Related papers: Tensor product approximations of high dimensional …
We present a tensor-structured algorithm for efficient large-scale DFT calculations by constructing a Tucker tensor basis that is adapted to the Kohn-Sham Hamiltonian and localized in real-space. The proposed approach uses an additive…
We provide a recursive method for constructing product formula approximations to exponentials of commutators, giving the first approximations that are accurate to arbitrarily high order. Using these formulas, we show how to approximate…
We analyze the problem of high-order polynomial approximation from a many-body physics perspective, and demonstrate the descriptive power of entanglement entropy in capturing model capacity and task complexity. Instantiated with a…
In this paper we describe a class of highly entangled subspaces of a tensor product of finite dimensional Hilbert spaces arising from the representation theory of free orthogonal quantum groups. We determine their largest singular values…
Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or…
There is a significant expansion in both volume and range of applications along with the concomitant increase in the variety of data sources. These ever-expanding trends have highlighted the necessity for more versatile analysis tools that…
The existing randomized algorithms need an initial estimation of the tubal rank to compute a tensor singular value decomposition. This paper proposes a new randomized fixedprecision algorithm which for a given third-order tensor and a…
After a review of how Boson Fock space (of arbitrary multiplicity) may be approximated by a countable Hilbert-space tensor product (known as toy Fock space) it is shown that vacuum-adapted multiple quantum Wiener integrals of bounded…
We prove some probabilistic estimates for tensor products of random vectors. As an application we obtain embeddings of certain matrix spaces into $L_1$.
Finite element methods usually construct basis functions and quadrature rules for multidimensional domains via tensor products of one-dimensional counterparts. While straightforward, this approach results in integration spaces larger than…
Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the…
Suitable discretizations through tensor product formulas of popular multidimensional operators (diffusion or diffusion--advection, for instance) lead to matrices with $d$-dimensional Kronecker sum structure. For evolutionary Partial…
In this paper we propose efficient randomized fixed-precision techniques for low tubal rank approximation of tensors. The proposed methods are faster and more efficient than the existing fixed-precision algorithms for approximating the…
Price promotion optimization is a computationally challenging problem central to supermarket operations, requiring simultaneous pricing decisions across multiple products and periods. This paper introduces a novel formulation for…
Running quantum algorithms often involves implementing complex quantum circuits with such a large number of multi-qubit gates that the challenge of tackling practical applications appears daunting. To date, no experiments have successfully…
Node elimination is a numerical approach to obtain cubature rules for the approximation of multivariate integrals. Beginning with a known cubature rule, nodes are selected for elimination, and a new, more efficient rule is constructed by…
Dimensionality reduction for high-order tensors is a challenging problem. In conventional approaches, higher order tensors are `vectorized` via Tucker decomposition to obtain lower order tensors. This will destroy the inherent high-order…
In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension $d$ by means of tensor product formulas. This is useful, for example, in the context of…
Numerical integration formulas in $n$-dimensional Euclidean space of degree three are discussed. For the integrals with permutation symmetry we present a method to construct its third-degree integration formulas with $2n$ real points. We…
In this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous…