Related papers: Contraction of matchgate tensor networks on non-pl…
A spanning tree T in a finite planar connected graph G determines a dual spanning tree T* in the dual graph G such that T and T* do not intersect. We show that it is not always possible to find T in G, such that the diameters of T and T*…
We study low rank approximation of tensors, focusing on the tensor train and Tucker decompositions, as well as approximations with tree tensor networks and more general tensor networks. For tensor train decomposition, we give a bicriteria…
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…
A tensor network is a type of decomposition used to express and approximate large arrays of data. A given data-set, quantum state or higher dimensional multi-linear map is factored and approximated by a composition of smaller multi-linear…
Let $G=(V,E)$ be a strongly connected graph with $|V|\geq 3$. For $T\subseteq V$, the strongly connected graph $G$ is $2$-T-connected if $G$ is $2$-edge-connected and for each vertex $w$ in $T$, $w$ is not a strong articulation point. This…
Tensor networks offer a variational formalism to efficiently represent wave-functions of extended quantum many-body systems on a lattice. In a tensor network N, the dimension \chi of the bond indices that connect its tensors controls the…
A fundamental process in the implementation of any numerical tensor network algorithm is that of contracting a tensor network. In this process, a network made up of multiple tensors connected by summed indices is reduced to a single tensor…
Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a…
Representing the nodes of continuous-time temporal graphs in a low-dimensional latent space has wide-ranging applications, from prediction to visualization. Yet, analyzing continuous-time relational data with timestamped interactions…
A $t$-spanner of a weighted undirected graph $G=(V,E)$, is a subgraph $H$ such that $d_H(u,v)\le t\cdot d_G(u,v)$ for all $u,v\in V$. The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all…
We study the recovery of the underlying graphs or permutations for tensors in the tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is $O(d\log d)$ for $d$-th…
The r-parity tensor of a graph is a generalization of the adjacency matrix, where the tensor's entries denote the parity of the number of edges in subgraphs induced by r distinct vertices. For r=2, it is the adjacency matrix with 1's for…
Given a set $P$ of $n$ points in the plane, we solve the problems of constructing a geometric planar graph spanning $P$ 1) of minimum degree 2, and 2) which is 2-edge connected, respectively, and has max edge length bounded by a factor of 2…
We show how to construct a tensor network representation of the path integral for reduced staggered fermions coupled to a non-abelian gauge field in two dimensions. The resulting formulation is both memory and computation efficient because…
A planar orthogonal drawing of a planar 4-graph G (i.e., a planar graph with vertex-degree at most four) is a crossing-free drawing that maps each vertex of G to a distinct point of the plane and each edge of $G$ to a sequence of horizontal…
Quantum circuits composed of CNOT and $R_z$ are fundamental building blocks of many quantum algorithms, so optimizing the synthesis of such quantum circuits is crucial. We address this problem from a theoretical perspective by studying the…
We prove a robust contraction decomposition theorem for $H$-minor-free graphs, which states that given an $H$-minor-free graph $G$ and an integer $p$, one can partition in polynomial time the vertices of $G$ into $p$ sets $Z_1,\dots,Z_p$…
Tensor network methods are a conceptually elegant framework for encoding complicated datasets, where high-order tensors are approximated as networks of low-order tensors. In practice, however, the numeric implementation of tensor network…
We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot maps to the partition function of a $q$-state Potts model defined as a planar…
We propose a new framework for the analysis of low-rank tensors which lies at the intersection of spectral graph theory and signal processing. As a first step, we present a new graph based low-rank decomposition which approximates the…