CNN-LRP: Understanding Convolutional Neural Networks Performance for Target Recognition in SAR Images

Target recognition is one of the most challenging tasks in synthetic aperture radar (SAR) image processing since it is highly affected by a series of pre-processing techniques which usually require sophisticated manipulation for different data and consume huge calculation resources. To alleviate this limitation, numerous deep-learning based target recognition methods are proposed, particularly combined with convolutional neural network (CNN) due to its strong capability of data abstraction and end-to-end structure. In this case, although complex pre-processing can be avoided, the inner mechanism of CNN is still unclear. Such a “black box” only tells a result but not what CNN learned from the input data, thus it is difficult for researchers to further analyze the causes of errors. Layer-wise relevance propagation (LRP) is a prevalent pixel-level rearrangement algorithm to visualize neural networks’ inner mechanism. LRP is usually applied in sparse auto-encoder with only fully-connected layers rather than CNN, but such network structure usually obtains much lower recognition accuracy than CNN. In this paper, we propose a novel LRP algorithm particularly designed for understanding CNN’s performance on SAR image target recognition. We provide a concise form of the correlation between output of a layer and weights of the next layer in CNNs. The proposed method can provide positive and negative contributions in input SAR images for CNN’s classification, viewed as a clear visual understanding of CNN’s recognition mechanism. Numerous experimental results demonstrate the proposed method outperforms common LRP.

Rearrangement Algorithm in Risk Aggregation

Sometimes there’s no closed-form analytical solutions for the risk measure of aggregate losses representing, say, a company’s losses in each country or city it operates in, a portfolio of losses subdivided by investment, or claims made by clients to an insurance company. Assuming there’s enough data to assign a distribution to those losses, we examine the Rearrangement Algorithm’s ability to numerically compute the Expected Shortfall and Exponential Premium Principle/Entropic Risk Measure of aggregate losses. A more efficient discretization scheme is introduced and the algorithm is extended to the Entropic Risk Measure which turns out to have a smaller uncertainty spread than the Expected Shortfall at least for the cases that we examined.

Rearrangement Algorithm in Risk Aggregation

Sometimes there’s no closed-form analytical solutions for the risk measure of aggregate losses representing, say, a company’s losses in each country or city it operates in, a portfolio of losses subdivided by investment, or claims made by clients to an insurance company. Assuming there’s enough data to assign a distribution to those losses, we examine the Rearrangement Algorithm’s ability to numerically compute the Expected Shortfall and Exponential Premium Principle/Entropic Risk Measure of aggregate losses. A more efficient discretization scheme is introduced and the algorithm is extended to the Entropic Risk Measure which turns out to have a smaller uncertainty spread than the Expected Shortfall at least for the cases that we examined.

A New Implementation of Genome Rearrangement Problem

Unsigned reverse genome rearrangement is an important part of bioinformatics research, which is widely used in biological similarity and homology analysis, revealing biological inheritance, variation, and evolution. Branch and bound, simulated annealing, and other algorithms in unsigned reverse genome rearrangement algorithm are rare in practical application because of their huge time and space consumption, and greedy algorithms are mostly used at present. By deeply analyzing the domain of unsigned reverse genome rearrangement algorithm based on greedy strategy (unsigned reverse genome rearrangement algorithm (URGRA) based on greedy strategy), the domain features are modeled, and the URGRA algorithm components are interactively designed according to the production programming method. With the support of the PAR platform, the algorithm component library of the URGRA is formally realized, and the concrete algorithm is generated by assembly, which improves the reliability of the assembly algorithm.

Implementing the Rearrangement Algorithm: An Example from Computational Risk Management

After a brief overview of aspects of computational risk management, the implementation of the rearrangement algorithm in R is considered as an example from computational risk management practice. This algorithm is used to compute the largest quantile (worst value-at-risk) of the sum of the components of a random vector with specified marginal distributions. It is demonstrated how a basic implementation of the rearrangement algorithm can gradually be improved to provide a fast and reliable computational solution to the problem of computing worst value-at-risk. Besides a running example, an example based on real-life data is considered. Bootstrap confidence intervals for the worst value-at-risk as well as a basic worst value-at-risk allocation principle are introduced. The paper concludes with selected lessons learned from this experience.

Extremal spectral gaps for periodic Schrödinger operators

The spectrum of a Schrödinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m, we consider the problem of maximizing the gap-to-midgap ratio for the mth spectral gap over the class of potentials which have fixed periodicity and are pointwise bounded above and below. We prove that the potential maximizing the mth gap-to-midgap ratio exists. In one dimension, we prove that the optimal potential attains the pointwise bounds almost everywhere in the domain and is a step-function attaining the imposed minimum and maximum values on exactly m intervals. Optimal potentials are computed numerically using a rearrangement algorithm and are observed to be periodic. In two dimensions, we develop an efficient rearrangement method for this problem based on a semi-definite formulation and apply it to study properties of extremal potentials. We show that, provided a geometric assumption about the maximizer holds, a lattice of disks maximizes the first gap-to-midgap ratio in the infinite contrast limit. Using an explicit parametrization of two-dimensional Bravais lattices, we also consider how the optimal value varies over all equal-volume lattices.