Significance Regression: Robust Regression for Collinear Data

This paper examines robust linear multivariable regression from collinear data. A brief review of M-estimators discusses the strengths of this approach for tolerating outliers and/or perturbations in the error distributions. The review reveals that M-estimation may be unreliable if the data exhibit collinearity. Next, significance regression (SR) is discussed. SR is a successful method for treating collinearity but is not robust. A new significance regression algorithm for the weighted-least-squares error criterion (SR-WLS) is developed. Using the weights computed via M-estimation with the SR-WLS algorithm yields an effective method that robustly mollifies collinearity problems. Numerical examples illustrate the main points

Experimentally exploring compressed sensing quantum tomography

In the light of the progress in quantum technologies, the task of verifying the correct functioning of processes and obtaining accurate tomographic information about quantum states becomes increasingly important. Compressed sensing, a machinery derived from the theory of signal processing, has emerged as a feasible tool to perform robust and significantly more resource-economical quantum state tomography for intermediate-sized quantum systems. In this work, we provide a comprehensive analysis of compressed sensing tomography in the regime in which tomographically complete data is available with reliable statistics from experimental observations of a multi-mode photonic architecture. Due to the fact that the data is known with high statistical significance, we are in a position to systematically explore the quality of reconstruction depending on the number of employed measurement settings, randomly selected from the complete set of data, and on different model assumptions. We present and test a complete prescription to perform efficient compressed sensing and are able to reliably use notions of model selection and cross-validation to account for experimental imperfections and finite counting statistics. Thus, we establish compressed sensing as an effective tool for quantum state tomography, specifically suited for photonic systems.Comment: 12 pages, 5 figure

Scalable reconstruction of density matrices

Recent contributions in the field of quantum state tomography have shown that, despite the exponential growth of Hilbert space with the number of subsystems, tomography of one-dimensional quantum systems may still be performed efficiently by tailored reconstruction schemes. Here, we discuss a scalable method to reconstruct mixed states that are well approximated by matrix product operators. The reconstruction scheme only requires local information about the state, giving rise to a reconstruction technique that is scalable in the system size. It is based on a constructive proof that generic matrix product operators are fully determined by their local reductions. We discuss applications of this scheme for simulated data and experimental data obtained in an ion trap experiment.Comment: 9 pages, 5 figures, replaced with published versio

Robust Principal Component Analysis?

This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the L1 norm. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. We discuss an algorithm for solving this optimization problem, and present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces

Electronic Spectra from TDDFT and Machine Learning in Chemical Space

Due to its favorable computational efficiency time-dependent (TD) density functional theory (DFT) enables the prediction of electronic spectra in a high-throughput manner across chemical space. Its predictions, however, can be quite inaccurate. We resolve this issue with machine learning models trained on deviations of reference second-order approximate coupled-cluster singles and doubles (CC2) spectra from TDDFT counterparts, or even from DFT gap. We applied this approach to low-lying singlet-singlet vertical electronic spectra of over 20 thousand synthetically feasible small organic molecules with up to eight CONF atoms. The prediction errors decay monotonously as a function of training set size. For a training set of 10 thousand molecules, CC2 excitation energies can be reproduced to within $\pm$0.1 eV for the remaining molecules. Analysis of our spectral database via chromophore counting suggests that even higher accuracies can be achieved. Based on the evidence collected, we discuss open challenges associated with data-driven modeling of high-lying spectra, and transition intensities