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SUMMARY:Adaptive Piecewise Polynomial Estimation via Trend Filtering - Rya
 n J. Tibshirani\, Carnegie Mellon University
DTSTART:20140530T150000Z
DTEND:20140530T160000Z
UID:TALK52747@talks.cam.ac.uk
CONTACT:20082
DESCRIPTION:We discuss trend filtering\, a recently proposed tool of Kim e
 t al. (2009) for nonparametric regression. The trend filtering estimate is
  defined as the minimizer of a penalized least squares criterion\, in whic
 h the penalty term sums the absolute kth order discrete\nderivatives over 
 the input points. Perhaps not surprisingly\, trend filtering estimates app
 ear to have the structure of kth degree spline functions\, with adaptively
  chosen knot points (we say "appear" here as\ntrend filtering estimates ar
 e not really functions over continuous domains\, and are only defined over
  the discrete set of inputs). This brings to mind comparisons to other non
 parametric regression tools\nthat also produce adaptive splines\; in parti
 cular\, we compare trend filtering to smoothing splines\, which penalize t
 he sum of squared derivatives across input points\, and to locally adaptiv
 e regression splines (Mammen & van de Geer 1997)\, which penalize the tota
 l\nvariation of the kth derivative.\n\nEmpirically\, trend filtering estim
 ates adapt to the local level of smoothness much better than smoothing spl
 ines\, and further\, they exhibit a remarkable similarity to locally adapt
 ive regression splines. Theoretically\, (suitably tuned) trend filtering e
 stimates converge to the true underlying function at the minimax rate over
  the class of functions whose kth derivative is of bounded variation. The 
 proof of this result follows from an asymptotic pairing of trend\nfilterin
 g and locally adaptive regression splines\, which have already been shown 
 to converge at the minimax rate (Mammen & van de Geer 1997). At the core o
 f this argument is a new result tying together the fitted values of two la
 sso problems that share the same outcome\nvector\, but have different pred
 ictor matrices.
LOCATION:MR12\,  Centre for Mathematical Sciences\, Wilberforce Road\, Cam
 bridge
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