PAC recursively partitions the data space to obtain rational initialization structure.

<p>Partition-based methods estimate data density by cutting the data space into smaller rectangles recursively. Shown in parts a-b are three clusters of points, the data marginal densities, and several partition scenarios. The data space (box) is partitioned in sequential steps denoted by numbers on the cut lines. Only the first three partition cuts are shown in parts a and b. (a) Bayesian Sequential Partition (BSP) is a Bayesian procedure that maximizes the posterior of the density estimation by dividing the data space via binary partitions; these partitions occur in the middle of the bounded region. On the other hand, Discrepancy Sequential Partition (DSP) performs division other than the mid-point; here, the division is guided by the discrepancy score through a series of tests of uniformity in point distribution, and the procedure stops when discrepancies are smaller than a set threshold. (b) In the (one-step) look-ahead version of BSP partition, the algorithm cuts the data space for all potential cuts plus one step more (steps 2 and 3), and it finds the optimal future scenario (after step 3). In comparison to the (sub-optimal) BSP scenario (one of many scenarios) illustrated in part (a), the scenario in (b) segregates the gold cluster much better, and it is a preferred cut to make in the continuation of partitioning procedure. In theory, BSP can produce sequential partitions for a pre-set number of steps ahead; however, to maintain computational feasibility, we implemented the one-step look-ahead BSP for this work. (c) The partitioning of simulated data space containing five subpopulations; the hyper-rectangles surround high-density areas, approximating the underlying distribution.