57,924 research outputs found
Fast Kernel Sparse Representation
Abstract—Two efficient algorithms are proposed to seek the sparse representation on high-dimensional Hilbert space. By proving that all the calculations in Orthogonal Match Pursuit (OMP) are essentially inner-product combinations, we modify the OMP algorithm to apply the kernel-trick. The proposed Kernel OMP (KOMP) is much faster than the existing methods, and illustrates higher accuracy in some scenarios. Furthermore, inspired by the success of group-sparsity, we enforce a rigid group-sparsity constraint on KOMP which leads to a non-iterative variation. The constrained cousin of KOMP, dubbed as Single-Step KOMP (S-KOMP), merely takes one step to achieve the sparse coefficients. A remarkable improvement (up to 2, 750 times) in efficiency is reported for S-KOMP, with only a negligible loss of accuracy. I
Simplex basis function based sparse least squares support vector regression
In this paper, a novel sparse least squares support vector regression algorithm, referred to as LSSVR-SBF, is
introduced which uses a new low rank kernel based on simplex basis function, which has a set of nonlinear parameters.
It is shown that the proposed model can be represented as a sparse linear regression model based on simplex basis
functions. We propose a fast algorithm for least squares support vector regression solution at the cost of O(N) by
avoiding direct kernel matrix inversion. An iterative estimation algorithm has been proposed to optimize the nonlinear parameters associated with the simplex basis functions with the aim of minimizing model mean square errors using the gradient descent algorithm. The proposed fast least square solution and the gradient descent algorithm are alternatively applied. Finally it is shown that the model has a dual representation as a piecewise linear model with respect to the
system input. Numerical experiments are carried out to demonstrate the effectiveness of the proposed approaches
Samplets: Construction and scattered data compression
We introduce the concept of samplets by transferring the construction of Tausch-White wavelets to scattered data. This way, we obtain a multiresolution analysis tailored to discrete data which directly enables data compression, feature detection and adaptivity. The cost for constructing the samplet basis and for the fast samplet transform, respectively, is , where is the number of data points. Samplets with vanishing moments can be used to compress kernel matrices, arising, for instance, kernel based learning and scattered data approximation. The result are sparse matrices with only remaining entries. We provide estimates for the compression error and present an algorithm that computes the compressed kernel matrix with computational cost . The accuracy of the approximation is controlled by the number of vanishing moments. Besides the cost efficient storage of kernel matrices, the sparse representation enables the application of sparse direct solvers for the numerical solution of corresponding linear systems. In addition to a comprehensive introduction to samplets and their properties, we present numerical studies to benchmark the approach. Our results demonstrate that samplets mark a considerable step in the direction of making large scattered data sets accessible for multiresolution analysis
Kernelized Sparse Self-Representation for Clustering and Recommendation
Sparse models have demonstrated substantial success in applications for data analysis such as clustering, classification and denoising. However, most of the current work is built upon the assumption that data is distributed in a union of subspaces, whereas limited work has been conducted on nonlinear datasets where data reside in a union of manifolds rather than a union of subspaces. To understand data nonlinearity using sparse models, in this paper, we propose to exploit the self-representation property of nonlinear data in an implicit feature space using kernel methods. We propose a kernelized sparse self-representation model, denoted as KSSR, and a novel Kernelized Fast Iterative Soft-Thresholding Algorithm, denoted as K-FISTA, to recover the underlying nonlinear structure among the data. We evaluate our method for clustering problems on both synthetic and real-world datasets, and demonstrate its superior performance compared to the other state-of-the-art methods. We also apply our method for collaborative filtering in recommender systems, and demonstrate its great potential for novel applications beyond clustering
The Sample Complexity of Dictionary Learning
A large set of signals can sometimes be described sparsely using a
dictionary, that is, every element can be represented as a linear combination
of few elements from the dictionary. Algorithms for various signal processing
applications, including classification, denoising and signal separation, learn
a dictionary from a set of signals to be represented. Can we expect that the
representation found by such a dictionary for a previously unseen example from
the same source will have L_2 error of the same magnitude as those for the
given examples? We assume signals are generated from a fixed distribution, and
study this questions from a statistical learning theory perspective.
We develop generalization bounds on the quality of the learned dictionary for
two types of constraints on the coefficient selection, as measured by the
expected L_2 error in representation when the dictionary is used. For the case
of l_1 regularized coefficient selection we provide a generalization bound of
the order of O(sqrt(np log(m lambda)/m)), where n is the dimension, p is the
number of elements in the dictionary, lambda is a bound on the l_1 norm of the
coefficient vector and m is the number of samples, which complements existing
results. For the case of representing a new signal as a combination of at most
k dictionary elements, we provide a bound of the order O(sqrt(np log(m k)/m))
under an assumption on the level of orthogonality of the dictionary (low Babel
function). We further show that this assumption holds for most dictionaries in
high dimensions in a strong probabilistic sense. Our results further yield fast
rates of order 1/m as opposed to 1/sqrt(m) using localized Rademacher
complexity. We provide similar results in a general setting using kernels with
weak smoothness requirements
Recent Progress in Image Deblurring
This paper comprehensively reviews the recent development of image
deblurring, including non-blind/blind, spatially invariant/variant deblurring
techniques. Indeed, these techniques share the same objective of inferring a
latent sharp image from one or several corresponding blurry images, while the
blind deblurring techniques are also required to derive an accurate blur
kernel. Considering the critical role of image restoration in modern imaging
systems to provide high-quality images under complex environments such as
motion, undesirable lighting conditions, and imperfect system components, image
deblurring has attracted growing attention in recent years. From the viewpoint
of how to handle the ill-posedness which is a crucial issue in deblurring
tasks, existing methods can be grouped into five categories: Bayesian inference
framework, variational methods, sparse representation-based methods,
homography-based modeling, and region-based methods. In spite of achieving a
certain level of development, image deblurring, especially the blind case, is
limited in its success by complex application conditions which make the blur
kernel hard to obtain and be spatially variant. We provide a holistic
understanding and deep insight into image deblurring in this review. An
analysis of the empirical evidence for representative methods, practical
issues, as well as a discussion of promising future directions are also
presented.Comment: 53 pages, 17 figure
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