Tame parametrised chain complexes

Persistent homology has proven to be a useful tool to extract information from data sets. Its method can be summarised by a standard workflow: start with data, build the chain complex of a simplicial complex modelling the data, apply homology obtaining the so-called persistent module, and retrieve topological information using invariants. Complete, and thus most discriminative, invariants are given by the indecomposables of the persistent modules. However, such invariants can be retrieved only for the objects of finite representation type whose decomposition is efficiently computed. In addition, homology might be an overkill, and some information may be lost while applying it to the chain complexes. The starting point of our investigation is the idea that a direct study of the chain complex can address these issues. Therefore, we investigate the category of tame parametrised chain complexes, which are chain complexes evolving according to one real parameter. Such a category is quite rich and includes many interesting types of objects, such as parametrised vector spaces, commutative ladders and zigzag modules. We define a model category structure on the category of tame parametrised chain complexes. This setting is quite natural since chain complexes admit a model category structure themselves. Moreover, we can exploit the rich theory of model category to extract invariants. In general, in a model category, there are special objects called cofibrant objects, that can be used to study any other object in the category by approximating it through them. After identifying the cofibrant objects in the category of tame parametrised chain complexes, we study their indecomposables. We find that, despite in general tame parametrised chain complexes are of wild representation type, the indecomposables of cofibrant objects can be fully described. We then approximate every tame parametrised chain complex using two cofibrant objects, called the minimal cover and the minimal representative. Such objects are crucial since they are invariants. In particular, the minimal cover is a homological invariant, and the minimal representative is a homotopical invariant. Thus, these two objects are retrieving all the topological information of the objects they are approximating. In conclusion, we prove that it is possible to analyse data using a new workflow: start with data, build the chain complex of a simplicial complex modelling the data, associate to it either a minimal cover or a minimal representative, and decompose the chosen one to retrieve a summary of the information in the data

Pruning vineyards: updating barcodes by removing simplices

The barcode computation of a filtration can be computationally expensive. Therefore, it is useful to have methods to update a barcode if the associated filtration undergoes small changes, such as changing the entrance order, or adding and removing simplices. There is already a rich literature on how to efficiently update a barcode in the first two cases, but the latter case has not been investigated yet. In this work, we provide an algorithm to update a reduced boundary matrix when simplices are removed. We show that the complexity of this algorithm is lower than recomputing the barcode from scratch, with both theoretical and experimental methods.Comment: 14 pages, 9 figure

Algorithmic decomposition of filtered chain complexes

We present an algorithm to decompose filtered chain complexes into sums of interval spheres. The algorithm's correctness is proved through principled methods from homotopy theory. Its asymptotic runtime complexity is shown to be cubic in the number of generators, e.g. the simplices of a simplicial complex, as it is based on the row reduction of the boundary matrix by Gaussian elimination. Applying homology to a filtered chain complex, one obtains a persistence module. So our method also provides a new algorithm for the barcode decomposition of persistence modules. The key differences with respect to the state-of-the-art persistent homology algorithms are that our algorithm uses row rather than column reductions, it intrinsically adopts both the clear and compress optimisation strategies, and, finally, it can process rows according to any random order

Amplitudes on abelian categories

The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates to data sets. While such distances are well-understood in the one-parameter case, the situation for multiparameter persistence modules is more challenging, since there exists no generalisation of the barcode. Here we introduce a general framework to study stability questions in multiparameter persistence. We introduce amplitudes -- invariants that arise from assigning a non-negative real number to each persistence module, and which are monotone and subadditive in an appropriate sense -- and then study different ways to associate distances to such invariants. Our framework is very comprehensive, as many different invariants that have been introduced in the Topological Data Analysis literature are examples of amplitudes, and furthermore many known distances for multiparameter persistence can be shown to be distances from amplitudes. Finally, we show how our framework can be used to prove new stability results.Comment: 49 pages, major revisions throughout; added section on preliminaries, reorganised/improved main sections in the paper, removed discussion about general finitely encoded module

Case report: complete long-lasting response to multimodal third line treatment with neurosurgical resection, carmustine wafer implantation and dabrafenib plus trametinib in a BRAFV600E mutated high-grade glioma

Dabrafenib plus trametinib is a promising new therapy for patients affected by BRAFV600E-mutant glioma, with high overall response and manageable toxicity. We described a complete and long-lasting response in a case of recurrent anaplastic pleomorphic xanthoastrocytoma CNS WHO-grade 3 BRAFV600E mutated. Due to very poor prognosis, there are a few described cases of high-grade glioma (HGG) patients treated with the combined target therapy as third-line treatment. The emergence of optimized sequencing strategies and targeted agents, including multimodal and systemic therapy with dabrafenib plus trametinib, will continue to broaden personalized therapy in HGG improving patient outcomes