Coreset for Rational Functions

We consider the problem of fitting a rational function $f:\mathbb{R}\to\mathbb{R}$ to a time-series $g:\{1,\cdots,n\}\to\mathbb{R}$. This is by minimizing the sum of distances (loss function) $\ell(f):=\sum_{i=1}^n |f(i)-g(i)|$, possibly with additional constraints and regularization terms that may depend on $f$. Our main motivation is to approximate such a time-series by a recursive sequence model $F_n=\sum_{i=1}^k \theta_i F_{n-i}$, e.g. a Fibonacci sequence, where $\theta\in \mathbb{R}^k$ are the model parameters, and $k\geq1$ is constant. For $\varepsilon\in(0,1)$, an $\varepsilon$-coreset for this problem is a small data structure that approximates $\ell(g)$ up to $1\pm\varepsilon$ multiplicative factor, for every rational function $g$ of constant degree. We prove that every signal has an $\varepsilon$-coreset of size $O(n^{0.001}/\varepsilon^2)$, and provide a construction algorithm that computes it in $O(n^{1.001})$ time. Open source code is provided, as well as extensive experimental results, on both real and synthetic datasets, which compare our method to existing solvers from Scipy.