Anomaly detection with density estimation

We leverage recent breakthroughs in neural density estimation to propose a new unsupervised ANOmaly detection with Density Estimation (ANODE) technique. By estimating the conditional probability density of the data in a signal region and in sidebands, and interpolating the latter into the signal region, a fully data-driven likelihood ratio of data versus background can be constructed. This likelihood ratio is broadly sensitive to overdensities in the data that could be due to localized anomalies. In addition, a unique potential benefit of the ANODE method is that the background can be directly estimated using the learned densities. Finally, ANODE is robust against systematic differences between signal region and sidebands, giving it broader applicability than other methods. We demonstrate the power of this new approach using the LHC Olympics 2020 R&D dataset. We show how ANODE can enhance the significance of a dijet bump hunt by up to a factor of 7 with a 10% accuracy on the background prediction. While the LHC is used as the recurring example, the methods developed here have a much broader applicability to anomaly detection in physics and beyond.

Figure 1

A graphical representation of searches for new particles in terms of the background and signal model dependence for achieving signal sensitivity (a) and background specificity (b). The Model Unspecific Search for New Physics (MUSiC) [20, 21] and general search [22, 23, 24] strategies are from CMS and ATLAS, respectively. LDA stands for latent dirichlet allocation [38, 80], ANOmaly detection with Density Estimation (ANODE) is the method presented in this paper, CWoLa stands for classification without labels [33, 34, 79], and SALAD stands for simulation assisted likelihood-free anomaly detection [39]. Direct density estimation is a form of sidebanding where the multidimensional feature space density is learned conditional on the resonant feature (see Sec. 3b).

Figure 2

Histograms for the invariant mass of the leading two jets for the Standard Model background as well as the injected signal. There are 1 million background events and 1000 signal events.

Figure 3

The four features used for classification: $m_{J_1}$ (top left), $m_{J_1}-m_{J_2}$ (top right), ${\tau }_{21}^{J_1}$ (bottom left), and ${\tau }_{21}^{J_2}$ (bottom right). These histograms are inclusive in $m_{JJ}$. There are 1 million background events and 1000 signal events for the mass histograms.

Figure 4

Scatter plot of $R(x|m)$ versus $\log p_{\text{background}}(x|m)$ across the test set in the SR. Background events are shown (as a two-dimensional histogram) in gray scale and individual signal events are shown in red.

Figure 5

Left: histogram of $R(x|m)$ evaluated on the test set; right: the integrated number of events that survive a threshold on $R(x|m)$. The two distributions are scaled to represent the rates for 500,000 total background events and 500 total signal events, as introduced in Sec. 4.

Figure 6

Distributions of $m_{J_1}$ (left) and $m_{J_2}-m_{J_1}$ (right) in the signal region after applying a threshold requirement on $R$.

Figure 7

Receiver operating characteristic (ROC) curve (left) and significance improvement characteristic (SIC) curve (right).

Figure 8

Left: the number of events after a threshold requirement $R>R_c$ using the two integration methods described in Sec. 3b, as well as the true background yield. Right: the ratio of the predicted and true background yields from the left plot, as a function of the actual number of events that survive the threshold requirement. The shaded bands around the central predictions are the $1\sigma$ statistical (Poisson) uncertainty derived from the observed background counts. The black dashed and dotted lines are 10% and 20% around a ratio of 1.

Figure 9

A comparison of the four features $x$ between the SR and two nearby sidebands defined by $m_{jj}\in [3.1,3.3]\mathrm{TeV}$ (lower sideband) and $m_{jj}\in [3.7,3.9]\mathrm{TeV}$ (upper sideband).

Figure 10

The lighter jet mass for the SR and the lower and upper sideband regions after the shift defined by Eq. (5.1).

Figure 11

ROC (left) and SIC (right) curves in the signal region using the shifted dataset specified by Eq. (5.1).

Figure 12

The same as Fig. 8, but for the shifted dataset. In particular, these plots compare the background prediction from two direct density estimation techniques with the true background yield after a threshold requirement $R(x|m)>R_c$.