On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants

Randomized trace estimation is a popular and well-studied technique that approximates the trace of a large-scale matrix B by computing the average of $$x^T Bx$$ x T B x for many samples of a random vector X. Often, B is symmetric positive definite (SPD) but a number of applications give rise to indefinite B. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite B with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite B, reducing the number of required samples by a factor n or even more, where n is the size of B. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.

Time-dependent selected reaction monitoring-based GC-MS/MS method for estimation of genotoxic impurities in new antibacterial agent: alalevonadifloxacin mesylate

Alalevonadifloxacin mesylate (ALA), pro-drug of levonadifloxacin is a new antibiotic approved in India to treat infections caused by Gram-positive bacteria. Alkyl mesylates (AMs) are known genotoxic impurities (GTI’s) formed in drug substances isolated as mesylate salts. Time-dependent selected reaction monitoring (t-SRM)-based gas chromatography tandem mass spectrometry (GC-MS/MS) method has been developed for trace estimation of AMs, namely, methyl methane sulfonate (MMS), ethyl methane sulfonate (EMS) and isopropyl methane sulfonate (IMS) in ALA. Liquid-liquid extraction (LLE) procedure using dichloromethane (DCM) as an extracting solvent was employed to extract AMs from the drug substance. Automatic selective reaction monitoring (auto-SRM) tool was applied to identify the most intense SRM pair of the ions to achieve the highest sensitivity. The method was validated in terms of specificity, linearity, sensitivity, precision, and accuracy. The limit of quantitation (LOQ) for the MMS, EMS and IMS were 5, 10, and 20 ng/g of ALA, respectively. For all analytes, the correlation coefficient (R) were greater than 0.9975 in the concentration range of 3.0–260 ng/mL. Mean recovery of all analytes was in the range of 91.77 to 97.65%.

2D Deconvolution Using Adaptive Kernel

An analysis tool using Adaptive Kernel to solve an ill-posed inverse problem for a 2D model space is introduced. It is applicable for linear and non-linear forward models, for example in tomography and image reconstruction. While an optimisation based on a Gaussian Approximation is possible, it becomes intractable for more than some hundred kernel functions. This is because the determinant of the Hessian of the system has be evaluated. The SVD typically used for 1D problems fails with increasing problem size. Alternatively Stochastic Trace Estimation can be used, giving a reasonable approximation. An alternative to searching for the MAP solution is to integrate using Marcov Chain Monte Carlo without the need to determine the determinant of the Hessian. This also allows to treat problems where a linear approximation is not justified.