Mixture Approximation Toolbox (mixbox)Mixture approximations are used to approximate \(L^1(\mathcal D)\) functions, \(\alpha: \mathcal D \rightarrow \mathbb R_{\geq 0}\). Typically, the kernel is assumed to be normalized such that \[ \int_{\mathcal D} \alpha(t) \, \mathrm dt = 1, \] which presumes that the integral of \(\alpha\) over the domain, \(\mathcal D\), is nonzero. In some cases, e.g., if \(\alpha\) is a probability density function, it is required to be non-negative: \[ 0 \leq \alpha(t). \] Finally, many convergence results require \(\alpha\) to be continuous. A mixture approximation is given by \[ \alpha(t) \approx \hat \alpha(t) = \sum_{m=0}^M c_m \ell_m(t), \] where \(\ell_m: \mathcal D \rightarrow \mathbb R_{\geq 0}\) is the \(m\)'th basis function, and \(c_m\) is the \(m\)'th weight. The basis functions must also be normalized, i.e., \[ \int_{\mathcal D} \ell_m(t) \, \mathrm dt = 1, \] and the coefficients must sum to one: \[ \sum_{m=0}^M c_m = 1. \] Depending on the domain, there exist the following three basic types of mixture approximations.
OverviewThe toolbox is composed of several GitHub repositories. However, not all functionality is implemented yet. The below list provides an overview of the current status of the toolbox.
Open problems |