Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal

We study the kernel instrumental variable algorithm of \citet{singh2019kernel}, a nonparametric two-stage least squares (2SLS) procedure which has demonstrated strong empirical performance. We provide a convergence analysis that covers both the identified and unidentified settings: when the structural function cannot be identified, we show that the kernel NPIV estimator converges to the IV solution with minimum norm. Crucially, our convergence is with respect to the strong $L_2$-norm, rather than a pseudo-norm. Additionally, we characterize the smoothness of the target function without relying on the instrument, instead leveraging a new description of the projected subspace size (this being closely related to the link condition in inverse learning literature). With the subspace size description and under standard kernel learning assumptions, we derive, for the first time, the minimax optimal learning rate for kernel NPIV in the strong $L_2$-norm. Our result demonstrates that the strength of the instrument is essential to achieve efficient learning. We also improve the original kernel NPIV algorithm by adopting a general spectral regularization in stage 1 regression. The modified regularization can overcome the saturation effect of Tikhonov regularization.

$\textit{FacialFilmroll}$: High-resolution multi-shot video editing

We present $\textit{FacialFilmroll}$, a solution for spatially and temporally consistent editing of faces in one or multiple shots. We build upon unwrap mosaic [Rav-Acha et al. 2008] by specializing it to faces. We leverage recent techniques to fit a 3D face model on monocular videos to (i) improve the quality of the mosaic for edition and (ii) permit the automatic transfer of edits from one shot to other shots of the same actor. We explain how $\textit{FacialFilmroll}$ is integrated in post-production facility. Finally, we present video editing results using $\textit{FacialFilmroll}$ on high resolution videos.