  We describe a novel subgradient following apparatus for calculating the
optimum of convex problems with variational penalties. In this setting, we
receive a sequence $y_i,\ldots,y_n$ and seek a smooth sequence
$x_1,\ldots,x_n$. The smooth sequence attains the minimum Bregman divergence to
an input sequence with additive variational penalties in the general form of
$\sum_i g_i(x_{i+1}-x_i)$. We derive, as special cases of our apparatus, known
algorithms for the fused lasso and isotonic regression. Our approach also
facilitates new variational penalties such as non-smooth barrier functions. We
next derive and analyze multivariate problems in which
$\mathbf{x}_i,\mathbf{y}_i\in\mathbb{R}^d$ and variational penalties that
depend on $\|\mathbf{x}_{i+1}-\mathbf{x}_i\|$. The norms we consider are
$\ell_2$ and $\ell_\infty$ which promote group sparsity. Last but not least, we
derive a lattice-based subgradient following for variational penalties
characterized through the output of arbitrary convolutional filters. This
paradigm yields efficient solvers for problems in which sparse high-order
discrete derivatives such as acceleration and jerk are desirable.
