We study the theoretical properties of the fused lasso procedure originally proposed by \cite{tibshirani2005sparsity} in the context of a linear regression model in which the regression coefficient are totally ordered and assumed to be sparse and piecewise constant. Despite its popularity, to the best of our knowledge, estimation error bounds in high-dimensional settings have only been obtained for the simple case in which the design matrix is the identity matrix. We formulate a novel restricted isometry condition on the design matrix that is tailored to the fused lasso estimator and derive estimation bounds for both the constrained version of the fused lasso assuming dense coefficients and for its penalised version. We observe that the estimation error can be dominated by either the lasso or the fused lasso rate, depending on whether the number of non-zero coefficient is larger than the number of piece-wise constant segments. Finally, we devise a post-processing procedure to recover the piecewise-constant pattern of the coefficients. Extensive numerical experiments support our theoretical findings.
翻译:我们研究了由\cite{tibshirani2005sparity} 最初在线性回归模型中提议的引信拉索程序的理论属性,该模型完全订购回归系数,并假定其稀疏和片状不变。尽管它很受欢迎,但据我们所知,仅在设计矩阵为身份矩阵的简单案例中,才获得高维设置的估计误差界限。我们在设计矩阵为特性矩阵的设计矩阵中设计了一个新的限制性参数。我们在设计矩阵上设计了一个新的限制性条件,该设计矩阵是专门为引信拉索测算器定制的,并且为假设密集系数的引信拉索的受限版本及其惩罚版本得出了估计界限。我们观察到,估计错误可以由拉索或引信拉索率来控制,取决于非零系数是否大于片状恒定的段数。最后,我们设计了一个后处理程序,以恢复系数的细调模式。广泛的数字实验支持我们的理论结论。