Riemannian data preprocessing in machine learning to focus on QCD color structure

Abstract

Identifying the quantum chromodynamics (QCD) color structure of processes provides additional information to enhance the reach for new physics searches at the large Hadron collider (LHC). Analyses of QCD color structure in the decay process of a boosted particle have been spotted as information becomes well localized in the limited phase space. While these kinds of a boosted jet analyses provide an efficient way to identify the color structure, the constrained phase space reduces the number of available data, resulting in a low significance. In this letter, we provide a simple but a novel data preprocessing method using a Riemann sphere to utilize a full phase space by decorrelating QCD structure from kinematics. We can achieve statistical stability by enlarging the size of testable data set with focusing on QCD structure effectively. We demonstrate the power of our method with the finite statistics of the LHC Run 2. Our method is complementary to conventional boosted jet analyses in utilizing QCD information over a wide range of a phase space.

Appendices

Appendix 1. The structure of CNN

For each event, we construct an image as a square array in the ($\eta \text {-}\phi$) plane with each pixel intensity given by the total hadrons $p_T$ deposited in the associated region in the calorimeter. The rectangular region between $-2.5\le \eta \le 2.5$ and $-\pi \le \phi \le \pi$ is discretized into $50\times 50$ pixels grid. As discussed earlier, in order to optimize the CNN performance and reduce the error, we applied the following preprocessing steps:

• 1. Image cleansing: removing all leptons and photons from the image.

• 2. Center: shift the center of the image from (0, 0) to $(\frac{\eta _{b^-}+\eta _{b^+}}{2},\frac{\phi _{b^-}+\phi _{b^+}}{2})$.

• 3. Normalization: normalize pixels intensity by diving each pixel in the image by the maximum pixel intensity value.

• 4. Inverse stereographic projection: project the image pixels in $(\eta \text {-}\phi )$ plane to a Riemann sphere by applying the inverse stereographic transformations. We fix the hot cores position in $\phi _R$ dimension to be at ($-\frac{\pi }{2},\frac{\pi }{2}$). The projected images represent sphere in three dimensions and thus we cannot use the normal CNN to classify them. One way is to use a spherical CNN approach as presented in [54]. Another way is to use the Mollweide projection as shown in Fig. 7 (left). In this work, we use Mollweide projected images as input to the CNN.

• 5. Momentum smearing: smear the momentum according to Gaussian distribution and correlate the neighboring pixels to decrease the sparsity in images [55]. Figure 7 illustrates the effect of Gaussian filter with standard deviation $\sigma =3$ on the reconstructed images for singlet, octet scalars and backgrounds.

Fig. 7

Mollweide projection for a single event before (left) and after (right) applying Gaussian kernel with standard deviation $\sigma =3$

Testing the CNN classification performance on two different data sets, Riemannian preprocessing and No preprocessing, requires to optimize the CNN structure for each data set individually. We found that CNN with six convolution layers with kernel size 3, two dense layers and one output layer with two neurons is the best choice for both cases. Each convolution layers pair is followed by a maxpooling layer of size 2 and a dropout layer with dropout rate of $20\%$. Dense layers are followed by dropout layer with dropout rate of $30\%$. The number of kernels in the first convolution layer is fixed to 16 and the activation functions is ReLU except the last dense layer (output layer) we use SoftMax function. To maintain the network stability and to avoid the covariate shift problem, we add batch normalization layers. In Riemannian preprocessing images, edges are of most important, thus we apply padding layer in the first convolution layer to keep image dimension intact. The Loss function, categorical cross entropy, is minimized using Adam optimizer with learning rate of 0.001. The number of kernels in the convolution layers and the number of neurons in the dense layers has been optimized using RandomizedSearchCV function in Scikit-learn. We use a balanced data set of 100K events for both processes, each data set is divided into $60\%$ training set, $20\%$ validation set and $20\%$ test set.

Appendix 2. Background rejection

Fig. 8

ROC curve in separating backgrounds from signal events by utilizing color-flow information in a resolved phase space

Background contribution in a $HZ\rightarrow b \bar{b} \ell \bar{\ell }$ channel comes from vector boson plus three or more jets, di-top production, single top production and di-boson production [40]; while we dropped the $W+jets$ and Wt processes as they have negligible contribution after passing all selection cuts. In [40], ATLAS analysis does not take an advantage of different color-flow information between a signal and backgrounds. Here, we add a CNN study as in the main text with/without Riemannian preprocessing. In Fig. 8, we present ROC performance using CNN analyses. We can achieve an extra factor 2 enhancement in a discovery significance using Riemannian preprocessing while we get only $25\%$ enhancement with a normal CNN analysis when we take a false positive as 0.2 for example.