In many applications in data clustering, it is desirable to find not just a single partition into clusters but a sequence of partitions describing the data at different scales, or levels of coarseness. A natural problem then is to analyse and compare the (not necessarily hierarchical) sequences of partitions that underpin such multiscale descriptions of data. Here, we introduce a filtration of abstract simplicial complexes, denoted the Multiscale Clustering Filtration (MCF), which encodes arbitrary patterns of cluster assignments across scales, and we prove that the MCF produces stable persistence diagrams. We then show that the zero-dimensional persistent homology of the MCF measures the degree of hierarchy in the sequence of partitions, and that the higher-dimensional persistent homology tracks the emergence and resolution of conflicts between cluster assignments across the sequence of partitions. To broaden the theoretical foundations of the MCF, we also provide an equivalent construction via a nerve complex filtration, and we show that in the hierarchical case, the MCF reduces to a Vietoris-Rips filtration of an ultrametric space. We briefly illustrate how the MCF can serve to characterise multiscale clustering structures in numerical experiments on synthetic data.

Extracting information from r\'esum\'es is typically formulated as a two-stage problem, where the document is first segmented into sections and then each section is processed individually to extract the target entities. Instead, we cast the whole problem as sequence labeling in two levels -- lines and tokens -- and study model architectures for solving both tasks simultaneously. We build high-quality r\'esum\'e parsing corpora in English, French, Chinese, Spanish, German, Portuguese, and Swedish. Based on these corpora, we present experimental results that demonstrate the effectiveness of the proposed models for the information extraction task, outperforming approaches introduced in previous work. We conduct an ablation study of the proposed architectures. We also analyze both model performance and resource efficiency, and describe the trade-offs for model deployment in the context of a production environment.

We introduce a family of complexity measures for the hypotheses of neural nets, based on a multilevel relative entropy. These complexity measures take into account the multilevel structure of neural nets, as opposed to the classical relative entropy (KL-divergence) term derived from PAC-Bayesian bounds [1] or mutual information bounds [2, 3]. We derive these complexity measures by combining the technique of chaining mutual information (CMI) [4], an algorithm-dependent extension of the classical chaining technique paired with the mutual information bound [2], with the multilevel architecture of neural nets. It is observed in this paper that if a neural net is regularized in a multilevel manner as defined in Section 4, then one can readily construct hierarchical coverings with controlled diameters for its hypothesis set, and exploit this to obtain new multi-scale and algorithm-dependent generalization bounds and, in turn, new regularizers and training algorithms. The effect of such multilevel regularizations on the representation ability of neural nets has also been recently studied in [5, 6] for the special case where layers are nearly-identity functions as for ResNets [7].