The paper introduces the "fused lasso," a penalized regression method that extends the lasso by incorporating both the L1-norm of the coefficients and their successive differences. This approach is particularly useful for problems with features that can be ordered in a meaningful way, such as protein mass spectroscopy and gene expression data. The fused lasso encourages sparsity in both the coefficients and their differences, leading to local constancy in the coefficient profile. The authors propose a computational approach using the two-phase active set algorithm SQOPT and discuss the asymptotic properties of the fused lasso. They also compare the fused lasso with the lasso and soft thresholding methods, and explore its application to real datasets, including prostate cancer and leukemia data. The paper concludes with a discussion on the computational challenges and potential generalizations of the fused lasso.The paper introduces the "fused lasso," a penalized regression method that extends the lasso by incorporating both the L1-norm of the coefficients and their successive differences. This approach is particularly useful for problems with features that can be ordered in a meaningful way, such as protein mass spectroscopy and gene expression data. The fused lasso encourages sparsity in both the coefficients and their differences, leading to local constancy in the coefficient profile. The authors propose a computational approach using the two-phase active set algorithm SQOPT and discuss the asymptotic properties of the fused lasso. They also compare the fused lasso with the lasso and soft thresholding methods, and explore its application to real datasets, including prostate cancer and leukemia data. The paper concludes with a discussion on the computational challenges and potential generalizations of the fused lasso.