arXiv subjects

Tobias Schmidt

Publications and source records attributed to Tobias Schmidt.

At least 37 records · Page 2Linked to original sources

arXiv2023

Robustness of Graph Neural Networks at Scale

Graph Neural Networks (GNNs) are increasingly important given their popularity and the diversity of applications. Yet, existing studies of their vulnerability to adversarial attacks rely on relatively small graphs. We address this gap and study how to attack and defend GNNs at scale. We propose two sparsity-aware first-order optimization attacks that maintain an efficient representation despite optimizing over a number of parameters which is quadratic in the number of nodes. We show that common surrogate losses are not well-suited for global attacks on GNNs. Our alternatives can double the attack strength. Moreover, to improve GNNs' reliability we design a robust aggregation function, Soft Median, resulting in an effective defense at all scales. We evaluate our attacks and defense with standard GNNs on graphs more than 100 times larger compared to previous work. We even scale one order of magnitude further by extending our techniques to a scalable GNN.

cs.LG↗

arXiv2022

Photometric IGM Tomography: Efficiently Mapping Quasar Light Echoes with Deep Narrow Band Imaging

In the standard picture, episodes of luminous quasar activity are directly related to supermassive black hole (SMBH) growth. The ionising radiation emitted over a quasar's lifetime alters the ionisation state of the surrounding intergalactic medium (IGM), enhancing the Ly$α$ forest transmission -- so-called proximity effect -- which can be observed in absorption spectra of background sources. Owing to the finite speed of light, the transverse direction of the proximity effect is sensitive to the quasar's radiative history, resulting in `light echoes' that encode the growth history of the SMBH on Myr-timescales. In this paper, we introduce a new technique to photometrically map this quasar light echoes using Ly$α$ forest tomography by using a carefully selected pair of narrow-band filters. A foreground narrow-band filter is used to measure Ly$α$ forest transmission along background galaxies selected as Ly$α$ emitters by a background narrow-band filter. This novel double narrow-band tomographic technique utilises the higher throughput and wider field of view of imaging over spectroscopy to efficiently reconstruct a two-dimensional map of Ly$α$ forest transmission around a quasar. We present a fully Bayesian framework to measure the luminous quasar lifetime of a SMBH from photometric IGM tomography, and examine the observational requirements. This new technique provides an efficient strategy to map a large area of the sky with a modest observing time and to identify interesting regions to be examined by further deep 3D follow-up spectroscopic Ly$α$ forest tomography.

astro-ph.GA↗

arXiv2021

Generic and Mod p Kazhdan-Lusztig Theory for GL_2

Let $F$ be a non-archimedean local field with residue field $\mathbb{F}_q$ and let $G = GL_{2/F}$. Let $\mathbf{q}$ be an indeterminate and let $H^{(1)}(\mathbf{q})$ be the generic pro-p Iwahori-Hecke algebra of the group $G(F)$. Let $V_{\widehat{G}}$ be the Vinberg monoid of the dual group $\widehat{G}$. We establish a generic version for $H^{(1)}(\mathbf{q})$ of the Kazhdan-Lusztig-Ginzburg spherical representation, the Bernstein map and the Satake isomorphism. We define the flag variety for the monoid $V_{\widehat{G}}$ and establish the characteristic map in its equivariant K-theory. These generic constructions recover the classical ones after the specialization $\mathbf{q} = q \in \mathbb{C}$. At $\mathbf{q} = q = 0 \in\overline{\mathbb{F}}_q$, the spherical map provides a dual parametrization of all the irreducible $H^{(1)}_{\overline{\mathbb{F}}_q}(0)$-modules.

math.NT↗

arXiv2021

Revealing asymmetrical dust distribution in the inner regions of HD 141569

We obtained polarimetric differential imaging of a gas-rich debris disk around HD 141569A with SPHERE in the H-band to compare the scattering properties of the innermost ring at 44 au with former observations in total intensity with the same instrument. In polarimetric imaging, we observed that the intensity of the ring peaks in the south-east, mostly in the forward direction, whereas in total intensity imaging, the ring is detected only at the south. This noticeable characteristic suggests a non-uniform dust density in the ring. We implemented a density function varying azimuthally along the ring and generated synthetic images both in polarimetry and in total intensity, which are then compared to the actual data. We find that the dust density peaks in the south-west at an azimuthal angle of $220^{\circ} \sim 238^{\circ}$ with a rather broad width of $61^{\circ} \sim 127^{\circ}$. Although there are still uncertainties that remain in the determination of the anisotropic scattering factor, the implementation of an azimuthal density variation to fit the data proved to be robust. Upon elaborating on the origin of this dust density distribution, we conclude that it could be the result of a massive collision when we account for the effect of the high gas mass that is present in the system on the dynamics of grains. Using the outcome of this modelization, we further measured the polarized scattering phase function for the observed scattering angle between 33$^{\circ}$ and 147$^{\circ}$ as well as the spectral reflectance of the southern part of the ring between 0.98 $μ$m and 2.1 $μ$m. We tentatively derived the grain properties by comparing these quantities with MCFOST models and assuming Mie scattering. Our preliminary interpretation indicates a mixture of porous sub-micron sized astro-silicate and carbonaceous grains.

astro-ph.EP↗

arXiv2021

Kashiwara's theorem for twisted arithmetic differential operators

We establish a version of Kashiwara's theorem for twisted sheaves of Berthelot's arithmetic differential operators for a closed immersion between smooth p-adic formal schemes. As an application, we construct simple modules for crystalline distribution algebras of reductive groups using the twisted version of the crystalline localisation theorem.

math.AG↗

arXiv2020

A semisimple mod $p$ Langlands correspondence in families for $GL_2(\mathbb{Q}_p)$

This is the sequel to arXiv:2007.01364v1. Let $F$ be any local field with residue characteristic $p>0$, and $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_p}$ be the mod $p$ pro-$p$-Iwahori Hecke algebra of $\mathbf{GL_2}(F)$. In arXiv:2007.01364v1 we have constructed a parametrization of the $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_p}$-modules by certain $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-Satake parameters, together with an antispherical family of $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_p}$-modules. Here we let $F=\mathbb{Q}_p$ (and $p\geq 5$) and construct a morphism from $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-Satake parameters to $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-Langlands parameters. As a result, we get a version in families of Breuil's semisimple mod $p$ Langlands correspondence for $\mathbf{GL_2}(\mathbb{Q}_p)$ and of Paškūnas' parametrization of blocks of the category of mod $p$ locally admissible smooth representations of $\mathbf{GL_2}(\mathbb{Q}_p)$ having a central character. The formulation of these results is possible thanks to the Emerton-Gee moduli space of semisimple $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-representations of the Galois group ${\rm Gal}(\overline{\mathbb{Q}}_p/ \mathbb{Q}_p)$.

math.NT↗

arXiv2020

Intermediate extensions and crystalline distribution algebras

Let G be a connected split reductive group over a complete discrete valuation ring of mixed characteristic. We use the theory of intermediate extensions due to Abe-Caro and arithmetic Beilinson-Bernstein localization to classify irreducible modules over the crystalline distribution algebra of G in terms of overconvergent isocrystals on locally closed subspaces in the (formal) flag variety of G. We treat the case of SL(2) as an example.

math.AG↗

arXiv2019

Mod p Hecke algebras and dual equivariant cohomology I: the case of $GL_2$

Let $F$ be a p-adic local field and $G=GL_2(F)$. Let $\mathcal{H}^{(1)}$ be the pro-p Iwahori-Hecke algebra of $G$ with coefficients in an algebraic closure of $\mathbb{F}_p$. We show that the supersingular irreducible $\mathcal{H}^{(1)}$-modules of dimension 2 can be realized through the equivariant cohomology of the flag variety of the mod p Langlands dual group of $G$.

math.NT↗

arXiv2019

Irreducible mod $p$ Lubin-Tate $(φ,Γ)$-modules

Let $F$ be a finite extension of $\mathbb{Q}_p$. We determine the Lubin-Tate $(φ,Γ)$-modules associated to the absolutely irreducible mod $p$ representations of the absolute Galois group ${\rm Gal}(\bar{F}/F)$.

math.NT↗

arXiv2019

Arithmetic structures for differential operators on formal schemes

Let ${\mathfrak o}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ and ${\mathfrak X}_0$ a smooth formal scheme over the formal spectrum of ${\mathfrak o}$. Given an admissible formal blow-up ${\mathfrak X}$ of ${\mathfrak X}_0$ we introduce sheaves of differential operators ${\mathscr D}^\dagger_{{\mathfrak X},k}$ on ${\mathfrak X}$, for every integer $k \ge k_{\mathfrak X}$, where $k_{\mathfrak X}$ depends on the blow-up morphism ${\mathfrak X}\rightarrow {\mathfrak X}_0$. This generalizes Berthelot's construction of sheaves of arit hmetic differential operators on ${\mathfrak X}_0$. The coherence of these sheaves and several other basic properties are proven. In the second part we study the projective limit sheaf ${\mathscr D}_{{\mathfrak X},\infty} = \varprojlim_k {\mathscr D}^\dagger_{{\mathfrak X},k}$ and so-called coadmissible modules for ${\mathscr D}_{{\mathfrak X},\infty}$. The inductive limit of the sheaves ${\mathscr D}_{{\mathfrak X},\infty}$, over all admissible blow-ups ${\mathfrak X}$ of ${\mathfrak X}_0$, gives rise to a sheaf ${\mathscr D}_{\langle {\mathfrak X}_0 \rangle}$ on the Zariski-Riemann space of ${\mathfrak X}_0$. Analogues of Theorems A and B are shown to hold in each of these settings, i.e., for ${\mathscr D}^\dagger_{{\mathfrak X},k}$, ${\mathscr D}_{{\mathfrak X},\infty}$, and ${\mathscr D}_{\langle {\mathfrak X}_0\rangle}$.

math.AG↗

arXiv2018

Functorial properties of generalised Steinberg representations

Let $G$ be the $F$-points of a connected reductive group over a non-archimedean local field $F$ of residue characteristic $p$ and $R$ be a commutative ring. Let $P=LU$ be a parabolic subgroup of $G$ and $Q$ be a parabolic subgroup of $G$ containing $P$. We study the functor $\mathrm{St}_Q^G$ taking a smooth $R$-representation $σ$ of $L$ which extends to a representation $\mathrm{e}_G(σ)$ of $G$ trivial on $U$ to the smooth $R$-representation $\mathrm{e}_G(σ) \otimes_R \mathrm{St}_Q^G(R)$ of $G$ where $\mathrm{St}_Q^G(R)$ is the generalised Steinberg representation.

math.RT↗

arXiv2018

Estimating Heterogeneous Consumer Preferences for Restaurants and Travel Time Using Mobile Location Data

This paper analyzes consumer choices over lunchtime restaurants using data from a sample of several thousand anonymous mobile phone users in the San Francisco Bay Area. The data is used to identify users' approximate typical morning location, as well as their choices of lunchtime restaurants. We build a model where restaurants have latent characteristics (whose distribution may depend on restaurant observables, such as star ratings, food category, and price range), each user has preferences for these latent characteristics, and these preferences are heterogeneous across users. Similarly, each item has latent characteristics that describe users' willingness to travel to the restaurant, and each user has individual-specific preferences for those latent characteristics. Thus, both users' willingness to travel and their base utility for each restaurant vary across user-restaurant pairs. We use a Bayesian approach to estimation. To make the estimation computationally feasible, we rely on variational inference to approximate the posterior distribution, as well as stochastic gradient descent as a computational approach. Our model performs better than more standard competing models such as multinomial logit and nested logit models, in part due to the personalization of the estimates. We analyze how consumers re-allocate their demand after a restaurant closes to nearby restaurants versus more distant restaurants with similar characteristics, and we compare our predictions to actual outcomes. Finally, we show how the model can be used to analyze counterfactual questions such as what type of restaurant would attract the most consumers in a given location.

econ.EM↗

arXiv2017

Deformation rings and parabolic induction

We study deformations of smooth mod $p$ representations (and their duals) of a $p$-adic reductive group $G$. Under some mild genericity condition, we prove that parabolic induction with respect to a parabolic subgroup $P=LN$ defines an isomorphism between the universal deformation rings of a supersingular representation $\barσ$ of $L$ and of its parabolic induction $\barπ$. As a consequence, we show that every Banach lift of $\barπ$ is induced from a unique Banach lift of $\barσ$.

math.RT↗

arXiv2017

D^\dagger-affinity of formal models of flag varieties

Let G be the group of L-rational points of a connected split reductive group over a finite extension L of Q_p. We show that formal models of the algebraic flag variety X of G are D-affine for certain sheaves of arithmetic differential operators. We then introduce the category of coadmissible G-equivariant arithmetic D-modules on the system of formal models of X and prove that it is anti-equivalent to the category of admissible locally L-analytic G-representations with trivial infinitesimal character. We compute the equivariant arithmetic D-modules of certain classes of representations.

math.RT↗

arXiv2017

$D$-modules arithmétiques, distributions et localisation

Let $p$ be a prime number, $V$ a discrete valuation ring of unequal caracteristics $(0,p)$, $G$ a smooth affine algebraic group over $Spec \,V$. Using partial divided powers techniques of Berthelot, we construct arithmetic distribution algebras, with level $m$, generalizing the classical construction of the distribution algebra. We also construct the weak completion of the classical distribution algebra. We then show that these distribution algebras can be identified with invariant arithmetic differential operators over $G$. We finally apply these constructions in the case of a reductive group and obtain a localization theorem for the sheaf of arithmetic differential operators on the formal flag variety obtained by $p$-adic completion, generalizing a previous result of Ardakov-Wadsley (for the level 0 and with some condition on $p$).

math.RT↗

arXiv2016

The UV-bright Quasar Survey (UVQS): DR1

We present the first data release (DR1) from our UV-bright Quasar Survey (UVQS) for new $z \sim 1$ active galactic nuclei (AGN) across the sky. Using simple GALEX UV and WISE near-IR color selection criteria, we generated a list of 1450 primary candidates with $FUV < 18.5$ mag. We obtained discovery spectra, primarily on 3m-class telescopes, for 1040 of these candidates and confirmed 86% as AGN with redshifts generally at $z>0.5$. Including a small set of observed secondary candidates, we report the discovery of 217 AGN with $FUV < 18$ mag that had no previously reported spectroscopic redshift. These are excellent potential targets for UV spectroscopy before the end of the {\it Hubble Space Telescope} mission. The main data products are publicly released through the Mikulski Archive for Space Telescopes.

astro-ph.GA↗

arXiv2015

Hecke algebras and affine flag varieties in characteristic p

Let G be a split semi-simple p-adic group and let H be its Iwahori-Hecke algebra with coefficients in the algebraic closure k of the finite field with p elements. Let F be the affine flag variety over k associated with G. We show, in the simply connected simple case, that a torus-equivariant K'-theory of F (with coefficients in k) admits an H-action by Demazure operators and that this provides a model for the regular representation of H.

math.RT↗

arXiv2015

One-electron self-interaction and the asymptotics of the Kohn-Sham potential: an impaired relation

One-electron self-interaction and an incorrect asymptotic behavior of the Kohn-Sham exchange-correlation potential are among the most prominent limitations of many present-day density functionals. However, a one-electron self-interaction-free energy does not necessarily lead to the correct long-range potential. This is here shown explicitly for local hybrid functionals. Furthermore, carefully studying the ratio of the von Weizsäcker kinetic energy density to the (positive) Kohn-Sham kinetic energy density, $τ_\mathrm{W}/τ$, reveals that this ratio, which frequently serves as an iso-orbital indicator and is used to eliminate one-electron self-interaction effects in meta-generalized-gradient approximations and local hybrid functionals, can fail to approach its expected value in the vicinity of orbital nodal planes. This perspective article suggests that the nature and consequences of one-electron self-interaction and some of the strategies for its correction need to be reconsidered.

physics.chem-ph↗