Related papers: Hackbusch Conjecture on tensor formats

Binary Neural Networks (BNNs) enable efficient deep learning by saving on storage and computational costs. However, as the size of neural networks continues to grow, meeting computational requirements remains a challenge. In this work, we…

Hypergraphs and tensors extend classic graph and matrix theory to account for multiway relationships, which are ubiquitous in engineering, biological, and social systems. While the Kronecker product is a potent tool for analyzing the…

Geometrical constructions using flexible cords have been known since the earliest days of recorded mathematics. In this paper we introduce rigorous definitions for two classes of string networks. A taut network is one in which all cords are…

We present an iterative method for the search of extreme entries in low-rank tensors which is based on a power iteration combined with a binary search. In this work we use the HT-format for low-rank tensors but other low-rank formats can be…

Tensor networks represent the state-of-the-art in computational methods across many disciplines, including the classical simulation of quantum many-body systems and quantum circuits. Several applications of current interest give rise to…

For multiqubit densities, the tensor of coherences (or Stokes tensor) is a real parameterization obtained by the juxtaposition of the affine Bloch vectors of each qubit. While it maintains the tensorial structure of the underlying space, it…

We discuss a possibility that the entire universe on its most fundamental level is a neural network. We identify two different types of dynamical degrees of freedom: "trainable" variables (e.g. bias vector or weight matrix) and "hidden"…

Deep neural networks have demonstrated state-of-the-art performance in a variety of real-world applications. In order to obtain performance gains, these networks have grown larger and deeper, containing millions or even billions of…

The tensor network representation of a state in higher dimensions, say a projected entangled-pair state (PEPS), is typically obtained indirectly through variational optimization or imaginary-time Hamiltonian evolution. Here, we propose a…

The existing tensor networks adopt conventional matrix product for connection. The classical matrix product requires strict dimensionality consistency between factors, which can result in redundancy in data representation. In this paper,…

The curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important tool in attempting to overcome this…

We describe the Algebraic Bethe Ansatz for the spin-1/2 XXX and XXZ Heisenberg chains with open and periodic boundary conditions in terms of tensor networks. These Bethe eigenstates have the structure of Matrix Product States with a…

Despite the latest prevailing success of deep neural networks (DNNs), several concerns have been raised against their usage, including the lack of intepretability the gap between DNNs and other well-established machine learning models, and…

Optimizing tensor networks with standard first-order methods often leads to slow convergence and entrapment in local minima. Although second-order optimization offers enhanced robustness, explicitly constructing the full Hessian matrix is…

Phylogenetic (evolutionary) trees and networks are leaf-labeled graphs that are widely used to represent the evolutionary relationships between entities such as species, languages, cancer cells, and viruses. To reconstruct and analyze…

Random forests on the one hand, and neural networks on the other hand, have met great success in the machine learning community for their predictive performance. Combinations of both have been proposed in the literature, notably leading to…

Lov\'asz (1967) showed that two graphs $G$ and $H$ are isomorphic if and only if they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph $F$, the number of homomorphisms from $F$ to $G$ equals the number…

In this article we show the duality between tensor networks and undirected graphical models with discrete variables. We study tensor networks on hypergraphs, which we call tensor hypernetworks. We show that the tensor hypernetwork on a…

The tensor network, as a facterization of tensors, aims at performing the operations that are common for normal tensors, such as addition, contraction and stacking. However, due to its non-unique network structure, only the tensor network…

Dynamic multilayer networks are frequently used to describe the structure and temporal evolution of multiple relationships among common entities, with applications in fields such as sociology, economics, and neuroscience. However,…