We show that the time-dependent variational principle provides a unifying framework for time-evolution methods and optimisation methods in the context of matrix product states. In particular, we introduce a new integration scheme for studying time-evolution, which can cope with arbitrary Hamiltonians, including those with long-range interactions. Rather than a Suzuki-Trotter splitting of the Hamiltonian, which is the idea behind the adaptive time-dependent density matrix renormalization group method or time-evolving block decimation, our method is based on splitting the projector onto the matrix product state tangent space as it appears in the Dirac-Frenkel time-dependent variational principle. We discuss how the resulting algorithm resembles the density matrix renormalization group (DMRG) algorithm for finding ground states so closely that it can be implemented by changing just a few lines of code and it inherits the same stability and efficiency. In particular, our method is compatible with any Hamiltonian for which DMRG can be implemented efficiently and DMRG is obtained as a special case of imaginary time evolution with infinite time step.Comment: 5 pages + 5 pages supplementary material (6 figures) (updated example, small corrections

The matrix completion problem consists of finding or approximating a low-rank matrix based on a few samples of this matrix. We propose a new algorithm for matrix completion that minimizes the least-square distance on the sampling set over the Riemannian manifold of fixed-rank matrices. The algorithm is an adaptation of classical non-linear conjugate gradients, developed within the framework of retraction-based optimization on manifolds. We describe all the necessary objects from differential geometry necessary to perform optimization over this lowrank matrix manifold, seen as a submanifold embedded in the space of matrices. In particular, we describe how metric projection can be used as retraction and how vector transport lets us obtain the conjugate search directions. Finally, we prove convergence of a regularized version of our algorithm under the assumption that the restricted isometry property holds for incoherent matrices throughout the iterations. The numerical experiments indicate that our approach scales very well for large-scale problems and compares favorably with the state-of-the-art, while outperforming most existing solvers.This report is the extended version of the manuscript [54]. It differs only by the addition of Appendix A.

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A robust and efficient time integrator for dynamical tensor approximation in the tensor train or matrix product state format is presented. The method is based on splitting the projector onto the tangent space of the tensor manifold. The algorithm can be used for updating time-dependent tensors in the given data-sparse tensor train / matrix product state format and for computing an approximate solution to high-dimensional tensor differential equations within this data-sparse format. The formulation, implementation and theoretical properties of the proposed integrator are studied, and numerical experiments with problems from quantum molecular dynamics and with iterative processes in the tensor train format are included.

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We extend results on the dynamical low-rank approximation for the treatment of time-dependent matrices and tensors (Koch and Lubich; see [SIAM to the recently proposed hierarchical Tucker (HT) tensor format (Hackbusch and Kühn; see [J. Fourier Anal. Appl., 15 (2009), pp. 706-722]) and the tensor train (TT) format (Oseledets; see [SIAM J. Sci. Comput., 33 (2011), pp. 2295-2317), which are closely related to tensor decomposition methods used in quantum physics and chemistry. In this dynamical approximation approach, the time derivative of the tensor to be approximated is projected onto the time-dependent tangent space of the approximation manifold along the solution trajectory. This approach can be used to approximate the solutions to tensor differential equations in the HT or TT format and to compute updates in optimization algorithms within these reduced tensor formats. By deriving and analyzing the tangent space projector for the manifold of HT/TT tensors of fixed rank, we obtain curvature estimates, which allow us to obtain quasi-best approximation properties for the dynamical approximation, showing that the prospects and limitations of the ansatz are similar to those of the dynamical low rank approximation for matrices. Our results are exemplified by numerical experiments. DYNAMICAL APPROXIMATION BY HT AND TT TENSORS 471embedded manifold M ⊆ V typically depending on much fewer parameters than the linear parametrization of V, the dynamical tensor approximation may be utilized: Assuming Y (0) ∈ M, an approximation Y (t) ∈ M is determined such that its derivative at every time t is the element of the tangent space T Y (t) M closest toȦ(t):In terms of the orthogonal projection P X onto the tangent space T X M at X ∈ M, the solution to (1.2) is equivalently characterized by projectingwhich results in a differential equation on the approximation manifold M.After an explicit parametrization of the manifold M under consideration, one obtains from (1.3) a set of nonlinear differential equations for the parameters of this parametrization, suitable for numerical integration. The above ansatz has been studied for manifolds of matrices of fixed rank k and tensors of fixed (multilinear) Tucker rank (k 1 , . . . , k d ) in [15,24,16], respectively. In the case of a tensor differential equation,Ȧ(t) = F (A(t)) is replaced by the approximate value F (Y (t)) on the right-hand side of (1.2). In the context of the time-dependent Schrödinger equation, the approach is known as the Dirac-Frenkel time-dependent variational principle [20].Classical tensor formats stemming from data analysis (that is, the canonical decomposition and the Tucker format [8,17]) exhibit certain structural weaknesses that make them unsuitable for the treatment of problems of the kind exemplified by (1.1) and (1.2) when the dimension d is large; see, e.g., the introduction in [11]. This motivated the development of recent tensor formats such as the HT format, in which a recursive, hierarchical construction of Tucker type is employed for tensor representati...

Dynamical low-rank approximation in the Tucker tensor format of given large timedependent tensors and of tensor differential equations is the subject of this paper. In particular, a discrete time integration method for rank-constrained Tucker tensors is presented and analyzed. It extends the known projector-splitting integrator for dynamical low-rank approximation of matrices to Tucker tensors and is shown to inherit the same favorable properties. The integrator is based on iteratively applying the matrix projector-splitting integrator to tensor unfoldings but with inexact solution in a substep. It has the property that it reconstructs time-dependent Tucker tensors of the given rank exactly. The integrator is also shown to be robust to the presence of small singular values in the tensor unfoldings. Numerical examples with problems from quantum dynamics and tensor optimization methods illustrate our theoretical results.

Circadian clocks play an important role in lipid homeostasis, with impact on various metabolic diseases. Due to the central role of skeletal muscle in whole-body metabolism, we aimed at studying muscle lipid profiles in a temporal manner. Moreover, it has not been shown whether lipid oscillations in peripheral tissues are driven by diurnal cycles of rest-activity and food intake or are able to persist in vitro in a cell-autonomous manner. To address this, we investigated lipid profiles over 24 h in human skeletal muscle in vivo and in primary human myotubes cultured in vitro. Glycerolipids, glycerophospholipids, and sphingolipids exhibited diurnal oscillations, suggesting a widespread circadian impact on muscle lipid metabolism. Notably, peak levels of lipid accumulation were in phase coherence with core clock gene expression in vivo and in vitro. The percentage of oscillating lipid metabolites was comparable between muscle tissue and cultured myotubes, and temporal lipid profiles correlated with transcript profiles of genes implicated in their biosynthesis. Lipids enriched in the outer leaflet of the plasma membrane oscillated in a highly coordinated manner in vivo and in vitro. Lipid metabolite oscillations were strongly attenuated upon siRNA-mediated clock disruption in human primary myotubes. Taken together, our data suggest an essential role for endogenous cell-autonomous human skeletal muscle oscillators in regulating lipid metabolism independent of external synchronizers, such as physical activity or food intake.lipid metabolism | circadian clock | human skeletal muscle | human primary myotubes | lipidomics C ircadian oscillations are daily cycles in behavior and physiology that are driven by the existence of underlying intrinsic biological clocks with near 24-h periods. This anticipatory mechanism has evolved to ensure that all aspects of behavior and physiology, including metabolic pathways, are temporally coordinated with daily cycles of rest-activity and feeding to provide the organism with an adaptive advantage (1). In mammals, circadian oscillations are driven by a master pacemaker, located in the suprachiasmatic nucleus (SCN) of the hypothalamus, which orchestrates subsidiary oscillators in peripheral organs via neuronal, endocrine, and metabolic signaling pathways (2, 3).Large-scale gene expression datasets suggest that, in mammals, the vast majority of circadian-gene expression is highly organ-specific (4-6). Key metabolic functions in peripheral organs are subject to daily oscillations, such as carbohydrate and lipid metabolism by the liver, skeletal muscle, and endocrine pancreas (7).Skeletal muscle is a major contributor of whole-body metabolism and is the main site of glucose uptake in the postprandial state (8). Therefore, perturbations in glucose sensing and metabolism in skeletal muscle are strongly associated with insulin resistance in type 2 diabetes (T2D) (9). Recent data support a fundamental role for the circadian muscle clock in the regulation of glucose uptake, with a significant re...