Structurally robust weak continuity

We pose the following optimization: Given y = {y(n)}_n=0^N-1 ε R^N, find a finite-alphabet x$+$/ = {x$+$/(n)}_n=0^N-1 ε A^N, that minimizes d(x,y) + g(x) subject to: x satisfies a hard structural (syntactic) constraint, e.g., x is piecewise constant of plateau run-length ≥ M, or locally monotonic of lomo -degree α. Here, d(x,y) = ∑_n=0^N-1 d_n (y(n), x(n)) measures fidelity to the data, and is known as the noise term, and g(x) = ∑_n=1^N-1 g_n(x(n),x(n - 1)) measures smoothness-complexity of the solution. This optimization represents the unification and outgrowth of several digital nonlinear filtering schemes, including, in particular, digital counterparts of Weak Continuity (WC) [6, 7, 2], and Minimum Description Length (MDL) [4] on one hand, and nonlinear regression, e.g., VORCA filtering [11], and Digital Locally Monotonic Regression [10], on the other. It is shown that the proposed optimization admits efficient Viterbi-type solution, and, in terms of performance, combines the best of both worlds.