<p>Recently, the use of cosmic ray muons in critical national security applications, e.g., nuclear nonproliferation and safeguards verification, has gained attention due to unique muon properties such as high energy and low attenuation even in very dense materials. Applications where muon tomography has been demonstrated include cargo screening for detection of special nuclear materials smuggling, source localization, material identification, determination of nuclear fuel debris location in nuclear reactors, etc. However, muon image reconstruction techniques are still limited in resolution mostly due to multiple Coulombscattering (MCS) within the target object. Improving and expanding muon tomography would require development of efficient & flexible physics-based algorithms to model the MCS process and accurately estimate the most probable trajectory of a muon as it traverses the target object. The present study introduces a novel algorithmic approach that utilizes Bayesian probability theory and a Gaussian approximation of MCS to estimate the most probable path of cosmic ray muons as they traverse uniform media.</p> <p>Using GEANT4, an investigation was conducted involving the trajectory of 10,000 muon particles that underwent bombardment from a point source parallel to the x-axis. The proposed algorithm was assessed through four types of simulations. In the first type, muons with energies of 1 GeV, 3 GeV, 10 GeV, and 100 GeV were utilized to evaluate the algorithms’ performance and accuracy. The second type of simulation involved the use of target cubes composed of different materials, including aluminum, iron, lead, and uranium. These simulations specifically focused on muons with an energy of 3 GeV. Next, the third type of simulation entailed employing target cubes with varying lengths, such as 10 cm, 20 cm, 40 cm, and 80 cm, specifically using muons with an energy of 3 GeV and a uranium target. Lastly, all the previous simulations were revised to accommodate a source of poly-energetic muons. This revision was undertaken to create a more realistic source scenario that aligns with the distribution of muon energies encountered in real-world situations.</p> <p>The results demonstrate significant improvements in precision and muon flux utilization when comparing different algorithms. The Generalized Muon Trajectory Estimation (GMTE) algorithm shows around 50% improvement in precision compared to currently used Straight Line Path (SLP) algorithm across all test scenarios. Additionally, GMTE algorithm exhibits around 38% improvement in precision compared to the extensively used Point of Closest Approach (PoCA) algorithm. Similarly for both mono and poly energetic source of muons, the GMTE algorithm shows 10%-35% increase in muon flux utilization for high Z materials and a 10%-15% increase for medium Z materials compared to the PoCA algorithm. Similarly, it demonstrates 6%-9% increase in muon flux utilization for both medium and high Z materials compared to the SLP algorithm across all test scenarios. These results highlight the enhanced performance and efficiency of GMTE algorithm in comparison to SLP and PoCA algorithms.</p> <p>Through these extensive simulations, our objective was to comprehensively evaluate the performance and effectiveness of the proposed algorithm across a range of variables, including energy levels, materials, and target geometries. The findings of our study demonstrate that the utilization of these algorithm enables improved resolution and reduced measurement time for cosmic ray muons when compared with current SLP and PoCA algorithm.</p>

<p dir="ltr">The generalization error of a classifier is related to the complexity of the set of functions among which the classifier is chosen. We study a family of low-complexity classifiers consisting of thresholding a random one-dimensional feature. The feature is obtained by projecting the data on a random line after embedding it into a higher-dimensional space parametrized by monomials of order up to k. More specifically, the extended data is projected n-times and the best classifier among those n, based on its performance on training data, is chosen. </p><p dir="ltr">We show that this type of classifier is extremely flexible, as it is likely to approximate, to an arbitrary precision, any continuous function on a compact set as well as any Boolean function on a compact set that splits the support into measurable subsets. In particular, given full knowledge of the class conditional densities, the error of these low-complexity classifiers would converge to the optimal (Bayes) error as k and n go to infinity. On the other hand, if only a training dataset is given, we show that the classifiers will perfectly classify all the training points as k and n go to infinity. </p><p dir="ltr">We also bound the generalization error of our random classifiers. In general, our bounds are better than those for any classifier with VC dimension greater than O(ln(n)). In particular, our bounds imply that, unless the number of projections n is extremely large, there is a significant advantageous gap between the generalization error of the random projection approach and that of a linear classifier in the extended space. Asymptotically, as the number of samples approaches infinity, the gap persists for any such n. Thus, there is a potentially large gain in generalization properties by selecting parameters at random, rather than optimization. </p><p dir="ltr">Given a classification problem and a family of classifiers, the Rashomon ratio measures the proportion of classifiers that yield less than a given loss. Previous work has explored the advantage of a large Rashomon ratio in the case of a finite family of classifiers. Here we consider the more general case of an infinite family. We show that a large Rashomon ratio guarantees that choosing the classifier with the best empirical accuracy among a random subset of the family, which is likely to improve generalizability, will not increase the empirical loss too much. </p><p dir="ltr">We quantify the Rashomon ratio in two examples involving infinite classifier families in order to illustrate situations in which it is large. In the first example, we estimate the Rashomon ratio of the classification of normally distributed classes using an affine classifier. In the second, we obtain a lower bound for the Rashomon ratio of a classification problem with a modified Gram matrix when the classifier family consists of two-layer ReLU neural networks. In general, we show that the Rashomon ratio can be estimated using a training dataset along with random samples from the classifier family and we provide guarantees that such an estimation is close to the true value of the Rashomon ratio.</p>