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We compare the properties of the stable rank and intrinsic dimension of real and complex matrices to those of the classical rank. Basic proofs and examples illustrate that the stable rank does not satisfy any of the fundamental rank properties, while the intrinsic dimension satisfies a few. In particular, the stable rank and intrinsic dimension of a submatrix can exceed those of the original matrix; adding a Hermitian positive semi-definite matrix can lower the intrinsic dimension of the sum; and multiplication by a nonsingular matrix can drastically change the stable rank and the intrinsic dimension. We generalize the concept of stable rank to the p-stable in a Schatten p-norm, thereby unifying the concepts of stable rank and intrinsic dimension: The stable rank is the 2-stable rank, while the intrinsic dimension is the 1-stable rank of a Hermitian positive semi-definite matrix. We derive sum and product inequalities for the pth root of the p-stable rank, and show that it is well-conditioned in the norm-wise absolute sense. The conditioning improves if the matrix and the perturbation are Hermitian positive semi-definite.

Computing low-rank approximations of kernel matrices is an important problem with many applications in scientific computing and data science. We propose methods to efficiently approximate and store low-rank approximations to kernel matrices that depend on certain hyperparameters. The main idea behind our method is to use multivariate Chebyshev function approximation along with the tensor train decomposition of the coefficient tensor. The computations are in two stages: an offline stage, which dominates the computational cost and is parameter-independent, and an online stage, which is inexpensive and instantiated for specific hyperparameters. A variation of this method addresses the case that the kernel matrix is symmetric and positive semi-definite. The resulting algorithms have linear complexity in terms of the sizes of the kernel matrices. We investigate the efficiency and accuracy of our method on parametric kernel matrices induced by various kernels, such as the Mat\'ern kernel, through various numerical experiments. Our methods have speedups up to $200\times$ in the online time compared to other methods with similar complexity and comparable accuracy., Comment: 29 pages, 3 tables, 2 figures

This paper tackles optimal sensor placement for Bayesian linear inverse problems, a popular version of the more general Optimal Experiment Design (OED) problem, using the D-optimality criterion. This is done by establishing connections between sensor placement and Column Subset Selection Problem (CSSP), which is a well-studied problem in Numerical Linear Algebra (NLA). In particular, we use the Golub-Klema-Stewart (GKS) approach which involves computing the truncated Singular Value Decomposition (SVD) followed by a pivoted QR factorization on the right singular vectors. The algorithms are further accelerated by using randomization to compute the low-rank approximation as well as for sampling the indices. The resulting algorithms are robust, computationally efficient, amenable to parallelization, require virtually no parameter tuning, and come with strong theoretical guarantees. One of the proposed algorithms is also adjoint-free which is beneficial in situations, where the adjoint is expensive to evaluate or is not available. Additionally, we develop a method for data completion without solving the inverse problem. Numerical experiments on model inverse problems involving the heat equation and seismic tomography in two spatial dimensions demonstrate the performance of our approaches., Comment: 31 pages, 9 figures

Strong Constraint 4D Variational (SC-4DVAR) is a data assimilation method that is widely used in climate and weather applications. SC-4DVAR involves solving a minimization problem to compute the maximum a posteriori estimate, which we tackle using the Gauss-Newton method. The computation of the descent direction is expensive since it involves the solution of a large-scale and potentially ill-conditioned linear system, solved using the preconditioned conjugate gradient (PCG) method. To address this cost, we efficiently construct scalable preconditioners using three different randomization techniques, which all rely on a certain low-rank structure involving the Gauss-Newton Hessian. The proposed techniques come with theoretical (probabilistic) guarantees on the condition number, and at the same time, are amenable to parallelization. We also develop an adaptive approach to estimate the sketch size and to choose between the reuse or recomputation of the preconditioner. We demonstrate the performance and effectiveness of our methodology on two representative model problems -- the Burgers and barotropic vorticity equation -- showing a drastic reduction in both the number of PCG iterations and the number of Gauss-Newton Hessian products (after including the preconditioner construction cost)., Comment: 28 pages