-norm of the parameters is used in order to achieve enhanced model sparsity, and it is suggested to minimize an approximate function of the zero-norm. It is shown that under certain condition, the kernel weights of the proposed pdf estimator based on the zero-norm approximation can be updated using the multiplicative nonnegative quadratic programming algorithm. Numerical examples are employed to demonstrate the efficacy of the proposed approach.