Tempered subanalytic topology on algebraic varieties

On a smooth algebraic variety over $\mathbb{C}$, we build the tempered subanalytic and Stein tempered subanalytic sites. We construct the sheaf of holomorphic functions tempered at infinity over these sites and study their relations with the sheaf of regular functions, proving in particular that these sheaves are isomorphic on Zariski open subsets. We show that these data allow to define the functors of tempered and Stein tempered analytifications. We study the relations between these two functors and the usual analytification functor. We also obtain algebraization results in the non-proper case and flatness results.Comment: 24 pages. Preliminary version. Comments are welcom

The Codimension-Three conjecture for holonomic DQ-modules

We prove an analogue for holonomic DQ-modules of the codimension-three conjecture for microdifferential modules recently proved by Kashiwara and Vilonen. Our result states that any holonomic DQ-module having a lattice extends uniquely beyond an analytic subset of codimension equal to or larger than three in a Lagrangian subvariety containing the support of the DQ-module.Comment: 37 pages, several minor correction

DG Affinity of DQ-modules

In this paper, we prove the dg affinity of formal deformation algebroid stacks over complex smooth algebraic varieties. For that purpose, we introduce the triangulated category of formal deformation modules which are cohomologically complete and whose associated graded module is quasi-coherent.Comment: 21 pages, references adde

A Riemann-Roch Theorem for dg Algebras

Given a smooth proper dg-algebra $A$, a perfect dg $A$-module $M$, and an endomorphism $f$ of $M$, we define the Hochschild class of the pair $(M,f)$ with values in the Hochschild homology of $A$. Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes.Comment: 26 pages. Many change

Visualization of AE's Training on Credit Card Transactions with Persistent Homology

Auto-encoders are among the most popular neural network architecture for dimension reduction. They are composed of two parts: the encoder which maps the model distribution to a latent manifold and the decoder which maps the latent manifold to a reconstructed distribution. However, auto-encoders are known to provoke chaotically scattered data distribution in the latent manifold resulting in an incomplete reconstructed distribution. Current distance measures fail to detect this problem because they are not able to acknowledge the shape of the data manifolds, i.e. their topological features, and the scale at which the manifolds should be analyzed. We propose Persistent Homology for Wasserstein Auto-Encoders, called PHom-WAE, a new methodology to assess and measure the data distribution of a generative model. PHom-WAE minimizes the Wasserstein distance between the true distribution and the reconstructed distribution and uses persistent homology, the study of the topological features of a space at different spatial resolutions, to compare the nature of the latent manifold and the reconstructed distribution. Our experiments underline the potential of persistent homology for Wasserstein Auto-Encoders in comparison to Variational Auto-Encoders, another type of generative model. The experiments are conducted on a real-world data set particularly challenging for traditional distance measures and auto-encoders. PHom-WAE is the first methodology to propose a topological distance measure, the bottleneck distance, for Wasserstein Auto-Encoders used to compare decoded samples of high quality in the context of credit card transactions.Comment: arXiv admin note: substantial text overlap with arXiv:1905.0989

A Comprehensive Framework for the Evaluation of Individual Treatment Rules From Observational Data

Individualized treatment rules (ITRs) are deterministic decision rules that recommend treatments to individuals based on their characteristics. Though ubiquitous in medicine, ITRs are hardly ever evaluated in randomized controlled trials. To evaluate ITRs from observational data, we introduce a new probabilistic model and distinguish two situations: i) the situation of a newly developed ITR, where data are from a population where no patient implements the ITR, and ii) the situation of a partially implemented ITR, where data are from a population where the ITR is implemented in some unidentified patients. In the former situation, we propose a procedure to explore the impact of an ITR under various implementation schemes. In the latter situation, on top of the fundamental problem of causal inference, we need to handle an additional latent variable denoting implementation. To evaluate ITRs in this situation, we propose an estimation procedure that relies on an expectation-maximization algorithm. In Monte Carlo simulations our estimators appear unbiased with confidence intervals achieving nominal coverage. We illustrate our approach on the MIMIC-III database, focusing on ITRs for dialysis initiation in patients with acute kidney injury