Related papers: Carving-width and contraction trees for tensor net…
Data tensors of orders 2 and greater are now routinely being generated. These data collections are increasingly huge and growing. Many scientific and medical data tensors are tensor fields (e.g., images, videos, geographic data) in which…
In this paper we study an emerging class of neural networks based on the morphological operators of dilation and erosion. We explore these networks mathematically from a tropical geometry perspective as well as mathematical morphology. Our…
The main purpose of the paper is to develop an approach to evaluation or estimation of the spanning tree congestion of planar graphs. This approach is used to evaluate the spanning tree congestion of triangular grids.
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…
Dynamical low-rank approximation by tree tensor networks is studied for the data-sparse approximation to large time-dependent data tensors and unknown solutions of tensor differential equations. A time integration method for tree tensor…
Efficient probabilistic inference by variable elimination in graphical models requires an optimal elimination order. However, finding an optimal order is a challenging combinatorial optimisation problem for models with a large number of…
Network design problems aim to compute low-cost structures such as routes, trees and subgraphs. Often, it is natural and desirable to require that these structures have small hop length or hop diameter. Unfortunately, optimization problems…
Contraction hierarchies are a simple hierarchical routing technique that has proved extremely efficient for static road networks. We explain how to generalize them to networks with time-dependent edge weights. This is the first hierarchical…
Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost…
Constrained least squares problems arise in many applications. Their memory and computation costs are expensive in practice involving high-dimensional input data. We employ the so-called "sketching" strategy to project the least squares…
A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy…
Spatial and temporal resource constraints are critical for both biological and artificial intelligent systems. Here we define differentiable cost terms for breadth, depth, and time within a recurrent convolutional neural network conceived…
Given a spatio-temporal network (ST network) where edge properties vary with time, a time-sub-interval minimum spanning tree (TSMST) is a collection of minimum spanning trees of the ST network, where each tree is associated with a time…
Knots are commonly represented and manipulated via diagrams, which are decorated planar graphs. When such a knot diagram has low treewidth, parameterized graph algorithms can be leveraged to ensure the fast computation of many invariants…
Among all characteristics exhibited by natural and man-made networks the small-world phenomenon is surely the most relevant and popular. But despite its significance, a reliable and comparable quantification of the question `how small is a…
We introduce a new neural architecture to learn the conditional probability of an output sequence with elements that are discrete tokens corresponding to positions in an input sequence. Such problems cannot be trivially addressed by…
Tree decompositions were developed by Robertson and Seymour. Since then algorithms have been developed to solve intractable problems efficiently for graphs of bounded treewidth. In this paper we extend tree decompositions to allow cycles to…
Linear sketching and recovery of sparse vectors with randomly constructed sparse matrices has numerous applications in several areas, including compressive sensing, data stream computing, graph sketching, and combinatorial group testing.…
In this paper, we consider the problem of compressing a trie while supporting the powerful \emph{locate} queries: to return the pre-order identifiers of all nodes reached by a path labeled with a given query pattern. Our result builds on…
This paper develops the sketching (i.e., randomized dimension reduction) theory for real algebraic varieties and images of polynomial maps, including, e.g., the set of low rank tensors and tensor networks. Through the lens of norming sets,…