Housing projects design in cities should be based on an understanding of the urban local climate; selection of fitto-purpose thermal comfort models and implementation corresponding design guidelines and best practices. In this context, we developed an analysis tool for bioclimatic design recommendations for architects in Madagascar. The aim of this tool is to support the decision-making process of architects and urban planners by proposing environmental design guidelines for Antananarivo and Toamasina, the two largest cities on the island. Firstly, we performed a climate zoning of the island based on altitude, solar irradiation and dry bulb temperature. Secondly, we developed a bioclimatic analysis based on temperature and humidity levels. The results show that ASHRAE adaptive comfort model is the best model for both cities because it can tolerate higher humidity limits of up to 80% or more. The natural comfort potential varies from 22 in Antananarivo to 45% in Toamasina. Results can be used to create informative bioclimatic analysis visualisations to better assess climate and determine thermal comfort models for other cities in hot-humid climates. The capabilities of the tool allow architects and urban planners to better understand the climate and propose practical design guidance.
Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come. In this article, by considering the space of persistence diagrams as a measure space, and by observing that its metrics can be expressed as solutions of optimal partial transport problems, we introduce a generalization of persistence diagrams, namely Radon measures supported on the upper half plane. Such measures naturally appear in topological data analysis when considering continuous representations of persistence diagrams (e.g. persistence surfaces) but also as limits for laws of large numbers on persistence diagrams or as expectations of probability distributions on the persistence diagrams space. We study the topological properties of this new space, which will also hold for the closed subspace of persistence diagrams. New results include a characterization of convergence with respect to transport metrics, the existence of Fréchet means for any distribution of diagrams, and an exhaustive description of continuous linear representations of persistence diagrams. We also showcase the usefulness of this framework to study random persistence diagrams by providing several statistical results made meaningful thanks to this new formalism.
Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data. Despite several appealing properties, incorporating PDs in learning pipelines can be challenging because their natural geometry is not Hilbertian. Indeed, this was recently exemplified in a string of papers which show that the simple task of averaging a few PDs can be computationally prohibitive. We propose in this article a tractable framework to carry out standard tasks on PDs at scale, notably evaluating distances, estimating barycenters and performing clustering. This framework builds upon a reformulation of PD metrics as optimal transport (OT) problems. Doing so, we can exploit recent computational advances: the OT problem on a planar grid, when regularized with entropy, is convex can be solved in linear time using the Sinkhorn algorithm and convolutions. This results in scalable computations that can stream on GPUs. We demonstrate the efficiency of our approach by carrying out clustering with diagrams metrics on several thousands of PDs, a scale never seen before in the literature.32nd Conference on Neural Information Processing Systems (NIPS 2018),
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