Searcharxiv⌕ Search

arXiv subjects

Philipp Frank

Publications and source records attributed to Philipp Frank.

48 records · Page 3Linked to original sources
arXiv2021

Dynamical field inference and supersymmetry

Knowledge on evolving physical fields is of paramount importance in science, technology, and economics. Dynamical field inference (DFI) addresses the problem of reconstructing a stochastically driven, dynamically evolving field from finite data. It relies on information field theory (IFT), the information theory for fields. Here, the relations of DFI, IFT, and the recently developed supersymmetric theory of stochastics (STS) are established in a pedagogical discussion. In IFT, field expectation values can be calculated from the partition function of the full space-time inference problem. The partition function of the inference problem invokes a functional Dirac function to guarantee the dynamics, as well as a field-dependent functional determinant, to establish proper normalization, both impeding the necessary evaluation of the path integral over all field configurations. STS replaces these problematic expressions via the introduction of fermionic ghost and bosonic Lagrange fields, respectively. The action of these fields has a supersymmetry, which means there exists an exchange operation between bosons and fermions that leaves the system invariant. In contrast to this, measurements of the dynamical fields do not adhere to this supersymmetry. The supersymmetry can also be broken spontaneously, in which case the system evolves chaotically. This affects the predictability of the system and thereby make DFI more challenging. We investigate the interplay of measurement constraints with the non-linear chaotic dynamics of a simplified, illustrative system with the help of Feynman diagrams and show that the Fermionic corrections are essential to obtain the correct posterior statistics over system trajectories.

quant-ph↗
arXiv2021

Field dynamics inference for local and causal interactions

Inference of fields defined in space and time from observational data is a core discipline in many scientific areas. This work approaches the problem in a Bayesian framework. The proposed method is based on statistically homogeneous random fields defined in space and time and demonstrates how to reconstruct the field together with its prior correlation structure from data. The prior model of the correlation structure is described in a non-parametric fashion and solely builds on fundamental physical assumptions such as space-time homogeneity, locality, and causality. These assumptions are sufficient to successfully infer the field and its prior correlation structure from noisy and incomplete data of a single realization of the process as demonstrated via multiple numerical examples.

physics.data-an↗
arXiv2021

Reconstructing non-repeating radio pulses with Information Field Theory

Particle showers in dielectric media produce radio signals which are used for the detection of both ultra-high energy cosmic rays and neutrinos with energies above a few PeV. The amplitude, polarization, and spectrum of these short, broadband radio pulses allow us to draw conclusions about the primary particles that caused them, as well as the mechanics of shower development and radio emission. However, confidently reconstructing the radio signals can pose a challenge, as they are often obscured by background noise. Information Field Theory offers a robust approach to this challenge by using Bayesian inference to calculate the most likely radio signal, given the recorded data. In this paper, we describe the application of Information Field Theory to radio signals from particle showers in both air and ice and demonstrate how accurately pulse parameters can be obtained from noisy data.

astro-ph.HE↗
arXiv2020

Probabilistic simulation of partial differential equations

Computer simulations of differential equations require a time discretization, which inhibits to identify the exact solution with certainty. Probabilistic simulations take this into account via uncertainty quantification. The construction of a probabilistic simulation scheme can be regarded as Bayesian filtering by means of probabilistic numerics. Gaussian prior based filters, specifically Gauss-Markov priors, have successfully been applied to simulation of ordinary differential equations (ODEs) and give rise to filtering problems that can be solved efficiently. This work extends this approach to partial differential equations (PDEs) subject to periodic boundary conditions and utilizes continuous Gaussian processes in space and time to arrive at a Bayesian filtering problem structurally similar to the ODE setting. The usage of a process that is Markov in time and statistically homogeneous in space leads to a probabilistic spectral simulation method that allows for an efficient realization. Furthermore, the Bayesian perspective allows the incorporation of methods developed within the context of information field theory such as the estimation of the power spectrum associated with the prior distribution, to be jointly estimated along with the solution of the PDE.

math.NA↗
arXiv2020

Bayesian decomposition of the Galactic multi-frequency sky using probabilistic autoencoders

All-sky observations of the Milky Way show both Galactic and non-Galactic diffuse emission, for example from interstellar matter or the cosmic microwave background (CMB). The different emitters are partly superimposed in the measurements, partly they obscure each other, and sometimes they dominate within a certain spectral range. The decomposition of the underlying radiative components from spectral data is a signal reconstruction problem and often associated with detailed physical modeling and substantial computational effort. We aim to build an effective and self-instructing algorithm detecting the essential spectral information contained Galactic all-sky data covering spectral bands from $γ$-ray to radio waves. Utilizing principles from information theory, we develop a state-of-the-art variational autoencoder specialized on the adaption to Gaussian noise statistics. We first derive a generic generative process that leads from a low-dimensional set of emission features to the observed high-dimensional data. We formulate a posterior distribution of these features using Bayesian methods and approximate this posterior with variational inference. The algorithm efficiently encodes the information of 35 Galactic emission data sets in ten latent feature maps. These contain the essential information required to reconstruct the initial data with high fidelity and are ranked by the algorithm according to their significance for data regeneration. The three most significant feature maps encode astrophysical components: (1) The dense interstellar medium (ISM), (2) the hot and dilute regions of the ISM and (3) the CMB. The machine-assisted and data-driven dimensionality reduction of spectral data is able to uncover the physical features encoding the input data. Our algorithm is able to extract the dense and dilute Galactic regions, as well as the CMB, from the sky brightness values only.

astro-ph.IM↗
arXiv2019

Unified Radio Interferometric Calibration and Imaging with Joint Uncertainty Quantification

The data reduction procedure for radio interferometers can be viewed as a combined calibration and imaging problem. We present an algorithm that unifies cross-calibration, self-calibration, and imaging. Being a Bayesian method, that algorithm does not only calculate an estimate of the sky brightness distribution, but also provides an estimate of the joint uncertainty which entails both the uncertainty of the calibration and the one of the actual observation. The algorithm is formulated in the language of information field theory and uses Metric Gaussian Variational Inference (MGVI) as the underlying statistical method. So far only direction-independent antenna-based calibration is considered. This restriction may be released in future work. An implementation of the algorithm is contributed as well.

astro-ph.IM↗
arXiv2018

Sharpening up Galactic all-sky maps with complementary data - A machine learning approach

Galactic all-sky maps at very disparate frequencies, like in the radio and $γ$-ray regime, show similar morphological structures. This mutual information reflects the imprint of the various physical components of the interstellar medium. We want to use multifrequency all-sky observations to test resolution improvement and restoration of unobserved areas for maps in certain frequency ranges. For this we aim to reconstruct or predict from sets of other maps all-sky maps that, in their original form, lack a high resolution compared to other available all-sky surveys or are incomplete in their spatial coverage. Additionally, we want to investigate the commonalities and differences that the ISM components exhibit over the electromagnetic spectrum. We build an $n$-dimensional representation of the joint pixel-brightness distribution of $n$ maps using a Gaussian mixture model and see how predictive it is: How well can one map be reproduced based on subsets of other maps? Tests with mock data show that reconstructing the map of a certain frequency from other frequency regimes works astonishingly well, predicting reliably small-scale details well below the spatial resolution of the initially learned map. Applied to the observed multifrequency data sets of the Milky Way this technique is able to improve the resolution of, e.g., the low-resolution Fermi LAT maps as well as to recover the sky from artifact-contaminated data like the ROSAT 0.855 keV map. The predicted maps generally show less imaging artifacts compared to the original ones. A comparison of predicted and original maps highlights surprising structures, imaging artifacts (fortunately not reproduced in the prediction), and features genuine to the respective frequency range that are not present at other frequency bands. We discuss limitations of this machine learning approach and ideas how to overcome them.

astro-ph.GA↗
arXiv2018

Separating diffuse from point-like sources - a Bayesian approach

We present the starblade algorithm, a method to separate superimposed point sources from auto-correlated, diffuse flux using a Bayesian model. Point sources are assumed to be independent from each other and to follow a power-law brightness distribution. The diffuse emission is described as a non-parametric log-normal model with a priori unknown correlation structure. This model enforces positivity of the underlying emission and allows for variation in the order of magnitudes. The correlation structure is recovered non-parametrically in addition to the diffuse flux and is used for the separation of the point sources. Additionally many measurement artifacts appear as point-like or quasi-point-like effects, not compatible with superimposed diffuse emission. An estimate of the separation uncertainty can be provided as well. We demonstrate the capabilities of the derived method on synthetic data and data obtained by the Hubble Space Telescope, emphasizing its effect on instrumental artifacts as well as physical sources. The performance of this method is compared to the background estimation of the SExtractor method, as well as to a denoising auto-encoder.

astro-ph.IM↗
arXiv2017

Field dynamics inference via spectral density estimation

Stochastic differential equations (SDEs) are of utmost importance in various scientific and industrial areas. They are the natural description of dynamical processes whose precise equations of motion are either not known or too expensive to solve, e.g., when modeling Brownian motion. In some cases, the equations governing the dynamics of a physical system on macroscopic scales occur to be unknown since they typically cannot be deduced from general principles. In this work, we describe how the underlying laws of a stochastic process can be approximated by the spectral density of the corresponding process. Furthermore, we show how the density can be inferred from possibly very noisy and incomplete measurements of the dynamical field. Generally, inverse problems like these can be tackled with the help of Information Field Theory (IFT). For now, we restrict to linear and autonomous processes. Though, this is a non-conceptual limitation that may be omitted in future work. To demonstrate its applicability we employ our reconstruction algorithm on a time-series and spatio-temporal processes.

stat.ME↗
arXiv2017

NIFTy 3 - Numerical Information Field Theory - A Python framework for multicomponent signal inference on HPC clusters

NIFTy, "Numerical Information Field Theory", is a software framework designed to ease the development and implementation of field inference algorithms. Field equations are formulated independently of the underlying spatial geometry allowing the user to focus on the algorithmic design. Under the hood, NIFTy ensures that the discretization of the implemented equations is consistent. This enables the user to prototype an algorithm rapidly in 1D and then apply it to high-dimensional real-world problems. This paper introduces NIFTy 3, a major upgrade to the original NIFTy framework. NIFTy 3 allows the user to run inference algorithms on massively parallel high performance computing clusters without changing the implementation of the field equations. It supports n-dimensional Cartesian spaces, spherical spaces, power spaces, and product spaces as well as transforms to their harmonic counterparts. Furthermore, NIFTy 3 is able to treat non-scalar fields. The functionality and performance of the software package is demonstrated with example code, which implements a real inference algorithm from the realm of information field theory. NIFTy 3 is open-source software available under the GNU General Public License v3 (GPL-3) at https://gitlab.mpcdf.mpg.de/ift/NIFTy/

astro-ph.IM↗
arXiv2016

SOMBI: Bayesian identification of parameter relations in unstructured cosmological data

This work describes the implementation and application of a correlation determination method based on Self Organizing Maps and Bayesian Inference (SOMBI). SOMBI aims to automatically identify relations between different observed parameters in unstructured cosmological or astrophysical surveys by automatically identifying data clusters in high-dimensional datasets via the Self Organizing Map neural network algorithm. Parameter relations are then revealed by means of a Bayesian inference within respective identified data clusters. Specifically such relations are assumed to be parametrized as a polynomial of unknown order. The Bayesian approach results in a posterior probability distribution function for respective polynomial coefficients. To decide which polynomial order suffices to describe correlation structures in data, we include a method for model selection, the Bayesian Information Criterion, to the analysis. The performance of the SOMBI algorithm is tested with mock data. As illustration we also provide applications of our method to cosmological data. In particular, we present results of a correlation analysis between galaxy and AGN properties provided by the SDSS catalog with the cosmic large-scale-structure (LSS). The results indicate that the combined galaxy and LSS dataset indeed is clustered into several sub-samples of data with different average properties (for example different stellar masses or web-type classifications). The majority of data clusters appear to have a similar correlation structure between galaxy properties and the LSS. In particular we revealed a positive and linear dependency between the stellar mass, the absolute magnitude and the color of a galaxy with the corresponding cosmic density field. A remaining subset of data shows inverted correlations, which might be an artifact of non-linear redshift distortions.

astro-ph.CO↗