Untangling Lariats: Subgradient Following of Variationally Penalized Objectives
CoRR(2024)
Abstract
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,…,y_n and seek a smooth sequence
x_1,…,x_n. The smooth sequence attains the minimum Bregman divergence to
an input sequence with additive variational penalties in the general form of
∑_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
𝐱_i,𝐲_i∈ℝ^d and variational penalties that
depend on 𝐱_i+1-𝐱_i. The norms we consider are
ℓ_2 and ℓ_∞ 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.
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