We propose a new algorithm for calculation of vibrational spectra of molecules using tensor train decomposition. Under the assumption that eigenfunctions lie on a low-parametric manifold of low-rank tensors we suggest using well-known iterative methods that utilize matrix inversion (locally optimal block preconditioned conjugate gradient method, inverse iteration) and solve corresponding linear systems inexactly along this manifold. As an application, we accurately compute vibrational spectra (84 states) of acetonitrile molecule CH3CN on a laptop in one hour using only 100 MB of memory to represent all computed eigenfunctions.
Alternatively to iterative methods, one can pose an optimization problem -- minimization of the Rayleigh quotient with the constraint on rank. This approach was recently considered in the matrix product state (MPS) community @cite_24 and independently in @cite_2 . The only disadvantages is that all eigenfunctions are treated in one basis, which leads to large ranks and the method becomes slow (calculation of the acetonitrile took several days). Nevertheless, this approach becomes very efficient and accurate when small number of eigenpairs are required.
@cite_24 @cite_2
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Minimization with orthogonality constraints (e.g., (X^ X = I )) and or spherical constraints (e.g., ( x _2 = 1 )) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with these difficulties, we apply the Cayley transform—a Crank-Nicolson-like update scheme—to preserve the constraints and based on it, develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics. The efficiency of the proposed algorithms is demonstrated on a variety of test problems. In particular, for the maxcut problem, it exactly solves a decomposition formulation for the SDP relaxation. For polynomial optimization, nearest correlation matrix estimation and extreme eigenvalue problems, the proposed algorithms run very fast and return solutions no worse than those from their state-of-the-art algorithms. For the quadratic assignment problem, a gap 0.842 to the best known solution on the largest problem “tai256c” in QAPLIB can be reached in 5 min on a typical laptop. The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast as a semidefinite programming problem, such an approach is computationally expensive to solve when the matrices are large. In this paper, we propose fixed point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 x 1000 matrices of rank 50 with a relative error of 1e-5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms. This paper concerns accurate computation of the singular value decomposition (SVD) of an (m n ) matrix (A ). As is well known, cross-product matrix based SVD algorithms compute large singular values accurately but generally deliver poor small singular values. A new novel cross-product matrix based SVD method is proposed: (a) Use a backward stable algorithm to compute the eigenpairs of (A^ T A ) and take the square roots of the large eigenvalues of it as the large singular values of (A ) ; (b) form the Rayleigh quotient of (A^ T A ) with respect to the matrix consisting of the computed eigenvectors associated with the computed small eigenvalues of (A^ T A ) ; (c) compute the eigenvalues of the Rayleigh quotient and take the square roots of them as the small singular values of (A ). A detailed quantitative error analysis is conducted on the method. It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff ( ). An algorithm is developed that is not only cheaper than the standard Golub–Reinsch and Chan SVD algorithms but also can update or downdate a new SVD by adding or deleting a row and compute certain refined Ritz vectors for large matrix eigenproblems at very low cost. Several variants of the algorithm are proposed that compute some or all parts of the SVD. Typical numerical examples confirm the high accuracy of our algorithm. The computation of a few singular triplets of large, sparse matrices is a challenging task, especially when the smallest magnitude singular values are needed in high accuracy. Most recent efforts try to address this problem through variations of the Lanczos bidiagonalization method, but they are still challenged even for medium matrix sizes due to the difficulty of the problem. We propose a novel SVD approach that can take advantage of preconditioning and of any well-designed eigensolver to compute both largest and smallest singular triplets. Accuracy and efficiency is achieved through a hybrid, two-stage meta-method, PHSVDS. In the first stage, PHSVDS solves the normal equations up to the best achievable accuracy. If further accuracy is required, the method switches automatically to an eigenvalue problem with the augmented matrix. Thus it combines the advantages of the two stages, faster convergence and accuracy, respectively. For the augmented matrix, solving the interior eigenvalue is facilitated by pr...