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In the 1-dimensional multiple changepoint detection problem, we prove thatany procedure with a fast enough $\ell_2$ error rate, in terms of itsestimation of the underlying piecewise constant mean vector, automatically hasan (approximate) changepoint screening property---specifically, each true jumpin the underlying mean vector has an estimated jump nearby. We also show, againassuming only knowledge of the $\ell_2$ error rate, that a simplepost-processing step can be used to eliminate spurious estimated changepoints,and thus delivers an (approximate) changepoint recoveryproperty---specifically, in addition to the screening property described above,we are assured that each estimated jump has a true jump nearby. As a specialcase, we focus on the application of these results to the 1-dimensional fusedlasso, i.e., 1-dimensional total variation denoising, and compare theimplications with existing results from the literature. We also studyextensions to related problems, such as changepoint detection over graphs.

Approximate Recovery in Changepoint Problems, from $\ell_2$ Estimation Error Rates