One of the most fascinating challenges in the context of parton density
function (PDF) is the determination of the best combined PDF uncertainty from
individual PDF sets. Since 2014 multiple methodologies have been developed to
achieve this goal. In this proceedings we first summarize the strategy adopted
by the PDF4LHC15 recommendation and then, we discuss about a new approach to
Monte Carlo PDF compression based on clustering through machine learning
algorithms.Comment: 4 pages, 4 figures, to appear in the proceedings of 50th Rencontres
  de Moriond, QCD and High Energy Interactions, La Thuile, Italy, March 201

Carrazza, Stefano

Latorre, José I.

English

arXiv

One of the most fascinating challenges in the context of parton density function (PDF) is the determination of the best combined PDF uncertainty from individual PDF sets. Since 2014 multiple methodologies have been developed to achieve this goal. In this proceedings we first summarize the strategy adopted by the PDF4LHC15 recommendation and then, we discuss about a new approach to Monte Carlo PDF compression based on clustering through machine learning algorithms.One of the most fascinating challenges in the context of parton density function (PDF) is the determination of the best combined PDF uncertainty from individual PDF sets. Since 2014 multiple methodologies have been developed to achieve this goal. In this proceedings we first summarize the strategy adopted by the PDF4LHC15 recommendation and then, we discuss about a new approach to Monte Carlo PDF compression based on clustering through machine learning algorithms

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Towards the compression of parton densities through machine learning algorithms

One of the most fascinating challenges in the context of parton density function (PDF) is the determination of the best combined PDF uncertainty from individual PDF sets. Since 2014 multiple methodologies have been developed to achieve this goal. In this proceedings we first summarize the strategy adopted by the PDF4LHC15 recommendation and then, we discuss about a new approach to Monte Carlo PDF compression based on clustering through machine learning algorithms

S. Carrazza

J. Latorre

AIR Universita degli studi di Milano

CERN-TH-2016-115
TOWARDS THE COMPRESSION OF PARTON DENSITIES THROUGH
MACHINE LEARNING ALGORITHMS
Stefano Carrazza1∗ and Jose´ I. Latorre2
1Theoretical Physics Department, CERN, Geneva, Switzerland, ∗Speaker
2Departament d’Estructura i Contituents de la Mate`ria, Universitat de Barcelona
One of the most fascinating challenges in the context of parton density function (PDF) is the
determination of the best combined PDF uncertainty from individual PDF sets. Since 2014
multiple methodologies have been developed to achieve this goal. In this proceedings we first
summarize the strategy adopted by the PDF4LHC15 recommendation and then, we discuss
about a new approach to Monte Carlo PDF compression based on clustering through machine
learning algorithms.
The PDF4LHC15 recommendation and tools for LHC Run II In October 2015 the
PDF4LHC Working Group released a new set of guidelines for the combination of PDF sets,
known as the “PDF4LHC15 recommendation” published in Ref. 1. This updated recommen-
dation proposes the construction of a combined prior PDF set of Monte Carlo (MC) replicas,
where each replica comes from global PDF determinations. The prior set is then compressed to
a minimal number of PDF members through reduction algorithms specialized in the removal of
information redundancy.
The PDF4LHC15 prior consists in Nrep = 900 MC replicas from NNPDF3.0
2, CT14 3 and
MMHT2014 4. Eigenvectors from CT14 and MMHT2014 are transformed into MC replicas
through the method developed by Watt and Thorne in Ref. 6 and implemented in the LHAPDF6 5
library. The PDF sets entering in the current combination satisfy requirements which guarantee
the consistency of results: use global datasets, compute theoretical predictions and DGLAP in
the GM-VFNS, set αs to the PDG average
7.
The subsequent step consists in removing the redundant information from the prior set
through reduction algorithms. For the PDF4LHC15 recommendation we have used 3 different
strategies: CMC-PDF 8, MC2H 9 and Meta-PDF 10. The CMC-PDF approach outputs a subset
of MC replicas which preserves the statistical properties of the prior set. The MC2H strategy
provides a symmetric Hessian PDF set obtained by using the MC replicas themselves as the
basis of the linear representation in combination with principal component analysis (PCA)
to reproduce the PDF covariance matrix with arbitrary precision. The Meta-PDF approach
refit each MC replica with a flexible meta-parametrization, from which the best constrained
combination are found by diagonalization of the covariance matrix on the PDF space.
The delivery of results in the MC representation is useful when considering regions where
predictions are non-Gaussian, such as searches at high-masses and generally wherever the PDF
is probed at large x. On the other hand, Hessian sets are useful for many experimental needs,
e.g. when using nuisance parameters, or when high accuracy is required. The PDF4LHC15
recommendation delivers sets at NLO and NNLO with nf = 4, 5, noted as: PDF4LHC15 mc, the
compressed Monte Carlo set with Nrep = 100 obtained with CMC-PDF; PDF4LHC15 100, the
symmetric Hessian set with Neig = 100 obtained with MC2H; PDF4LHC15 30, the symmetric
Hessian set with Neig = 30 obtained with Meta-PDF.
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Figure 1 – Illustrative example of the clustering idea. A parliament is composed by several elements (left), for
each of them it is possible to determine the most representative exemplars and the fraction of elements similar to
them (right).
Finally, thanks to developments for the PDF4LHC15 recommendation, a recent Hessian
reduction algorithm called Specialized Minimal PDF (SMPDF) was published in Ref. 11. The
SMPDF methodology constructs PDFs designed to provide an accurate representation of PDF
uncertainties for specific processes or classes of processes with a minimal number of PDF error
sets.
Monte Carlo PDF compression through machine learning In the next paragraphs we
present a new concept of compression approach for MC PDFs based on clustering through
machine learning algorithms.
Let us consider a generic PDF set composed by a large number of MC replicas. Starting from
a simple visual inspection we observe that groups of replicas have similar shapes, positions and
lengths. This observation suggests that in the PDF space there is a limited number of shapes and
directions privileged by replicas and so, if we want to reduce the number of members contained
in a PDF set we should extract the most important replicas and their respective weights. This
observation is illustrated in Figure 1 by the analogy of the politicians and their parties in a
parliament which in our case study are identified to PDF replicas. The left plot shows the initial
distribution of elements. In this example similar objects are identified by a color, and we have 5
groups. The right plot shows the most representative exemplars of each group and their weight.
The next step is to setup a clustering algorithm able to identify the number of groups, the most
representative exemplars and their weights.
Here we use Affinity Propagation (AP), by messaging passing algorithm presented in Ref. 12
where the authors show its impressive capability of grouping data with complex structure. The
choice of this particular algorithm is motivated by its capability of determining automatically
the number of final clusters and its members without requiring as input an a prior knowledge
or guess of the number of clusters. The only requirement of AP is to set a distance definition
to quantify the similarity between elements of a given ensemble of PDF replicas. In the AP
approach, we construct a similarity matrix, defined as:
Si,j = −d(`i, `j), (1)
where d(`i, `j) is the distance estimator defined by the user. We performed the current analysis
with the squared euclidean distance between the arc-length of replicas defined as:
`k =
nf∑
α=−nf
∫ 1
0
√√√√1 + (df (k)α (x,Q)
dx
)2
dx, (2)
where k is the replica index and α runs over the nf independent PDF flavors at the factorization
scale Q. We observed that similar results are also obtained when using just the spatial euclidean
distance between replicas.
In Figure 2 we show the results of this clustering procedure for the NNPDF3.0 NLO set
with Nrep = 1000 replicas. The AP algorithm identifies 14 clusters which are represented by
different colors for the down (left plot) and strange (right plot) PDFs. The final step consists in
computing the weight associated to each cluster center exemplar. For each cluster i we define
its associated weight, wi, as:
wi = Ni/Nrep,
∑
i
wi = 1, (3)
where Ni is the number of elements contained in the cluster i. The output of this procedure is
a MC set of PDFs with Nrep = 14 MC replicas and a list of Nrep weights.
In Figure 3 we compare the central value and its uncertainty for the down and strange PDFs
for the NNPDF prior and the compressed set obtained with AP. For the AP PDF set we plot the
weighted mean and standard deviation. In general, we observe that a good level of agreement
is obtained. Furthermore, in Figure 4 we compare theoretical predictions of both sets for the
ATLAS inclusive jets setup with |η| < 0.3 from Ref. 14 (left plot) and a tt¯ rapidity distribution
at LHC with
√
s = 13 TeV (right plot). Also in this case, the level of agreement is satisfactory.
Similar results are obtained when using the PDF4LHC prior set. This approach has two
advantages in comparison to the CMC-PDF method: the instantaneous computation time, and
the possibility to compress to a very lower number of replicas due to the flexibility of weights.
We are confident that this or similar approaches based on the idea of weighting MC replicas are
the right future direction to obtain fast and outstanding compression performance of MC PDF
sets.
Acknowledgments S.C. is supported by the HICCUP ERC Consolidator grant (614577).
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10-3 10-2 10-1
x
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All 14 clusters NNPDF Prior- xd(x,Q=1 GeV)
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Figure 2 – Examples of clustering of MC replicas using affinity propagation and arc-length distance metrics.
10-3 10-2 10-1
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AP Compression Nrep=14
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Figure 3 – Comparisons for central values and uncertainties between the prior PDF set and the affinity propagation
clustering compression.
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Figure 4 – Comparisons of theoretical predictions for the prior PDF set and the affinity propagation clustering
compression sets. Plots obtained with SMPDF.


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Towards the compression of parton densities through machine learning algorithms

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