
Machine Learning for the LHC
LHC physics, like many other field in and around particle physics is living through an exciting, datadriven era. As theorists we can come up with ideas for physics beyond the Standard Model, motivated by dark matter or the matterantimatter asymmetry or whatever else keeps us up at night, and immediately try it on LHC data. Even if we do not have access to the actual data we can devise search strategies using a simulation chain based on first principles and then hand our ideas and tools over to our experimental friends. The scientific challenge is that we have to understand the basic structure of LHC data, its (precision) simulation, and the best ways of using huge amounts of data. This immediately brings us to machine learning as a great tool for particle physicists. In our group we develop and apply modern data analysis tools in four ways, where this number should be increasing. Here is a list of papers we published on machine learning applications:
Jet and Event Classification
Multivariate analysis methods have a long tradition in particle physics. If we use subjet physics as an example, we want to ask the question what kind of quark or gluon initiated an observed jet. Traditionally, this question is asked in relation to bottom quarks, which can be identified by a displaced decay vertex and one or more leptons inside the hadronic jets. From a theory and experimental perspective the cleanest signature are boosted top quarks. The original HEPTopTagger approach asks for two mass drops, one from the top decay and one from the Wdecay. We know that this tagger can be hugely improved when we include a wealth of kinematic observables, some directly and some algorithmically derived from the energies and the momenta of the subjets. The obvious question is why we need to construct these highlevel observables and if we cannot just feed the 4momenta of the subjets into a classification algorithm. As a matter of fact, we can do exactly that. The next question is if we really need to train our classification on data where we know the truth label, or if we can just look for jets which look as little like QCD as possible. That is the second aspect we are working on these days. This work has been done with Sven Bollweg, Anja Butter, Sascha Diefenbacher, Barry Dillon, Hermann Frost, Manuel Haussmann, Theo Heimel, Gregor Kasieczka, Nicholas Kiefer, Michel Luchmann, Michael Russell, Torben Schell, Christof Sauer, Peter Sorrenson, Jennie Thompson, and more students and postdocs.
Constructing a symmetric autoencoder with better latent spaces (2021). We know from our earlier study that it is possible to extract top jets from a QCD jet sample, essentially based on the compressibility of the encoded information. To apply unsupervized classification the other way around, we need to define a nice latent space and access this latent space rather than the reconstruction loss. And then it works.  
Measurement of jet momenta with uncertainties and calibration of regression tools with Bayesian networks (2020). The measurement by the regression network tracks statistical and systematic uncertainties from the training data. We propose to calibrate the network in a straightforward way through the smearing introduced by the measurement of labels.  
Jet and event classification with capsule networks (2019). Such capsule networks are a natural extension of (scalar) convolutional networks and can be used to analyse sparse sets of detector objects, each represented by a sparse calorimeter image.  
Top tagging with uncertainties using Bayesian classification networks (2019). Such a tagger provides a classification output and a jetwise error estimate on the classification outcome. While statistical uncertainties from a limited training sample are easily traced, systematic uncertainties lead to a correlation of the central value and the uncertainties, all the way to adversarial examples.  
Toptagging community paper (2019) comparing a wide range of top taggers, from imagebased to 4vectorbased and theorymotivated tagging approaches. We show their respective performances for a standard data set and find that there are many ways of contructing highly performing taggers ready to hit LHC data.  
LoLabased quarkgluon tagger (2018) showing that the same 4vectorbased architecture can be used to distinguish hard processes with preferably quark or gluon jets. The issue is how to design a highperformance tagger in the presence of detector effects.  
Jet autoencoder (2018) based on the DeepTop and LoLa taggers. We show howtop jets are less compressible than QCD jets, how they can be tagged, and how we can decorrelate the jet mass using an adversary. We also test the autoencoder on new physics in jets.  
4vectorbased LoLa tagger (2017) which allows us to combine information from the calorimeter and the tracker, accounting for the different resolutions. It can be thought of a graph network over Minkowski space, where we use the fact that we know the Minkowski metric. For a cross check this is one of the few papers which quotes the Minkowski metric with an error bar.  
Imagebased DeepTop tagger (2017) showing that we can use convolutional networks on calorimeter images to identify boosted tops. We showed how this deeplearning tagger compares to classic multivariate methods and how it is possible to interpret the intermediate layers and the output of the tagging network. 
Event Generation
Firstprinciple simulations LHC events based on quantum field theory (mostly QCD) is one of the unique features of LHC physics. Many groups are investing a huge effort into computing the underlying predictions in perturbative quantum field theory. These prediction can be included in multipurpose event generators like Pythia, Sherpa, Madgraph, or Herwig. These simulations are based on Monte Carlo simulations and are extremely efficient. The question is if we can use machine learning tools to improve them further or to get access to information that is usually lost in the simulation framework. One example is the use of the hard matrix element in a hypothesis test, usually called the matrix element method. Our group includes Marco Bellagente, Anja Butter, Sascha Diefenbacher, Manuel Haussmann, Gregor Kasieczka, Ulli Kothe, Michel Luchmann, Armand Rousselot, and Ramon Winterhalder, and it is still expanding.
Brandnew Bayesian generative network, specifically a normalizing flow network or INN (2021). It learns not only a density map over phase space, but also an uncertainty map, and it provides this information in terms of events with an uncertainty weight. That setup is probably the first time, someone has attached uncertainties to a generative network, a crucial step for LHC applications. Moreover, we can study how the network learns the density and uncertainty maps in parallel and confirm that the flow network really works like some kind of functional fit.  
Study of superresolution on jet images (2020). The mismatch of calorimeter and tracking resolution on jet constituents is one of the leading themes of jet classification. We show how this measurement structure motivates applying a generative superresolution network. As usual, we look at QCD vs topdecay jets and study the model dependence for these two kinds of jets.  
GAN study on event unweighting (2020). The transformation of weighted events into unweighted events is the numerical bottleneck of many LHC simulations. We show how the classic hitandmiss algorithm can be improved significantly by applying a generative network which gets trained on weighted events and produces unweighted events.  
Generative networks review article (2020). We given an overview of the many ways generative networks are developed for LHC event generation. This includes many different architectures and applications.  
GAN study on the statistical gain (2020). An open question, crucial to LHC applications, is how much we can gain beyond the statistical power of the training sample by training a generatve network for more events. We define an amplification factor for simple multidimensional toy distributions and first find that, just like a fit, the structure of the GAN adds information to the discrete set of training events. Second, the GANned events do not have the same individual statistical power as a sampled event. The GAN applification factor becomes larger for sparcely distributed training data in highdimensional phase spaces.  
Conditional INN application to detector and QCD unfolding (2020). Invertible networks based on normalizing flows and built out of coupling layers can also serve as generative networks for LHC events. Their greatest advantage is that in a conditional setup they are built to generate spreadout probability distributions in the target space. For LHC unfolding this means we can construct a probability distribution over partonlevel phase space for a single detectorlevel event. We expand the detector unfolding to also unfold jet radiation to a predefined hard process.  
Fully conditional GAN application to detector unfolding (2019). If we train a conditional GAN on matched event samples we can use it to invert a MonteCarlobased simulation of, for instance, detector effects. This unfolding is not limited to one or lowdimensional distribution, but covers the entire phase space. The matching of local structures in the two data sets reduces the model dependence of the unfolding procedure.  
GAN application to event subtraction (2019). A general problem in dealing with event samples is that there is no efficient way to subtract event sample from each other, or combine events with positive and negative weights. We show how GANs can do that avoid statistical limitations from the usual binning procedure. Applications could be subtraction terms in Monte Carlo simulations or background subtraction in analyses.  
GAN application to event generation (2019). We show that a GAN, supplemented with a dedicated MMD loss, can generate top pair production events at the LHC. We model all kinematic distributions all the way to the three top decay jets. We find that regions with large systematic uncertainties on the GAN are directly linked to sparse training data. 
Inference
Finally, LHC measurements can benefit conceptionally from machine learning methods. To start with, information geometry is not exactly machine learning, but it is a concept which benefits from machine learning when we apply it to LHC physics. The question we are trying to answer is what kind of information is available to an LHC analysis, what observables capture it best, and what the limiting factors in an analysis might be. The machine learning aspect only comes in once we ask these questions beyond the parton level, and two former Heidelberg students (Johann Brehmer and Felix Kling) have worked with Kyle Cranmer's NYU group on developing the corresponding MadMiner program. Other collaborators on this topic include Sally Dawson and Sam Homiller for Higgs applications, or Ulli Kothe and Stefan Radev for invertible networks.
Proposing a measurement of QCD splitting kernels using a conditional INN (2020). We start by defining a new interpretation framework for QCD jet measurements, leading to an actual measurement of QCDparameters. The complexity of lowlevel data and the highly nonlinear relation between splitting kernels and jet data make this measurement an excellent case for a machine learning application.  
Validation of simplfied template cross sections for VH production (2019). We analyse how these observables compare to an analysis of the full phase space, including detector effects and mixxing transverse momentum. This is where machine learning enters.  
Information geometry of Higgs CP in the SMEFT framework (2017). We compare different Higgs production and decay signatures, all based on the amplitude with four additional fermions, in their potential to test the CP properties of the Higgs coupling to intermediate gauge bosons. We link this approach to the established optimal observables.  
Information geometry of Higgs signatures in the SMEFT framework (2016). We compute the information available over the entire partonic phase space of Higgs signatures and compare for instance the impact of the QBF tagging jet kinematics with the Higgs decay kinematics. 
Global SFitter analyses
Obviously, machine learning should be able to improve global analyses of LHC data, where we interpret a large number of measurements in a highdimensional parameter space. Our SFitter tool has serves us well in the last 10 years, so we are working on improving it with machine learning applications...