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In speech and image processing, many good algorithms do not perform well in noisy conditions. Recently, robust algorithms have been proposed to improve the performance and robustness of those algorithms. However, many of them are either computationally very expensive or break down quickly. A generalized maximum-likelihood-based robust algorithm called Subset Least Squares (SLS) has been proposed by Kashyap to deal with contaminated data with less complexity and higher breakdown. In this paper, we first illustrate the basic framework of SLS and the use of SLS for robust regression. We then present the use of SLS method in several image and speech applications. Finally, we discuss the possible use of SLS in future applications.

Salt-and-pepper noise consists of outlier pixel values which significantly impair image structure and quality. Multiparent fractal image coding (MFIC) methods substantially exploit image redundancy by utilizing multiple domain blocks to approximate the range block, partially compensating for the information loss caused by noise. Motivated by this, we propose two novel image restoration methods based on MFIC to remove salt-and-pepper noise. The first method integrates Huber M-estimation into MFIC, resulting in an improved anti-salt-and-pepper noise robust fractal coding approach. The second method incorporates MFIC into a total variation (TV) regularization model, including a data fidelity term, an MFIC term and a TV regularization term. An alternative iterative method based on proximity operator is developed to effectively solve the proposed model. Experimental results demonstrate that these two proposed approaches achieve significantly enhanced performance compared to traditional fractal coding methods.

Since fuzzy linear regression was introduced by Tanaka et al., fuzzy regression analysis has been widely studied and applied in various areas. Diamond proposed the fuzzy least squares method to eliminate disadvantages in the Tanaka et al method. In this paper, we propose a modified fuzzy least squares regression analysis. When independent variables are crisp, the dependent variable is a fuzzy number and outliers are present in the data set. In the proposed method, the residuals are ranked as the comparison of fuzzy sets, and the weight matrix is defined by the membership function of the residuals. To illustrate how the proposed method is applied, two examples are discussed and compared in methods from the literature. Results from the numerical examples using the proposed method give good solutions.

It is known that to alleviate the performance deterioration caused by the outliers, the robust support vector (SV) regression is proposed, which is essentially a convex optimization problem associated with a non-convex loss function. The theory analysis for its performance cannot be finished by the usual convex analysis approach. For a robust SV regression algorithm containing two homotopy parameters, a non-convex method is developed with the quasiconvex analysis theory and the error estimate is given. An explicit convergence rate is provided, and the effect degree of outliers on the performance is quantitatively shown.

Financial anomalies have been studied in the U.S. Recent evidence suggests that financial anomalies have diminished in the U.S. and possibly in non-U.S. portfolios. Have the anomalies changed and are they persistent? Have historical and earnings forecasting data been a consistent and highly statistically significant source of excess returns? We test many financial anomalies of the 1980–1990s and report that several models and strategies continue to produce statistically significant excess returns. We test a large set of U.S. and global variables over the past 16 years. We report that many of these fundamental, earnings forecasts, revisions, and breadth and momentum, and cash deployment strategies maintained their statistical significance during the 2003–2018 time period. Moreover, the earnings forecasting model and robust regression estimated that composite model excess returns are greater in non-U.S. and global markets than in the U.S. markets.

In speech and image processing, many good algorithms do not perform well in noisy conditions. Recently, robust algorithms have been proposed to improve the performance and robustness of those algorithms. However, many of them are either computationally very expensive or break down quickly. A generalized maximum-likelihood-based robust algorithm called Subset Least Squares (SLS) has been proposed by Kashyap to deal with contaminated data with less complexity and higher breakdown. In this paper, we first illustrate the basic framework of SLS and the use of SLS for robust regression. We then present the use of SLS method in several image and speech applications. Finally, we discuss the possible use of SLS in future applications.