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Francois Petit

Publications and source records attributed to Francois Petit.

18 recordsLinked to original sources
arXiv2026

Evaluating the influence of treatment-effect heterogeneity on discrimination

Analyzing the heterogeneity of treatment effects is crucial in personalized medicine to identify which patients will benefit from specific treatments. The performance of a conditional average treatment effects model to guide treatment decisions can be assessed in different ways, with an important one being the model's ability to effectively discriminate between individuals who benefit from the treatment and those who do not. While many methods and algorithms have been proposed to develop conditional average treatment effects models and individualized treatment rules, little is known about the discriminative ability that can be achieved according to the population's underlying distribution of treatment effects. In this work, we computed the discrimination that can be achieved under oracle CATE for a panel of 20 distributions with varying average treatment effects and levels of heterogeneity. The assessment included the following discrimination metrics: the c-statistic for benefit, the concentration of benefit, and the population average prescriptive effect (PAPE). Results showed that the three metrics employed in this study did not require the same levels of treatment effect heterogeneity to lead to high discrimination results. Notably, achieving high c-statistic for benefit and PAPE values required greater heterogeneity than obtaining high concentration of benefit values. The three metrics considered behave very differently across the distributions. For instance, the concentration of benefit can indicate perfect discrimination in settings with negligible treatment-effects heterogeneity.

stat.AP↗
arXiv2024

Choosing Covariate Balancing Methods for Causal Inference: Practical Insights from a Simulation Study

Background: Inverse probability of treatment weighting (IPTW) is used for confounding adjustment in observational studies. Newer weighting methods include energy balancing (EB), kernel optimal matching (KOM), and tailored-loss covariate balancing propensity scores (TLF), but practical guidance remains limited. We evaluate their performance when implemented according to published recommendations. Methods: We conducted Monte Carlo simulations across 36 scenarios varying sample size, treatment prevalence, and a complexity factor increasing confounding and reducing overlap. Data generation used predominantly categorical covariates with some correlation. Average treatment effect and average treatment effect on the treated were estimated using IPTW, EB, KOM, and TLF combined with weighted least squares and, when supported, a doubly robust (DR) estimators. Inference followed published recommendations for each method when feasible, using standard alternatives otherwise. \textsc{PROBITsim} dataset used for illustration. Results: DR reduced sensitivity to the weighting scheme with an outcome regression adjusted for all confounders, despite functional-form misspecification. EB and KOM were most reliable; EB was tuning-free but scale dependent, whereas KOM required kernel and penalty choices. IPTW was variance sensitive when treatment prevalence was far from 50\%. TLF traded lower variance for higher bias, producing an RMSE plateau and sub-nominal confidence interval coverage. \textsc{PROBITsim} results mirrored these patterns. Conclusions: Rather than identifying a best method, our findings highlight failure modes and tuning choices to monitor. When the outcome regression adjusts for all confounders, DR estimation can be dependable across weighting schemes. Incorporating weight-estimation uncertainty into confidence intervals remains a key challenge for newer approaches.

stat.ME↗
arXiv2024

Computation of $\gamma$-linear projected barcodes for multiparameter persistence

The $\gamma$-linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules along linear forms in the polar of some fixed cone $\gamma$. So far, the computation of the $\gamma$-linear projected barcode has only been studied in the functional setting, in which persistence modules come from the persistent cohomology of $\mathbb{R}^n$-valued functions. Here we develop a method that works in the algebraic setting directly, for any multiparameter persistence module over $\mathbb{R}^n$ that is given via a finite free resolution. Our approach is similar to that of RIVET: first, it pre-processes the resolution to build an arrangement in the dual of $\mathbb{R}^n$ and a barcode template in each face of the arrangement; second, given any query linear form $u$ in the polar of $\gamma$, it locates $u$ within the arrangement to produce the corresponding barcode efficiently. While our theoretical complexity bounds are similar to the ones of RIVET, our arrangement turns out to be simpler thanks to the linear structure of the space of linear forms. Our theoretical analysis combines sheaf-theoretic and module-theoretic techniques, showing that multiparameter persistence modules can be converted into a special type of complexes of sheaves on vector spaces called conic-complexes, whose derived pushforwards by linear forms have predictable barcodes.

math.AT↗
arXiv2022

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.

stat.ME↗
arXiv2022

Projected distances for multi-parameter persistence modules

Relying on sheaf theory, we introduce the notions of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto $\mathbb{R}$. Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). We conduct a systematic study of the stability of projected barcodes and show that the fibered barcode is a particular instance of projected barcodes. We prove that the ISM and the SCD provide lower bounds for the convolution distance. Furthermore, we show that the $\gamma$-linear ISM and the $\gamma$-linear SCD which are projected distances tailored for $\gamma$-sheaves can be computed using TDA software dedicated to one-parameter persistence modules. Moreover, the time and memory complexity required to compute these two metrics are advantageous since our approach does not require computing nor storing an entire $n$-persistence module.

math.AT↗
arXiv2021

Thickening of the diagonal and interleaving distance

Given a topological space $X$, a thickening kernel is a monoidal presheaf on $(\mathbb{R}_{\geq0},+)$ with values in the monoidal category of derived kernels on $X$. A bi-thickening kernel is defined on $(\mathbb{R},+)$. To such a thickening kernel, one naturally associates an interleaving distance on the derived category of sheaves on $X$. We prove that a thickening kernel exists and is unique as soon as it is defined on an interval containing $0$, allowing us to construct (bi-)thickenings in two different situations. First, when $X$ is a ``good'' metric space, starting with small usual thickenings of the diagonal. The associated interleaving distance satisfies the stability property and Lipschitz kernels give rise to Lipschitz maps. Second, by using [GKS12], when $X$ is a manifold and one is given a non-positive Hamiltonian isotopy on the cotangent bundle. In case $X$ is a complete Riemannian manifold having a strictly positive convexity radius, we prove that it is a good metric space and that the two bi-thickening kernels of the diagonal, one associated with the distance, the other with the geodesic flow, coincide.

math.AT↗
arXiv2021

A property of the interleaving distance for sheaves

Let $X$ be a real analytic manifold endowed with a distance satisfying suitable properties and let $\mathbf{k}$ be a field. In [PS20], the authors construct a pseudo-distance on the derived category of sheaves of $\mathbf{k}$-modules on $X$, generalizing a previous construction of [KS18]. Here, we prove that if the distance between two constructible sheaves with compact support (or more generally, constructible sheaves up to infinity) on $X$ is zero, then these two sheaves are isomorphic. This answers in particular a question of [KS18].

math.AG↗
arXiv2020

Ephemeral persistence modules and distance comparison

We provide a definition of ephemeral multi-persistent modules and prove that the quotient of persistent modules by the ephemeral ones is equivalent to the category of $γ$-sheaves. In the case of one-dimensional persistence, our definition agrees with the usual one showing that the observable category and the category of $γ$-sheaves are equivalent. We also establish isometry theorems between the category of persistent modules and $γ$-sheaves both endowed with their interleaving distance. Finally, we compare the interleaving and convolution distances.

math.AT↗
arXiv2020

Integral representation theorems for DQ-modules

We identify the type of $\mathbb{C}[[\hbar]]$-linear structure inherent in the $\infty$-categories which arise in the theory of Deformation Quantization modules. Using this structure, we show that the $\infty$-category of quasicoherent cohomologically complete DQ-modules is a deformation of the $\infty$-category of quasicoherent sheaves. We also obtain integral representation results for DQ-modules similar to the ones of Toën and Ben-Zvi-Nadler-Francis, stating that suitably linear functors between $\infty$-categories of DQ-modules are integral transforms.

math.AG↗
arXiv2019

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.

cs.LG↗
arXiv2019

Holomorphic Frobenius actions for DQ-modules

Given a complex manifold endowed with a $\mathbb{C}^\times$-action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the $\mathbb{C}^\times$-action is free and proper, then the category of F-equivariant DQ-modules is equivalent to the category of modules over the sheaf of invariant sections of the DQ-algebra. As an application, we deduce the codimension three conjecture for formal microdifferential modules from the one for DQ-modules on a symplectic manifold.

math.AG↗
arXiv2017

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.

math.AG↗
arXiv2015

Quantization of spectral curves and DQ-modules

Given an holomorphic Higgs bundle on a compact Riemann surface of genus greater than one, we construct a DQ-module supported by the spectral curve associated to this bundle. Then, we relate quantum curves arising in various situations (quantization of spectral curves of Higgs Bundles, quantization of the $A$-polynomial...) and DQ-modules and show that DQ-modules provide a suitable framework to study the quantization of spectral curves.

math.AG↗
arXiv2014

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.

math.AG↗
arXiv2013

Fourier-Mukai transform in the quantized setting

We prove that a coherent DQ-kernel induces an equivalence between the derived categories of DQ-modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves.

math.AG↗
arXiv2012

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.

math.AG↗
arXiv2011

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.

math.AG↗