Tensor denoising with trend filtering

We extend the notion of trend filtering to tensors by considering $k\mathrm{th}$-order Vitali variation – a discretized version integral absolute value total derivative. prove adaptive $\ell^0$-rates and not-so-slow $\ell^1$-rates for tensor denoising with filtering. For $k={1,2,3,4}$ we that $d$-dimensional margin can be estimated at $\ell^0$-rate $n^{-1}$, up logarithmic terms, if underlying is product $(k-1)\mathrm{th}$-order polynomials on constant number hyperrectangles. general $k$ $\ell^1$-rate estimation $n^{- \frac{H(d)+2k-1}{2H(d)+2k-1}}$, where $H(d)$ $d\mathrm{th}$ harmonic number. Thanks an ANOVA-type decomposition apply these results lower dimensional margins bounds whole tensor. Our tools are interpolating bound effective sparsity $\ell^0$-rates, mesh grids and, in background, projection arguments Dalalyan, Hebiri, Lederer (2017).