Due to the challenge posed by multi-source and heterogeneous data collected from diverse environments, causal relationships among features can exhibit variations influenced by different time spans, regions, or strategies. This diversity makes a single causal model inadequate for accurately representing complex causal relationships in all observational data, a crucial consideration in causal learning. To address this challenge, the nonlinear Causal Kernel Clustering method is introduced for heterogeneous subgroup causal learning, highlighting variations in causal relationships across diverse subgroups. The main component for clustering heterogeneous subgroups lies in the construction of the $u$-centered sample mapping function with the property of unbiased estimation, which assesses the differences in potential nonlinear causal relationships in various samples and supported by causal identifiability theory. Experimental results indicate that the method performs well in identifying heterogeneous subgroups and enhancing causal learning, leading to a reduction in prediction error.

Controlling the internal representation space of a neural network is a desirable feature because it allows to generate new data in a supervised manner. In this paper we will show how this can be achieved while building a low-dimensional mapping of the input stream, by deriving a generalized algorithm starting from Self Organizing Maps (SOMs). SOMs are a kind of neural network which can be trained with unsupervised learning to produce a low-dimensional discretized mapping of the input space. They can be used for the generation of new data through backward propagation of interpolations made from the mapping grid. Unfortunately the final topology of the mapping space of a SOM is not known before learning, so interpolating new data in a supervised way is not an easy task. Here we will show a variation from the SOM algorithm consisting in constraining the update of prototypes so that it is also a function of the distance of its prototypes from extrinsically given targets in the mapping space. We will demonstrate how such variants, that we will call Supervised Topological Maps (STMs), allow for a supervised mapping where the position of internal representations in the mapping space is determined by the experimenter. Controlling the internal representation space in STMs reveals to be an easier task than what is currently done using other algorithms such as variational or adversarial autoencoders.

Multi-instance learning (MIL) is useful for tackling labeling ambiguity in learning tasks, by allowing a bag of instances to share one label. Recently, bag mapping methods, which transform a bag to a single instance in a new space via instance selection, have drawn significant attentions. To date, most existing works are developed based on the original space, i.e., utilizing all instances for bag mapping, and instance selection is indirectly tied to the MIL objective. As a result, it is hard to guarantee the distinguish capacity of the selected instances in the new bag mapping space for MIL. In this paper, we propose a direct discriminative mapping approach for multi-instance learning (MILDM), which identifies instances to directly distinguish bags in the new mapping space. Experiments and comparisons on real-world learning tasks demonstrate the algorithm performance.