Although helpful, this approach is labour-intensive and not necessarily optimal a robust machine-learning method would obviate the need for this additional step and capture all of the available classification power directly from the raw data. These are generally nonlinear functions of the input features that capture physical insights about the data. A common approach is to combine shallow classifiers with high-level features that are derived manually from the raw features. Circuit complexity theory tells us that deep neural networks (DN) have the potential to compute complex functions much more efficiently (fewer hidden units), but in practice they are notoriously difficult to train due to the vanishing gradient problem 4, 5 the adjustments to the weights in the early layers of a DN rapidly approach zero during training. Although any function can theoretically be represented by a ‘shallow’ classifier, such as a neural network with a single hidden layer 3, an intractable number of hidden units may be required. The relative likelihood function is a complicated function in a high-dimensional space. Machine-learning classifiers such as neural networks provide a powerful way to solve this learning problem. The high dimensionality of data, referred to as the feature space, makes it intractable to generate enough simulated collisions to describe the relative likelihood in the full feature space, and machine-learning tools are used for dimensionality reduction. Often this relative likelihood function cannot be expressed analytically, so simulated collision data generated with Monte Carlo methods are used as a basis for approximation of the likelihood function. For this reason, the critical element of the search for new particles and forces in high-energy physics is the computation of the relative likelihood, the ratio of the sample likelihood functions in the two considered hypotheses, shown by Neyman and Pearson 2 to be the optimal discriminating quantity. To discover a new particle, physicists must isolate a subspace of their high-dimensional data in which the hypothesis of a new particle or force gives a significantly different prediction than the null hypothesis, allowing for an effective statistical test. Given the limited quantity and expensive nature of the data, improvements in analytical tools directly boost particle discovery potential. Such discoveries require powerful statistical methods, and machine-learning tools have a critical role. Observing these particles and measuring their properties may yield critical insights about the very nature of matter 1. The primary tools of experimental high-energy physicists are modern accelerators, which collide protons and/or antiprotons to create exotic particles that occur only at extremely high-energy densities. By investigating the structure of matter and the laws that govern its interactions, this field strives to discover the fundamental properties of the physical universe. The field of high-energy physics is devoted to the study of the elementary constituents of matter. This demonstrates that deep-learning approaches can improve the power of collider searches for exotic particles. Here, using benchmark data sets, we show that deep-learning methods need no manually constructed inputs and yet improve the classification metric by as much as 8% over the best current approaches. Recent advances in the field of deep learning make it possible to learn more complex functions and better discriminate between signal and background classes. Progress on this problem has slowed, as a variety of techniques have shown equivalent performance. Standard approaches have relied on ‘shallow’ machine-learning models that have a limited capacity to learn complex nonlinear functions of the inputs, and rely on a painstaking search through manually constructed nonlinear features. Finding these rare particles requires solving difficult signal-versus-background classification problems, hence machine-learning approaches are often used. Collisions at high-energy particle colliders are a traditionally fruitful source of exotic particle discoveries.
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