# Classification of multidimensional data to multidimensional clusters with a varying subcluster structure

I have a large dataset with mixed (numerical, categorical, textual) data that I need to classify. The clusters are well-defined, but multidimensional (i.e. vector-valued) and have a varying structure as follows:

• Cluster Structure 1: $$(x_1, * , * , ... , * )$$

• Cluster Structure 2: $$(x_1, x_2, * , ... , * )$$

• Cluster Structure 3: $$(x_1, * , x_3, ..., * )$$

• ...

• Cluster Structure N: $$(x_1, x_2, x_3, ... , x_m)$$

where asterisks ($$*$$) indicate missing component entries, and each component, when present, (e.g. $$x_2$$) is a cluster to be identified. The Cluster Structures are disjoint in the sense that each member of the data set is only assigned to one Cluster Structure, but Cluster Structures can share components. For example, Cluster Structure 1 only consists of one "subcluster" ($$x_1$$) to be identified, whereas Cluster Structures 2 and 3 have two subclusters to be identified, with one of the clustering subproblems ($$x_1$$) shared and the other ($$x_2$$ vs. $$x_3$$) being different. The subcluster $$x_1$$ (and only that) is present in all Cluster Structures.

For each member of the dataset, I need to classify both the Cluster Structure and the associated tuplet of component clusters $$x_i$$, but I am not sure if it is better to treat these clustering problems sequentially (first structure then subclusters) or independently of each other (separate classifier for each structure). I could, of course, also define a "metacluster" as $$(S, x_1, x_2, x_3, ..., x_m)$$ where $$S$$ is the Cluster Structure id to be classified, but then I'd be treating each cluster structure identically when in fact classification within Cluster Structure 1 (once identified) is potentially much easier than in Cluster Structure N as there is only one subcluster to be identified (namely $$x_1$$).

I would be interested in any existing solutions (algorithms, research papers etc) that would apply, even remotely, to this problem.

• Thanks, but could you please elaborate on the first option? Maybe I was not clear enough in the first description, but the clusters $x_i$ are not directly observable from data set. Hence they are not inputs, but outputs (or classification results) stated as a tuplet of varying length and composition. Or did you mean that I should "explode" the universe of all clusters (behind $x_i$) and consider it as a multilabel problem? If so, then I think I'd again be losing information, as explained in the description. Feb 26 at 9:09
• I'm trying to assign a data sample to one of the cluster structures (classification assignment #1), and I'm trying to find the most likely clusters $x_i$ for that structure and the sample (classification assignment #2). The catch is that the cluster structures are "intertwined", in the sense that they may share subproblems from assignment #2. Feb 26 at 9:15
• For instance, classification problem $x_1$ is shared by all solutions of assignment #1. Feb 26 at 9:18