The Eigenspace Spectral Regularization Method for Solving Discrete Ill-Posed Systems

This paper shows that discrete linear equations with Hilbert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, TST matrix operator, and sparse matrix operator are ill-posed in the sense of Hadamard. Gauss least square method (GLSM), QR factorization method (QRFM), Cholesky decomposition method (CDM), and singular value decomposition (SVDM) failed to regularize these ill-posed problems. This paper introduces the eigenspace spectral regularization method (ESRM), which solves ill-posed discrete equations with Hilbert matrix operator, circulant matrix operator, conference matrix operator, and banded and sparse matrix operator. Unlike GLSM, QRFM, CDM, and SVDM, the ESRM regularizes such a system. In addition, the ESRM has a unique property, the norm of the eigenspace spectral matrix operator κ K = K − 1 K = 1 . Thus, the condition number of ESRM is bounded by unity, unlike the other regularization methods such as SVDM, GLSM, CDM, and QRFM.

Epistatic Net allows the sparse spectral regularization of deep neural networks for inferring fitness functions

Despite recent advances in high-throughput combinatorial mutagenesis assays, the number of labeled sequences available to predict molecular functions has remained small for the vastness of the sequence space combined with the ruggedness of many fitness functions. While deep neural networks (DNNs) can capture high-order epistatic interactions among the mutational sites, they tend to overfit to the small number of labeled sequences available for training. Here, we developed Epistatic Net (EN), a method for spectral regularization of DNNs that exploits evidence that epistatic interactions in many fitness functions are sparse. We built a scalable extension of EN, usable for larger sequences, which enables spectral regularization using fast sparse recovery algorithms informed by coding theory. Results on several biological landscapes show that EN consistently improves the prediction accuracy of DNNs and enables them to outperform competing models which assume other priors. EN estimates the higher-order epistatic interactions of DNNs trained on massive sequence spaces-a computational problem that otherwise takes years to solve.

A Photogrammetric-Photometric Stereo Method for High-Resolution Lunar Topographic Mapping Using Yutu-2 Rover Images

Topographic products are important for mission operations and scientific research in lunar exploration. In a lunar rover mission, high-resolution digital elevation models are typically generated at waypoints by photogrammetry methods based on rover stereo images acquired by stereo cameras. In case stereo images are not available, the stereo-photogrammetric method will not be applicable. Alternatively, photometric stereo method can recover topographic information with pixel-level resolution from three or more images, which are acquired by one camera under the same viewing geometry with different illumination conditions. In this research, we extend the concept of photometric stereo to photogrammetric-photometric stereo by incorporating collinearity equations into imaging irradiance model. The proposed photogrammetric-photometric stereo algorithm for surface construction involves three steps. First, the terrain normal vector in object space is derived from collinearity equations, and image irradiance equation for close-range topographic mapping is determined. Second, based on image irradiance equations of multiple images, the height gradients in image space can be solved. Finally, the height map is reconstructed through global least-squares surface reconstruction with spectral regularization. Experiments were carried out using simulated lunar rover images and actual lunar rover images acquired by Yutu-2 rover of Chang’e-4 mission. The results indicate that the proposed method achieves high-resolution and high-precision surface reconstruction, and outperforms the traditional photometric stereo methods. The proposed method is valuable for ground-based lunar surface reconstruction and can be applicable to surface reconstruction of Earth and other planets.

Divergence-free quantum electrodynamics in locally conformally flat space–time

This paper describes an approach to quantum electrodynamics (QED) in curved space–time obtained by considering infinite-dimensional algebra bundles associated to a natural principal bundle [Formula: see text] associated with any locally conformally flat space–time, with typical fibers including the Fock space and a space of fermionic multiparticle states which forms a Grassmann algebra. Both these algebras are direct sums of generalized Hilbert spaces. The requirement of [Formula: see text] covariance associated with the geometry of space–time, where [Formula: see text] is the structure group of [Formula: see text], leads to the consideration of [Formula: see text] intertwining operators between various spaces. Scattering processes are associated with such operators and are encoded in an algebra of kernels. Intertwining kernels can be generated using [Formula: see text] covariant matrix-valued measures. Feynman propagators, fermion loops and the electron self-energy can be given well-defined interpretations as such measures. Divergence-free calculations in QED can be carried out by computing the spectra of these measures and kernels (a process called spectral regularization). As an example of the approach the precise Uehling potential function for the [Formula: see text] atom is calculated without requiring renormalization from which the Uehling contribution to the Lamb shift can be calculated exactly.

Construction and Monte Carlo Estimation of Wavelet Frames Generated by a Reproducing Kernel

We introduce a construction of multiscale tight frames on general domains. The frame elements are obtained by spectral filtering of the integral operator associated with a reproducing kernel. Our construction extends classical wavelets as well as generalized wavelets on both continuous and discrete non-Euclidean structures such as Riemannian manifolds and weighted graphs. Moreover, it allows to study the relation between continuous and discrete frames in a random sampling regime, where discrete frames can be seen as Monte Carlo estimates of the continuous ones. Pairing spectral regularization with learning theory, we show that a sample frame tends to its population counterpart, and derive explicit finite-sample rates on spaces of Sobolev and Besov regularity. Our results prove the stability of frames constructed on empirical data, in the sense that all stochastic discretizations have the same underlying limit regardless of the set of initial training samples.

The Eigenspace Spectral Regularization Method for solving Discrete Ill-Posed Systems

In this paper, it is shown that discrete equations with Hilb ert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, and sparse matrix operator are ill-posed in the sense of Hadamard. These ill-posed problems cannot be regularized by Gauss Least Square Method (GLSM), QR Factorization Method (QRFM), Cholesky Decomposition Method (CDM) and Singular Value Decomposition (SVDM). To overcome the limitations of these methods of regularization, an Eigenspace Spectral Regularization Method (ESRM) is introduced which solves ill-p os ed discrete equations with Hilb ert matrix operator, circulant matrix operator, conference matrix operator, banded matrix operator, and sparse matrix operator. Unlike GLSM, QRFM, CDM, and SVDM, the ESRM regularize such a system. In addition, the ESRM has a unique property, the norm of the eigenspace spectral matrix operator κ (K) = ||K − 1K|| = 1. Thus, the condition number of ESRM is bounded by unity unlike the other regularization methods such as SVDM, GLSM, CDM, and QRFM.

SAR Target Classification Based on Sample Spectral Regularization

Synthetic Aperture Radar (SAR) target classification is an important branch of SAR image interpretation. The deep learning based SAR target classification algorithms have made remarkable achievements. But the acquisition and annotation of SAR target images are time-consuming and laborious, and it is difficult to obtain sufficient training data in many cases. The insufficient training data can make deep learning based models suffering from over-fitting, which will severely limit their wide application in SAR target classification. Motivated by the above problem, this paper employs transfer-learning to transfer the prior knowledge learned from a simulated SAR dataset to a real SAR dataset. To overcome the sample restriction problem caused by the poor feature discriminability for real SAR data. A simple and effective sample spectral regularization method is proposed, which can regularize the singular values of each SAR image feature to improve the feature discriminability. Based on the proposed regularization method, we design a transfer-learning pipeline to leverage the simulated SAR data as well as acquire better feature discriminability. The experimental results indicate that the proposed method is feasible for the sample restriction problem in SAR target classification. Furthermore, the proposed method can improve the classification accuracy when relatively sufficient training data is available, and it can be plugged into any convolutional neural network (CNN) based SAR classification models.