Related papers: Estimates for the quantized tensor train ranks for…
In the Quantum-Train (QT) framework, mapping quantum state measurements to classical neural network weights is a critical challenge that affects the scalability and efficiency of hybrid quantum-classical models. The traditional QT framework…
In this paper, we obtain asymptotic formulas for an infinite class of rank generating functions. As an application, we solve a conjecture of Andrews and Lewis on inequalities between certain ranks.
There are quantum algorithms for finding a function $f$ satisfying a set of conditions, such as solving partial differential equations, and these achieve exponential quantum speedup compared to existing classical methods, especially when…
We provide a unified method for obtaining upper bounds for certain functional integrals appearing in quantum mechanics and non-relativistic quantum field theory, functionals of the form $E\left[\exp(A_T)\right]$, the (effective) action…
The technique of quantum electrodynamics (QED) calculations of energy levels in the helium atom is reviewed. The calculations start with the solution of the Schr\"odinger equation and account for relativistic and QED effects by perturbation…
Decompositions of tensors into factor matrices, which interact through a core tensor, have found numerous applications in signal processing and machine learning. A more general tensor model which represents data as an ordered network of…
Tree tensor networks such as the tensor train format are a common tool for high dimensional problems. The associated multivariate rank and accordant tuples of singular values are based on different matricizations of the same tensor. While…
Recently, the \textit{Tensor Nuclear Norm~(TNN)} regularization based on t-SVD has been widely used in various low tubal-rank tensor recovery tasks. However, these models usually require smooth change of data along the third dimension to…
We derive asymptotic estimates for the coefficient of $z^{k}$ in $\left( f\left( z\right) \right) ^{n}$ when $n\rightarrow \infty $ and $k$ is of order $n^{\delta }$, where $0<\delta <1,$ and $f\left( z\right) $ is a power series satisfying…
This work proposes a novel general-purpose estimator for supervised machine learning (ML) based on tensor trains (TT). The estimator uses TTs to parametrize discretized functions, which are then optimized using Riemannian gradient descent…
This article is concerned with obtaining the standard tau function descriptions of integrable equations (in particular, here the KdV and Ernst equations are considered) from the geometry of their twistor correspondences. In particular, we…
Point-mass filters solve Bayesian recursive relations by approximating probability density functions of a system state over grids of discrete points. The approach suffers from the curse of dimensionality. The exponential increase of the…
The tensor-train (TT) decomposition is widely used to compress large tensors into a more compact form by exploiting their inherent data structures. A fundamental approach for constructing the TT format is the well-known TT-SVD method, which…
Approximating a tensor in the tensor train (TT) format has many important applications in scientific computing. Rounding a TT tensor involves further compressing a tensor that is already in the TT format. This paper proposes new randomized…
The modified q-Bessel functions and the q-Bessel-Macdonald functions of the first and second kind are introduced. Their definition is based on representations as power series. Recurrence relations, the q-Wronskians, asymptotic…
This paper is concerned with exponential moments of integral-of-quadratic functions of quantum processes with canonical commutation relations of position-momentum type. Such quadratic-exponential functionals (QEFs) arise as robust…
The nonequilibrium Green's function formalism provides a versatile and powerful framework for numerical studies of nonequilibrium phenomena in correlated many-body systems. For calculations starting from an equilibrium initial state, a…
We give reconstruction algorithms for subclasses of depth-3 arithmetic circuits. In particular, we obtain the first efficient algorithm for finding tensor rank, and an optimal tensor decomposition as a sum of rank-one tensors, when given…
We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called…
Tensor Train (TT) decompositions provide a powerful framework to compress grid-structured data, such as sampled function values, on regular Cartesian grids. Such high compression, in turn, enables efficient high-dimensional computations.…