Recovering positive examples for data that mixes disjoint sparsity with popularity - based positive structure For our final synthetic experiment , we construct a scenario that combines aspects of our two previous experiments . In this experiment , we consider three disjoint groups . For each disjoint group we use a categorical distribution with nonuniform event probabilities for sampling items within baskets , which induces a positive correlation structure within each group . Therefore , the oracle will expect to see a high negative correlation for disjoint pairs , compared to all other non - disjoint pairs within a particular disjoint group . For items with a high co - occurrence probability , we expect the symmetric DPP to recover a near zero negative correlation , and the nonsymmetric DPP to recover a positive correlation . Furthermore , we expect both the nonsymmetric and symmetric models to recover higher marginal probabilities , or K ii values , for more popular items . The determinantal volumes for positive pairs containing popular items will thus tend to be larger than the volumes of negative pairs . Therefore , for baskets containing popular items , we expect that both the nonsymmetric   for high - sparsity data . 14 disjoint groups are used for data generation . and symmetric models will be able to easily discriminate between positive and negative baskets . When constructing positive baskets , popular items are sampled with high probability , proportional to their popularity . We therefore expect that both models will be able to recover some signal about the correlation structure of the data within each disjoint group , resulting in a predictive AUC higher than 0.5 , since the popularity - based positive correlation structure within each group allows the model to recover some structure about correlations among item pairs within each group . However , we expect that the nonsymmetric model will provide better predictive performance than the symmetric model , since its properties enable recovery of disjoint structure ( as discussed previously ) . We see the expected results in Figure 3 , which are further confirmed by the predictive AUC results : 0.7 for the symmetric model , and 0.75 for the nonsymmetric model .