Search

In this note, we introduce a family of "power sum" kernels and the corresponding Gaussian processes on symmetric groups $\mathrm{S}_n$. Such processes are bi-invariant: the action of $\mathrm{S}_n$ on itself from both sides does not change their finite-dimensional distributions. We show that the values of power sum kernels can be efficiently calculated, and we also propose a method enabling approximate sampling of the corresponding Gaussian processes with polynomial computational complexity. By doing this we provide the tools that are required to use the introduced family of kernels and the respective processes for statistical modeling and machine learning.

Given a rational pointed $n$-dimensional cone $C$, we study the integer Carath\'{e}odory rank $\operatorname{CR}(C)$ and its asymptotic form $\operatorname{CR^{\rm a}}(C)$, where we consider ``most'' integer vectors in the cone. The main result significantly improves the previously known upper bound for $\operatorname{CR^{\rm a}}(C)$. We also study bounds on $\operatorname{CR}(C)$ in terms of $\Delta$, the maximal absolute $n\times n$ minor of the matrix given in an integral polyhedral representation of $C$. If $\Delta\in\lbrace 1,2\rbrace$, we show $\operatorname{CR}(C) = n$, and prove upper bounds for simplicial cones, improving the best known upper bound on $\operatorname{CR}(C)$ for $\Delta\leq n$.

Gaussian processes are arguably the most important class of spatiotemporal models within machine learning. They encode prior information about the modeled function and can be used for exact or approximate Bayesian learning. In many applications, particularly in physical sciences and engineering, but also in areas such as geostatistics and neuroscience, invariance to symmetries is one of the most fundamental forms of prior information one can consider. The invariance of a Gaussian process' covariance to such symmetries gives rise to the most natural generalization of the concept of stationarity to such spaces. In this work, we develop constructive and practical techniques for building stationary Gaussian processes on a very large class of non-Euclidean spaces arising in the context of symmetries. Our techniques make it possible to (i) calculate covariance kernels and (ii) sample from prior and posterior Gaussian processes defined on such spaces, both in a practical manner. This work is split into two parts, each involving different technical considerations: part I studies compact spaces, while part II studies non-compact spaces possessing certain structure. Our contributions make the non-Euclidean Gaussian process models we study compatible with well-understood computational techniques available in standard Gaussian process software packages, thereby making them accessible to practitioners.

X-ray photoemission spectroscopy (XPS) provides direct information on the atomic composition and stoichiometry by measuring core electron binding energies. Moreover, according to the shift of the binding energy, so-called chemical shift, the precise chemical type of bonds can be inferred, which brings additional information on the local structure. In this work, we present a theoretical study of the chemical shift firstly by comparing different theories, from Hartree-Fock (HF) and density-functional theory (DFT) to many-body perturbation theory (MBPT) approaches like the GW approximation and its static version (COHSEX). The accuracy of each theory is assessed by benchmarking against the experiment on the chemical shift of the carbon 1s electron in a set of molecules. More importantly, by decomposing the chemical shift into different contributions according to terms in the total Hamiltonian, the physical origin of the chemical shift is identified as classical electrostatics., Comment: 8 pages, 5 figures 1 table

We demonstrate a three-node quantum network for C-band photon pairs using 2 pairs of 59 km of deployed fiber between Fermi and Argonne National Laboratories. The C-band pairs are directed to nodes using a standard telecommunication switch and synchronized to picosecond-scale timing resolution using a coexisting O- or L-band optical clock distribution system. We measure a reduction of coincidence-to-accidental ratio (CAR) of the C-band pairs from 51 $\pm$ 2 to 5.3 $\pm$ 0.4 due to Raman scattering of the O-band clock pulses. Despite this reduction, the CAR is nevertheless suitable for quantum networks.

This article surveys the Weierstrass representation of surfaces in the three- and four-dimensional spaces, with an emphasis on its relation to the Willmore functional. We also describe an application of this representation to constructing a new type of solutions to the Davey-Stewartson II equation. They have regular initial data, gain one-point singularities at certain moments of time, and extend to smooth solutions for the remaining times., Comment: 18 pages

The worldwide increase of antimicrobial resistance (AMR) is a serious threat to human health. To avert the spread of AMR, fast reliable diagnostics tools that facilitate optimal antibiotic stewardship are an unmet need. In this regard, Raman spectroscopy promises rapid label- and culture-free identification and antimicrobial susceptibility testing (AST) in a single step. However, even though many Raman-based bacteria-identification and AST studies have demonstrated impressive results, some shortcomings must be addressed. To bridge the gap between proof-of-concept studies and clinical application, we have developed machine learning techniques in combination with a novel data-augmentation algorithm, for fast identification of minimally prepared bacteria phenotypes and the distinctions of methicillin-resistant (MR) from methicillin-susceptible (MS) bacteria. For this we have implemented a spectral transformer model for hyper-spectral Raman images of bacteria. We show that our model outperforms the standard convolutional neural network models on a multitude of classification problems, both in terms of accuracy and in terms of training time. We attain more than 96$\%$ classification accuracy on a dataset consisting of 15 different classes and 95.6$\%$ classification accuracy for six MR-MS bacteria species. More importantly, our results are obtained using only fast and easy-to-produce training and test data, Comment: 14 pages, 5 figures

We demonstrate the imaging capability of a frequency modulated continuous wave lidar based on a fiber bundle. The lidar constructs velocity and range images for hard targets at a rate of 60 Hz. The sensing range is up to 30 m with 20 mW of output power. The instrument employs custom electronics with seven parallel heterodyne receivers. An example of image recovery is presented on 6-pixel "pictures" of a spinning disk and a drone hovering in the air. In experiments, we also tested the laser tuning linearity correction with a phase-locked loop. We see the practicality of such a low-resolution system as a boost in scanning rate of conventional lidars or for direct target imaging with a further upgrade of pixel count., Comment: This work has been submitted to the IEEE for possible publication

X-ray photoelectron spectroscopy (XPS) measures electron removal energies, providing direct access to core and valence electron binding energies, hence probing the electronic structure. In this work, we benchmark for the first time the ab initio many-body GW approximation on the complete electron binding energies of noble gas atoms (He-Rn), which spans 100~keV. Our results demonstrate that GW achieves an accuracy within 1.2% in XPS binding energies, by systematically restoring the underestimation from density-functional theory (DFT, error of 14%) or the overestimation from Hartree-Fock (HF, error of 4.7%). Such results also imply the correlations of $d$ electrons are very well described by GW., Comment: 7 pages, 3 figures, 5 tables

RegretNet is a recent breakthrough in the automated design of revenue-maximizing auctions. It combines the flexibility of deep learning with the regret-based approach to relax the Incentive Compatibility (IC) constraint (that participants prefer to bid truthfully) in order to approximate optimal auctions. We propose two independent improvements of RegretNet. The first is a neural architecture denoted as RegretFormer that is based on attention layers. The second is a loss function that requires explicit specification of an acceptable IC violation denoted as regret budget. We investigate both modifications in an extensive experimental study that includes settings with constant and inconstant number of items and participants, as well as novel validation procedures tailored to regret-based approaches. We find that RegretFormer consistently outperforms RegretNet in revenue (i.e. is optimal-er) and that our loss function both simplifies hyperparameter tuning and allows to unambiguously control the revenue-regret trade-off by selecting the regret budget., Comment: NeurIPS 2022

We compute the trace formula for the magnetic Laplacian on a compact hyperbolic surface of constant curvature with constant magnetic field for energies above the Mane critical level of the corresponding magnetic geodesic flow. We discuss the asymptotic behavior of the coefficients of the trace formula when the energy approaches the Mane critical level., Comment: 20 pages, v3. Final version

The Moutard transform is constructed for the solutions of the Davey-Stewartson II equation. It is geometrically interpreted using the spinor (Weierstrass) representation of surfaces in four-dimensional Euclidean space. Using the Moutard transformation and minimal surfaces, examples of solutions are constructed that have smooth, rapidly decreasing initial data and lose their regularity in a finite time., Comment: 18 pages

Parallel black box optimization consists in estimating the optimum of a function using $\lambda$ parallel evaluations of $f$. Averaging the $\mu$ best individuals among the $\lambda$ evaluations is known to provide better estimates of the optimum of a function than just picking up the best. In continuous domains, this averaging is typically just based on (possibly weighted) arithmetic means. Previous theoretical results were based on quadratic objective functions. In this paper, we extend the results to a wide class of functions, containing three times continuously differentiable functions with unique optimum. We prove formal rate of convergences and show they are indeed better than pure random search asymptotically in $\lambda$. We validate our theoretical findings with experiments on some standard black box functions.

Let $f(z) = \sum_{n=1}^\infty a_f(n)q^n$ be a holomorphic cuspidal newform with even integral weight $k\geq 2$, level $N$, trivial nebentypus, and no complex multiplication (CM). For all primes $p$, we may define $\theta_p\in [0,\pi]$ such that $a_f(p) = 2p^{(k-1)/2}\cos \theta_p$. The Sato-Tate conjecture states that the angles $\theta_p$ are equidistributed with respect to the probability measure $\mu_{\textrm{ST}}(I) = \frac{2}{\pi}\int_I \sin^2 \theta \; d\theta$, where $I\subseteq [0,\pi]$. Using recent results on the automorphy of symmetric-power $L$-functions due to Newton and Thorne, we explicitly bound the error term in the Sato-Tate conjecture when $f$ corresponds to an elliptic curve over $\mathbb{Q}$ of arbitrary conductor or when $f$ has squarefree level. In these cases, if $\pi_{f,I}(x) := \#\{ p \leq x : p \nmid N, \theta_p\in I\}$, and $\pi(x) := \# \{ p \leq x \}$, we prove the following bound: $$\left| \frac{\pi_{f,I}(x)}{\pi(x)} - \mu_{\textrm{ST}}(I)\right| \leq 58.1\frac{\log((k-1)N \log{x})}{\sqrt{\log{x}}} \qquad \text{for} \quad x \geq 3.$$ As an application, we give an explicit bound for the number of primes up to $x$ that violate the Atkin-Serre conjecture for $f$., Comment: 31 pages; minor improvements in exposition; minor typos fixed

In biomedical optics, it is often of interest to statistically model the amplitude of the speckle using some distributional models with their parameters acting as biomarkers. In this paper, a paradigm shift is being advocated in which a distributional model-free approach is used. Specifically, a range of distances, evaluated in different domains, between an empirical nonparametric distribution of the normalized speckle amplitude sample and the benchmark Rayleigh distribution, is considered. Using OCT images from phantoms, two ex-vivo experiments with porcine corneas and an in-vivo experiment with human corneas, an evidence is provided that the distributional model-free approach, despite its simplicity, could lead to better results than the best-fitted (among a range of considered models) distributional model. Concluding, in practice, the distributional model-free approach should be considered as the first choice to speckle modeling before a distributional-based approach is utilized.