Related papers: Dimension of Tensor Network varieties
Tensor Networks are non-trivial representations of high-dimensional tensors, originally designed to describe quantum many-body systems. We show that Tensor Networks are ideal vehicles to connect quantum mechanical concepts to machine…
We invent the notion of a {\it dimension of a variety} $V$ as the cardinality of all its proper {\it derived} subvarieties (of the same type). The dimensions of varieties of lattices, varieties of regular bands and other general algebraic…
We characterize the variational power of quantum circuit tensor networks in the representation of physical many-body ground-states. Such tensor networks are formed by replacing the dense block unitaries and isometries in standard tensor…
Tensor networks (TNs) have been gaining interest as multiway data analysis tools owing to their ability to tackle the curse of dimensionality and to represent tensors as smaller-scale interconnections of their intrinsic features. However,…
In this paper, we extend some classes of structured matrices to higher order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a…
The comprehensive characterization of the structure of complex networks is essential to understand the dynamical processes which guide their evolution. The discovery of the scale-free distribution and the small world property of real…
We investigate robust linear consensus over networks under capacity-constrained communication. The capacity of each edge is encoded as an upper bound on the number of state variables that can be communicated instantaneously. When the edge…
We study the real rank of points with respect to a real variety $X$. This is a generalization of various tensor ranks, where $X$ is in a specific family of real varieties like Veronese or Segre varieties. The maximal real rank can be…
Tensor networks provide a powerful new framework for classifying and simulating correlated and topological phases of quantum matter. Their central premise is that strongly correlated matter can only be understood by studying the underlying…
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…
Tensors decompositions are a class of tools for analysing datasets of high dimensionality and variety in a natural manner, with the Canonical Polyadic Decomposition (CPD) being a main pillar. While the notion of CPD is closely intertwined…
We propose a novel measure of degree heterogeneity, for unweighted and undirected complex networks, which requires only the degree distribution of the network for its computation. We show that the proposed measure can be applied to all…
Neural networks with rectified linear unit activations are essentially multivariate linear splines. As such, one of many ways to measure the "complexity" or "expressivity" of a neural network is to count the number of knots in the spline…
The results of Strassen and Raz show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds. We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we…
We consider an edge-weighted uniform random graph with a given degree sequence (Repeated Configuration Model) which is a useful approximation for many real-world networks. It has been observed that the vertices which are separated from the…
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…
We introduce a flexible setup allowing for a neural network to learn both its size and topology during the course of a standard gradient-based training. The resulting network has the structure of a graph tailored to the particular learning…
A tensor network is a diagram that specifies a way to "multiply" a collection of tensors together to produce another tensor (or matrix). Many existing algorithms for tensor problems (such as tensor decomposition and tensor PCA), although…
Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this enterprise is to find the…
Finding the rank of a tensor is a problem that has many applications. Unfortunately it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of…