Related papers: Graphical Calculus for products and convolutions
We investigate the algebraic complexity of tensor calulus. We consider a generalization of iterated matrix product to tensors and show that the resulting formulas exactly capture VP, the class of polynomial families efficiently computable…
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…
We describe a graphical calculus for completely positive maps and in doing so review the theory of open quantum systems and other fundamental primitives of quantum information theory using the language of tensor networks. In particular we…
A tensor is a multidimensional array of numbers that can be used to store data, encode a computational relation and represent quantum entanglement. In this sense a tensor can be viewed as valuable resource whose transformation can lead to…
Gauss' and Stokes' theorems are fundamental results in vector calculus and important tools in physics and engineering. When students are asked to describe the meaning of Gauss' divergence theorem, they often use statements like this: "The…
Graphical techniques provide a very useful practical device for calculations involving the so-called spin network states, which encode the quantum degrees of freedom of spatial geometry in loop quantum gravity. Graphical calculus of SU(2),…
Computational Group Theory is applied to indexed objects (tensors, spinors, and so on) with dummy indices. There are two groups to consider: one describes the intrinsic symmetries of the object and the other describes the interchange of…
The convolution operator at the core of many modern neural architectures can effectively be seen as performing a dot product between an input matrix and a filter. While this is readily applicable to data such as images, which can be…
In mathematics, many notations have been invented for the concise representation of mathematical formulae. Tensor index notation is one of such notations and has been playing a crucial role in describing formulae in mathematical physics.…
Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of…
This paper proposes graphical representations of data and rationale provenance in workflows that convert both category labels and associated numeric data between distinct but semantically related taxonomies. We motivate the graphical…
Axon is a language that enables shape and rank inference for tensors in a Deep Learning graphs. It aims to make shapes implicit and inferred, in a similar manner to how types are implicit and inferred in many functional programming…
In this paper we explore the design of sequent calculi operating on graphs. For this purpose, we introduce a set of logical connectives allowing us to extend the correspondence between cographs and classical propositional formulas to any…
Tensors, or multidimensional arrays, are data structures that can naturally represent visual data of multiple dimensions. Inherently able to efficiently capture structured, latent semantic spaces and high-order interactions, tensors have a…
Dynamic graphs (DG) describe dynamic interactions between entities in many practical scenarios. Most existing DG representation learning models combine graph convolutional network and sequence neural network, which model spatial-temporal…
Graphical calculi are vital tools for representing and reasoning about quantum circuits and processes. Some are not only graphically intuitive but also logically complete. The best known of these is the ZX-calculus, which is an industry…
Graphical functions are single-valued complex functions which arise from Feynman amplitudes. We study their properties and use their connection to multiple polylogarithms to calculate Feynman periods. For the zig-zag and two more families…
Graphical languages are a convenient shorthand to represent computation, with rewrite rules relating one graph to another. In contrast, proof assistants rely heavily on inductive datatypes, particularly when giving semantics to embedded…
Many irregular domains such as social networks, financial transactions, neuron connections, and natural language constructs are represented using graph structures. In recent years, a variety of graph neural networks (GNNs) have been…
Many problems in high-dimensional statistics appear to have a statistical-computational gap: a range of values of the signal-to-noise ratio where inference is information-theoretically possible, but (conjecturally) computationally…