Andrew Knyazev

Andrew Knyazev, Abram Jujunashvili, Merico Argentati

We define angles from-to and between infinite dimensional subspaces of a Hilbert space, inspired by the work of E. J. Hannan, 1961/1962 for general canonical correlations of stochastic processes. The spectral theory of selfadjoint operators is used to investigate the properties of the angles, e.g., to establish connections between the angles corresponding to orthogonal complements. The classical gaps and angles of Dixmier and Friedrichs are characterized in terms of the angles. We introduce principal invariant subspaces and prove that they are connected by an isometry that appears in the polar decomposition of the product of corresponding orthogonal projectors. Point angles are defined by analogy with the point operator spectrum. We bound the Hausdorff distance between the sets of the squared cosines of the angles corresponding to the original subspaces and their perturbations. We show that the squared cosines of the angles from one subspace to another can be interpreted as Ritz values in the Rayleigh-Ritz method, where the former subspace serves as a trial subspace and the orthogonal projector of the latter subspace serves as an operator in the Rayleigh-Ritz method. The Hausdorff distance between the Ritz values, corresponding to different trial subspaces, is shown to be bounded by a constant times the gap between the trial subspaces. We prove a similar eigenvalue perturbation bound that involves the gap squared. Finally, we consider the classical alternating projectors method and propose its ultimate acceleration, using the conjugate gradient approach. The corresponding convergence rate estimate is obtained in terms of the angles. We illustrate a possible acceleration for the domain decomposition method with a small overlap for the 1D diffusion equation.

Andrew Knyazev, Peizhen Zhu

The absolute change in the Rayleigh quotient (RQ) for a Hermitian matrix with respect to vectors is bounded in terms of the norms of the residual vectors and the angle between vectors in [\doi{10.1137/120884468}]. We substitute multidimensional subspaces for the vectors and derive new bounds of absolute changes of eigenvalues of the matrix RQ in terms of singular values of residual matrices and principal angles between subspaces, using majorization. We show how our results relate to bounds for eigenvalues after discarding off-diagonal blocks or additive perturbations.

Andrew Knyazev, Alexander Malyshev

We propose fast O(N) preconditioning, where N is the number of gridpoints on the prediction horizon, for iterative solution of (non)-linear systems appearing in model predictive control methods such as forward-difference Newton-Krylov methods. The Continuation/GMRES method for nonlinear model predictive control, suggested by T. Ohtsuka in 2004, is a specific application of the Newton-Krylov method, which uses the GMRES iterative algorithm to solve a forward difference approximation of the optimality equations on every time step.

David Zhuzhunashvili, Andrew Knyazev

Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is demonstrated to efficiently solve eigenvalue problems for graph Laplacians that appear in spectral clustering. For static graph partitioning, 10-20 iterations of LOBPCG without preconditioning result in ~10x error reduction, enough to achieve 100% correctness for all Challenge datasets with known truth partitions, e.g., for graphs with 5K/.1M (50K/1M) Vertices/Edges in 2 (7) seconds, compared to over 5,000 (30,000) seconds needed by the baseline Python code. Our Python code 100% correctly determines 98 (160) clusters from the Challenge static graphs with 0.5M (2M) vertices in 270 (1,700) seconds using 10GB (50GB) of memory. Our single-precision MATLAB code calculates the same clusters at half time and memory. For streaming graph partitioning, LOBPCG is initiated with approximate eigenvectors of the graph Laplacian already computed for the previous graph, in many cases reducing 2-3 times the number of required LOBPCG iterations, compared to the static case. Our spectral clustering is generic, i.e. assuming nothing specific of the block model or streaming, used to generate the graphs for the Challenge, in contrast to the base code. Nevertheless, in 10-stage streaming comparison with the base code for the 5K graph, the quality of our clusters is similar or better starting at stage 4 (7) for emerging edging (snowballing) streaming, while the computations are over 100-1000 faster.

Andrew Knyazev

Since introduction [A. Knyazev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method, SISC (2001) DOI:10.1137/S1064827500366124] and efficient parallel implementation [A. Knyazev et al., Block locally optimal preconditioned eigenvalue xolvers (BLOPEX) in HYPRE and PETSc, SISC (2007) DOI:10.1137/060661624], LOBPCG has been used is a wide range of applications in mechanics, material sciences, and data sciences. We review its recent implementations and applications, as well as extensions of the local optimality idea beyond standard eigenvalue problems.

Andrew Knyazev, Alexander Malyshev

Signal reconstruction from a sample using an orthogonal projector onto a guiding subspace is theoretically well justified, but may be difficult to practically implement. We propose more general guiding operators, which increase signal components in the guiding subspace relative to those in a complementary subspace, e.g., iterative low-pass edge-preserving filters for super-resolution of images. Two examples of super-resolution illustrate our technology: a no-flash RGB photo guided using a high resolution flash RGB photo, and a depth image guided using a high resolution RGB photo.

Andrew Knyazev, Akshay Gadde, Hassan Mansour et al.

An axiomatic approach to signal reconstruction is formulated, involving a sample consistent set and a guiding set, describing desired reconstructions. New frame-less reconstruction methods are proposed, based on a novel concept of a reconstruction set, defined as a shortest pathway between the sample consistent set and the guiding set. Existence and uniqueness of the reconstruction set are investigated in a Hilbert space, where the guiding set is a closed subspace and the sample consistent set is a closed plane, formed by a sampling subspace. Connections to earlier known consistent, generalized, and regularized reconstructions are clarified. New stability and reconstruction error bounds are derived, using the largest nontrivial angle between the sampling and guiding subspaces. Conjugate gradient iterative reconstruction algorithms are proposed and illustrated numerically for image magnification.

Eugene Vecharynski, Andrew Knyazev

This paper addresses the question of what exactly is an analogue of the preconditioned steepest descent (PSD) algorithm in the case of a symmetric indefinite system with an SPD preconditioner. We show that a basic PSD-like scheme for an SPD-preconditioned symmetric indefinite system is mathematically equivalent to the restarted PMINRES, where restarts occur after every two steps. A convergence bound is derived. If certain information on the spectrum of the preconditioned system is available, we present a simpler PSD-like algorithm that performs only one-dimensional residual minimization. Our primary goal is to bridge the theoretical gap between optimal (PMINRES) and PSD-like methods for solving symmetric indefinite systems, as well as point out situations where the PSD-like schemes can be used in practice.

Andrew Knyazev, Alexander Malyshev

Denoising filters, such as bilateral, guided, and total variation filters, applied to images on general graphs may require repeated application if noise is not small enough. We formulate two acceleration techniques of the resulted iterations: conjugate gradient method and Nesterov's acceleration. We numerically show efficiency of the accelerated nonlinear filters for image denoising and demonstrate 2-12 times speed-up, i.e., the acceleration techniques reduce the number of iterations required to reach a given peak signal-to-noise ratio (PSNR) by the above indicated factor of 2-12.

Andrew Knyazev

In [DOI:10.1109/ICMEW.2014.6890711], a graph-based denoising is performed by projecting the noisy image to a lower dimensional Krylov subspace of the graph Laplacian, constructed using nonnegative weights determined by distances between image data corresponding to image pixels. We~extend the construction of the graph Laplacian to the case, where some graph weights can be negative. Removing the positivity constraint provides a more accurate inference of a graph model behind the data, and thus can improve quality of filters for graph-based signal processing, e.g., denoising, compared to the standard construction, without affecting the costs.

Andrew Knyazev, Alexander Malyshev

Graph-based spectral denoising is a low-pass filtering using the eigendecomposition of the graph Laplacian matrix of a noisy signal. Polynomial filtering avoids costly computation of the eigendecomposition by projections onto suitable Krylov subspaces. Polynomial filters can be based, e.g., on the bilateral and guided filters. We propose constructing accelerated polynomial filters by running flexible Krylov subspace based linear and eigenvalue solvers such as the Block Locally Optimal Preconditioned Conjugate Gradient (LOBPCG) method.

Dong Tian, Hassan Mansour, Andrew Knyazev et al.

In 3D image/video acquisition, different views are often captured with varying noise levels across the views. In this paper, we propose a graph-based image enhancement technique that uses a higher quality view to enhance a degraded view. A depth map is utilized as auxiliary information to match the perspectives of the two views. Our method performs graph-based filtering of the noisy image by directly computing a projection of the image to be filtered onto a lower dimensional Krylov subspace of the graph Laplacian. We discuss two graph spectral denoising methods: first using Chebyshev polynomials, and second using iterations of the conjugate gradient algorithm. Our framework generalizes previously known polynomial graph filters, and we demonstrate through numerical simulations that our proposed technique produces subjectively cleaner images with about 1-3 dB improvement in PSNR over existing polynomial graph filters.

Andrew Knyazev, Alexander Malyshev

The most efficient signal edge-preserving smoothing filters, e.g., for denoising, are non-linear. Thus, their acceleration is challenging and is often performed in practice by tuning filter parameters, such as by increasing the width of the local smoothing neighborhood, resulting in more aggressive smoothing of a single sweep at the cost of increased edge blurring. We propose an alternative technology, accelerating the original filters without tuning, by running them through a special conjugate gradient method, not affecting their quality. The filter non-linearity is dealt with by careful freezing and restarting. Our initial numerical experiments on toy one-dimensional signals demonstrate 20x acceleration of the classical bilateral filter and 3-5x acceleration of the recently developed guided filter.