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Register a new userA Data-driven Approach to X-ray Spectral Fitting: Quasi-Deconvolution
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X-ray spectral fitting of astronomical sources requires convolving the intrinsic spectrum or model with the instrumental response. Standard forward modeling techniques have proven success in recovering the underlying physical parameters in moderate to high signal-to-noise regimes; however, they struggle to achieve the same level of accuracy in low signal-to-noise regimes. Additionally, the use of machine learning techniques on X-ray spectra requires access to the intrinsic spectrum. Therefore, the measured spectrum must be effectively deconvolved from the instrumental response. In this note, we explore numerical methods for inverting the matrix equation describing X-ray spectral convolution. We demonstrate that traditional methods are insufficient to recover the intrinsic X-ray spectrum and argue that a novel approach is required.
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The new generation of radio interferometers is characterized by high sensitivity, wide fields of view and large fractional bandwidth. To synthesize the deepest images enabled by the high dynamic range of these instruments requires us to take into account the direction-dependent Jones matrices, while estimating the spectral properties of the sky in the imaging and deconvolution algorithms. In this paper we discuss and implement a wide-band wide-field spectral deconvolution framework (DDFacet) based on image plane faceting, that takes into account generic direction-dependent effects. Specifically, we present a wide-field co-planar faceting scheme, and discuss the various effects that need to be taken into account to solve for the deconvolution problem (image plane normalization, position-dependent PSF, etc). We discuss two wide-band spectral deconvolution algorithms based on hybrid matching pursuit and sub-space optimisation respectively. A few interesting technical features incorporated in our imager are discussed, including baseline dependent averaging, which has the effect of improving computing efficiency. The version of DDFacet presented here can account for any externally defined Jones matrices and/or beam patterns.
Inspired by the quantum computing algorithms for Linear Algebra problems [HHL,TaShma] we study how the simulation on a classical computer of this type of Phase Estimation algorithms performs when we apply it to solve the Eigen-Problem of Hermitian matrices. The result is a completely new, efficient and stable, parallel algorithm to compute an approximate spectral decomposition of any Hermitian matrix. The algorithm can be implemented by Boolean circuits in $O(log^2 n)$ parallel time with a total cost of $O(n^{omega+1})$ Boolean operations. This Boolean complexity matches the best known rigorous $O(log^2 n)$ parallel time algorithms, but unlike those algorithms our algorithm is (logarithmically) stable, so further improvements may lead to practical implementations. All previous efficient and rigorous approaches to solve the Eigen-Problem use randomization to avoid bad condition as we do too. Our algorithm makes further use of randomization in a completely new way, taking random powers of a unitary matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian perturbation and a random polynomial power are sufficient to ensure almost pairwise independence of the phases $(mod (2pi))$ is the main technical contribution of this work. This randomization enables us, given a Hermitian matrix with well separated eigenvalues, to sample a random eigenvalue and produce an approximate eigenvector in $O(log^2 n)$ parallel time and $O(n^omega)$ Boolean complexity. We conjecture that further improvements of our method can provide a stable solution to the full approximate spectral decomposition problem with complexity similar to the complexity (up to a logarithmic factor) of sampling a single eigenvector.
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