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This paper studies a generic sparse regression problem with a customizable sparsity pattern matrix, motivated by, but not limited to, a supervised gene clustering problem in microarray data analysis. The clustered lasso method is proposed with the l1-type penalties imposed on both the coefficients and their pairwise differences. Somewhat surprisingly, it behaves differently than the lasso or the fused lasso – the exact clustering effect expected from the l1 penalization is rarely seen in applications. An asymptotic study is performed to investigate the power and limitations of the l1-penalty in sparse regression. We propose to combine data-augmentation and weights to improve the l1 technique. To address the computational issues in high dimensions, we successfully generalize a popular iterative algorithm both in practice and in theory and propose an ‘annealing’ algorithm applicable to generic sparse regressions (including the fused/clustered lasso). Some effective accelerating techniques are further investigated to boost the convergence. The accelerated annealing (AA) algorithm, involving only matrix multiplications and thresholdings, can handle a large design matrix as well as a large sparsity pattern matrix.

Reversible integer mapping (or integer transform) is a useful way to realize lossless coding, and this technique has been used for multi-component image compression in the new international image compression standard JPEG 2000. For any nonsingular linear transform of finite dimension, its integer transform can be implemented by factorizing the transform matrix into 3 triangular elementary reversible matrices (TERMs) or a series of single-row elementary reversible matrices (SERMs). To speed up and parallelize integer transforms, we study block TERM and SERM factorizations in this paper. First, to guarantee flexible scaling manners, the classical determinant (det) is generalized to a matrix function, DET , which is shown to have many important properties analogous to those of det . Then based on DET , a generic block TERM factorization, BLUS , is presented for any nonsingular block matrix. Our conclusions can cover the early optimal point factorizations and provide an efficient way to implement integer transforms for large matrices.