Topics in Trend Filtering with Poisson Loss

Many types of data come as counts — disease cases per day, website visits per hour, or pixel intensities in images. A common goal is to recover the smooth trend underlying these noisy counts. Trend filtering is a nonparametric method that fits flexible, piecewise polynomial curves which adapt automatically to abrupt changes in the signal without prespecifying where they occur. However, existing methods assume Gaussian noise, whereas count data follow Poisson-type models whose variability grows with the signal magnitude, making the effective noise heteroscedastic.

This dissertation develops scalable algorithms for trend filtering under Poisson loss and methodology for solving real-life applications. We propose two proximal algorithms that extend the estimator from simple time series to general graph structures. We apply this framework to epidemic surveillance, producing an R package (rtestim) that estimates time-varying reproduction numbers with principled, cross-validated tuning. We further identify and resolve a numerical instability in the linear system solvers that arise as inner subproblems of these algorithms, by recasting the system as a linear Gaussian state-space model, yielding a solver that is both stable and efficient. Finally, an ongoing work of ours shows that observation-dependent penalty weights can recover minimax optimal rates under the heteroscedastic noise inherent in exponential-family models.