Abstract This paper reports on our study of a sparse neural network regression method based on a single hidden layer architecture and sparsity‐inducing penalties. To determine the size of network in a data‐adaptive way, we adopt the lasso and group lasso penalty functions to simultaneously induce sparsity at node and predictor levels. We also devise several techniques to improve performance of the neural network regression estimator. We adopt a B‐spline activation function with a compact support to identify local trends of data. In addition, we develop an algorithm based on node addition process, in which additional nodes come into the network as a complexity parameter decreases. At each value of the complexity parameter, the algorithm initializes the additional nodes along the direction that best captures the unexplained functional relationship between the response and predictors. The optimization step is conducted based on an efficient coordinate descent algorithm. Numerical studies based on simulated and real datasets illustrate that the combination of the aforementioned devices significantly improves the performance of the neural network estimator.

In this study, we develop a penalized additive regression estimation method based on a neural network architecture. An additive neural network model is constructed by using a linear combination of univariate neural networks, or equivalently functional components. We use a B-spline activation function, which is useful to capture local features of data, for nodes that constitute the model. A penalty function is adopted to induce sparsity in functional components and nodes on each component simultaneously. This enables us to obtain a sparse representation, which in turn improves accountability of the model. To implement the proposed estimation method, we devise an efficient iterative algorithm based on a coordinate-wise updating process. An initialization scheme specialized for the B-spline activation function is proposed. The initialization approach enables the proposed method to achieve better performance compared with random initialization scheme. Numerical studies show that the fitted functional components of our estimator adapt to local and sparse structures based on a given dataset.

This paper examines stochastic deconvolution over noncommutative compact Lie groups. This involves Fourier analysis on compact Lie groups as well as convolution products over such groups. An observation process consisting of a known impulse response function convolved with an unknown signal with additive white noise is assumed. Data collected through the observation process then allow us to construct an estimator of the signal. Signal recovery is then assessed through integrated mean squared error for which the main results show that asymptotic minimaxity depends on smoothness properties of the impulse response function. Thus, if the Fourier transform of the impulse response function is bounded polynomially, then the asymptotic minimax signal recovery is polynomial, while if the Fourier transform of the impulse response function is exponentially bounded, then the asymptotic minimax signal recovery is logarithmic. Such investigations have been previously considered in both the engineering and statistics literature with applications in among others, medical imaging, robotics, and polymer science.

MOTIVATION: The problem of class prediction has received a tremendous amount of attention in the literature recently. In the context of DNA microarrays, where the task is to classify and predict the diagnostic category of a sample on the basis of its gene expression profile, a problem of particular importance is the diagnosis of cancer type based on microarray data. One method of classification which has been very successful in cancer diagnosis is the support vector machine (SVM). The latter has been shown (through simulations) to be superior in comparison with other methods, such as classical discriminant analysis, however, SVM suffers from the drawback that the solution is implicit and therefore is difficult to interpret. In order to remedy this difficulty, an analysis of variance decomposition using structured kernels is proposed and is referred to as the structured polychotomous machine. This technique utilizes Newton-Raphson to find estimates of coefficients followed by the Rao and Wald tests, respectively, for addition and deletion of import vectors. RESULTS: The proposed method is applied to microarray data and simulation data. The major breakthrough of our method is efficiency in that only a minimal number of genes that accurately predict the classes are selected. It has been verified that the selected genes serve as legitimate markers for cancer classification from a biological point of view. AVAILABILITY: All source codes used are available on request from the authors.

This paper addresses the issue of obtaining the optimal rotation to match two functions on the sphere by minimizing the squared error norm and the Kullback–Leibler information criteria. In addition, the accuracy in terms of the band-limited approximations in both cases are also discussed. Algorithms for fast and accurate rotational matching play a significant role in many fields ranging from computational biology to spacecraft attitude estimation. In electron microscopy, peaks in the so-called "rotation function" determine correlations in orientation between density maps of macromolecular structures when the correspondence between the coordinates of the structures is not known. In X-ray crystallography, the rotational matching of Patterson functions in Fourier space is an important step in the determination of protein structures. In spacecraft attitude estimation, a star tracker compares observed patterns of stars with rotated versions of a template that is stored in its memory. Many algorithms for computing and sampling the rotation function have been proposed over the years. These methods usually expand the rotation function in a bandlimited Fourier series on the rotation group. In some contexts the highest peak of this function is interpreted as the optimal rotation of one structure into the other, and in other contexts multiple peaks describe symmetries in the functions being compared. Prior works on rotational matching seek to maximize the correlation between two functions on the sphere. We also consider the use of the Kullback–Leibler information criteria. A gradient descent algorithm is proposed for obtaining the optimal rotation, and a measure is defined to compare the convergence of this procedure applied to the maximal correlation and Kullback–Leibler information criteria.

We consider a two-dimensional (2-D) problem of positron-emission tomography (PET) where the random mechanism of the generation of the tomographic data is modeled by Poisson processes. The goal is to estimate the intensity function which corresponds to emission density. Using the wavelet-vaguelette decomposition (WVD), we propose an estimator based on thresholding of empirical vaguelette coefficients which attains the minimax rates of convergence on Besov function classes. Furthermore, we construct an adaptive estimator which attains the optimal rate of convergence up to a logarithmic term.

This paper studies the issue of optimal deconvolution density estimation using wavelets. The approach taken here can be considered as orthogonal series estimation in the more general context of the density estimation. We explore the asymptotic properties of estimators based on thresholding of estimated wavelet coefficients. Minimax rates of convergence under the integrated square loss are studied over Besov classes B/sub /spl sigma/pq/ of functions for both ordinary smooth and supersmooth convolution kernels. The minimax rates of convergence depend on the smoothness of functions to be deconvolved and the decay rate of the characteristic function of convolution kernels. It is shown that no linear deconvolution estimators can achieve the optimal rates of convergence in the Besov spaces with p<2 when the convolution kernel is ordinary smooth and super smooth. If the convolution kernel is ordinary smooth, then linear estimators can be improved by using thresholding wavelet deconvolution estimators which are asymptotically minimax within logarithmic terms. Adaptive minimax properties of thresholding wavelet deconvolution estimators are also discussed.

We conducted positron emission tomography (PET) image reconstruction experiments using the wavelet transform. The Wavelet-Vaguelette decomposition was used as a framework from which expressions for the necessary wavelet coefficients might be derived, and then the wavelet shrinkage was applied to the wavelet coefficients for the reconstruction (WVS). The performances of WVS were evaluated and compared with those of the filtered back-projection (FBP) using software phantoms, physical phantoms, and human PET studies. The results demonstrated that WVS gave stable reconstruction over the range of shrinkage parameters and provided better noise and spatial resolution characteristics than FBP.