Density estimation via mixture discrepancy and moments

With the aim of generalizing histogram statistics to higher dimensional cases, density estimation via discrepancy based sequential partition (DSP) has been proposed to learn an adaptive piecewise constant approximation defined on a binary sequential partition of the underlying domain, where the star discrepancy is adopted to measure the uniformity of particle distribution. However, the calculation of the star discrepancy is NP-hard and it does not satisfy the reflection invariance and rotation invariance either. To this end, we use the mixture discrepancy and the comparison of moments as a replacement of the star discrepancy, leading to the density estimation via mixture discrepancy based sequential partition (DSP-mix) and density estimation via moment-based sequential partition (MSP), respectively. Both DSP-mix and MSP are computationally tractable and exhibit the reflection and rotation invariance. Numerical experiments in reconstructing Beta mixtures, Gaussian mixtures and heavy-tailed Cauchy mixtures up to 30 dimension are conducted, demonstrating that MSP can maintain the same accuracy compared with DSP, while gaining an increase in speed by a factor of two to twenty for large sample size, and DSP-mix can achieve satisfactory accuracy and boost the efficiency in low-dimensional tests ($d \le 6$), but might lose accuracy in high-dimensional problems due to a reduction in partition level.