/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

8. Subset Selection in Regression (Monographs on Statistics and Applied Probability, no. 40). By A. J. Miller. ISBN 0 412 35380 6. Chapman and Hall, London, 1990. 240 pp. £25.00.

Subset Selection in Regression

In this paper, a fast algorithm for overcomplete sparse decomposition, called SL0, is proposed. The algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations, and its applications include underdetermined sparse component analysis (SCA), atomic decomposition on overcomplete dictionaries, compressed sensing, and decoding real field codes. Contrary to previous methods, which usually solve this problem by minimizing the l 1 norm using linear programming (LP) techniques, our algorithm tries to directly minimize the l 1 norm. It is experimentally shown that the proposed algorithm is about two to three orders of magnitude faster than the state-of-the-art interior-point LP solvers, while providing the same (or better) accuracy.

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A Fast Approach for Overcomplete Sparse Decomposition Based on Smoothed $\ell ^{0}$ Norm

Convex Analysis: (pms-28)

In this paper, we consider sparse decomposition (SD) of two-dimensional (2D) signals on overcomplete dictionaries with separable atoms. Although, this problem can be solved by converting it to the SD of one-dimensional (1D) signals, this approach requires a tremendous amount of memory and computational cost. Moreover, the uniqueness constraint obtained by this approach is too restricted. Then in the paper, we present an algorithm to be used directly for sparse decomposition of 2D signals on dictionaries with separable atoms. Moreover, we will state another uniqueness constraint for this class of decomposition. Our algorithm is obtained by modifying the Smoothed L0 (SL0) algorithm, and hence we call it two-dimensional SL0 (2D-SL0).

/pdf/sparse-decomposition-of-two-dimensional-signals-3dvpe5vgbg.pdf

Sparse decomposition of two dimensional signals

In this paper, a new algorithm for Sparse Component Analysis (SCA) or atomic decomposition on over-complete dictionaries is presented. The algorithm is essentially a method for obtaining sufficiently sparse solutions of underdetermined systems of linear equations. The solution obtained by the proposed algorithm is compared with the minimum l1-norm solution achieved by Linear Programming (LP). It is experimentally shown that the proposed algorithm is about two orders of magnitude faster than the state-of-the-art l1-magic, while providing the same (or better) accuracy.

/pdf/fast-sparse-representation-based-on-smoothed-l-o-norm-5bo996yo2y.pdf

Fast sparse representation based on smoothed l o norm

A dictionary learning algorithm learns a set of atoms from some training signals in such a way that each signal can be approximated as a linear combination of only a few atoms. Most dictionary learning algorithms use a two-stage iterative procedure. The first stage is to sparsely approximate the training signals over the current dictionary. The second stage is the update of the dictionary. In this paper we develop some atom-by-atom dictionary learning algorithms, which update the atoms sequentially. Specifically, we propose an efficient alternative to the well-known K-SVD algorithm, and show by various experiments that the proposed algorithm is much faster than K-SVD while its results are better. Moreover, we propose a novel algorithm that instead of alternating between the two dictionary learning stages", "performs only the second stage. While in K-SVD each atom is updated along with the nonzero entries of its associated row vector in the coefficient matrix (which we name it its profile), in the new algorithm", "each atom is updated along with the whole entries of its profile. As a result, contrary to K-SVD, the support of each profile can be changed while updating the dictionary. To further accelerate the convergence of this algorithm and to have a control on the cardinality of the representations, we then propose its two-stage counterpart by adding the sparse approximation stage. Experimental results on recovery of a known synthetic dictionary and dictionary learning for a class of auto-regressive signals demonstrate the promising performance of the proposed algorithms.

/pdf/learning-overcomplete-dictionaries-based-on-atom-by-atom-1qr6jcufg5.pdf

Learning Overcomplete Dictionaries Based on Atom-by-Atom Updating

In this paper an extended Kalman filter (EKF) has been proposed for the filtering of noisy ECG signals. The method is based on a modified nonlinear dynamic model, previously introduced for the generation of synthetic ECG signals. An automatic parameter selection method has also been suggested, to adapt the model with a vast variety of normal and abnormal ECG signals. The results show that the EKF output is able to track the original ECG signal shape even in the most noisiest epochs of the ECG signal. The proposed method may serve as an efficient filtering procedure for applications such as the noninvasive extraction of fetal cardiac signals from maternal abdominal signals

/pdf/filtering-noisy-ecg-signals-using-the-extended-kalman-filter-4x487glwfr.pdf

Massoud Babaie-Zadeh

Papers

A Fast Approach for Overcomplete Sparse Decomposition Based on Smoothed $\ell ^{0}$ Norm

Sparse decomposition of two dimensional signals

Fast sparse representation based on smoothed l o norm

Learning Overcomplete Dictionaries Based on Atom-by-Atom Updating

Filtering noisy ECG signals using the extended kalman filter based on a modified dynamic ECG model