In an increasing number of applications, it is of interest to recover an approximately low-rank data matrix from noisy observations. This paper develops an unbiased risk estimate-holding in a Gaussian model-for any spectral estimator obeying some mild regularity assumptions. In particular, we give an unbiased risk estimate formula for singular value thresholding (SVT), a popular estimation strategy which applies a soft-thresholding rule to the singular values of the noisy observations. Among other things, our formulas offer a principled and automated way of selecting regularization parameters in a variety of problems. In particular, we demonstrate the utility of the unbiased risk estimation for SVT-based denoising of real clinical cardiac MRI series data. We also give new results concerning the differentiability of certain matrix-valued functions.

In compressed sensing, one takes n < N samples of an N-dimensional vector x 0 using an n × N matrix A, obtaining undersampled measurements y = Ax 0 . For random matrices with independent standard Gaussian entries, it is known that, when x 0 is k-sparse, there is a precisely determined phase transition: for a certain region in the (k/n,n/N)-phase diagram, convex optimization min || x || 1 subject to y = Ax, x ∈ X N typically finds the sparsest solution, whereas outside that region, it typically fails. It has been shown empirically that the same property-with the same phase transition location-holds for a wide range of non-Gaussian random matrix ensembles. We report extensive experiments showing that the Gaussian phase transition also describes numerous deterministic matrices, including Spikes and Sines, Spikes and Noiselets, Paley Frames, Delsarte-Goethals Frames, Chirp Sensing Matrices, and Grassmannian Frames. Namely, for each of these deterministic matrices in turn, for a typical k-sparse object, we observe that convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian random matrices. Our experiments considered coefficients constrained to X N for four different sets X ∈ {[0, 1], R + , R, C}, and the results establish our finding for each of the four associated phase transitions.sparse recovery | universality in random matrix theory equiangular tight frames | restricted isometry property | coherence C ompressed sensing aims to recover a sparse vector x 0 ∈ X N from indirect measurements y = Ax 0 ∈ X n with n < N, and therefore, the system of equations y = Ax 0 is underdetermined. Nevertheless, it has been shown that, under conditions on the sparsity of x 0 , by using a random measurement matrix A with Gaussian i.i.d entries and a nonlinear reconstruction technique based on convex optimization, one can, with high probability, exactly recover x 0 (1, 2). The cleanest expression of this phenomenon is visible in the large n; N asymptotic regime. We suppose that the object x 0 is k-sparse-has, at most, k nonzero entries-and consider the situation where k ∼ ρn and n ∼ δN. Fig. 1A depicts the phase diagram ðρ; δ; Þ ∈ ð0; 1Þ 2 and a curve ρ*ðδÞ separating a success phase from a failure phase. Namely, if ρ < ρ*ðδÞ, then with overwhelming probability for large N, convex optimization will recover x 0 exactly; however, if ρ > ρ*ðδÞ, then with overwhelming probability for large N convex optimization will fail. [Indeed, Fig. 1 depicts four curves ρ*ðδjXÞ of this kind for X ∈ f½0; 1; R + ; R; Cg-one for each of the different types of assumptions that we can make about the entries of x 0 ∈ X N (details below).]How special are Gaussian matrices to the above results? It was shown, first empirically in ref. 3 and recently, theoretically in ref. 4, that a wide range of random matrix ensembles exhibits precisely the same behavior, by which we mean the same phenomenon of separation into success and failure phases with the same phase boundary. Such universality, if exhib...

Purpose Quantitative T1, T2, T2*, and fat fraction (FF) maps are promising imaging biomarkers for the assessment of liver disease, however these are usually acquired in sequential scans. Here we propose an extended MR fingerprinting (MRF) framework enabling simultaneous liver T1, T2, T2*, and FF mapping from a single ~14 s breath‐hold scan. Methods A gradient echo (GRE) liver MRF sequence with nine readouts per TR, low flip angles (5‐15°), varying magnetisation preparation and golden angle radial trajectory is acquired at 1.5T to encode T1, T2, T2*, and FF simultaneously. The nine‐echo time‐series are reconstructed using a low‐rank tensor constrained reconstruction and used to fit T2*, B0 and to separate the water and fat signals. Water‐ and fat‐specific T1, T2, and M0 are obtained through dictionary matching, whereas FF estimation is extracted from the M0 maps. The framework was evaluated in a standardized T1/T2 phantom, a water‐fat phantom, and 12 subjects in comparison to reference methods. Preliminary clinical feasibility is shown in four patients. Results The proposed water T1, water T2, T2*, and FF maps in phantoms showed high coefficients of determination (r2 > 0.97) relative to reference methods. Measured liver MRF values in vivo (mean ± SD) for T1, T2, T2*, and FF were 671 ± 60 ms, 43.2 ± 6.8 ms, 29 ± 6.6 ms, and 3.2 ± 2.6% with biases of 92 ms, −7.1 ms, −1.4 ms, and 0.63% when compared to conventional methods. Conclusion A nine‐echo liver MRF sequence allows for quantitative multi‐parametric liver tissue characterization in a single breath‐hold scan of ~14 s. Future work will aim to validate the proposed approach in patients with liver disease.

We propose a novel statistical learning framework for automatically and efficiently building reduced kinetic Monte Carlo (KMC) models of large-scale elementary reaction networks from data generated by a single or few molecular dynamics simulations (MD).

We developed a novel methodology to decompose the WSS vector in WSS and WSS in 3D domains, using 4D flow MRI data. Our method provides a more robust quantification of WSS and WSS in comparison with other reported methods. Magn Reson Med 79:2816-2823, 2018. © 2017 International Society for Magnetic Resonance in Medicine.