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Title: Pointwise error bounds for fused lasso Authors:  Sabyasachi Chatterjee - University of Illinois at Urbana Champaign (United States) [presenting]
Abstract: An element-wise error bound is obtained for the Fused Lasso estimator for any general convex loss function $\rho$. We then focus on the special cases when either $\rho$ is the square loss function (for mean regression) or is the quantile loss function (for quantile regression) for which we derive new point-wise error bounds. Even though error bounds for the usual Fused Lasso estimator and its quantile version have been studied before, our bound appears to be new. This is because all previous works bound a global loss function like the sum of squared error or a sum of Huber losses in the case of quantile regression. Clearly, element-wise bounds are stronger than global loss error bounds as it reveals how the loss behaves locally at each point. Our element-wise error bound also has a clean and explicit dependence on the tuning parameter, which informs the user of a good choice of $\rho$. In addition, our bound is non-asymptotic with explicit constants and is able to recover almost all the known results for Fused Lasso (both mean and quantile regression) with additional improvements in some cases.