In mathematics, a tensor is an algebraic object that describes a linear mapping from one set of algebraic objects to another. According to Ars Technica, “Basically, a tensor is a matrix of equations, instead of a matrix of pure numbers. Tensor mathematics is the manipulation of these equation matrices as a method of solving ALL of the involved equations.” Tensor can be considered as a multi-linear map. Given a set of coordinates (or expand out to functions or other objects), each of these coordinates can be transformed according to a set of rules (linear transformations) into a new set of coordinates. The key here is that each coordinate can have a unique transformation.

Tensors were conceived in 1900 by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who first published their theory of tensor calculus, which is also known as absolute differential calculus. The importance of tensor calculus became apparent in 1915 when physicist Albert Einstein revealed that he had found it indispensable for the gravitational field equations used in his theory of general relativity.

One way to understand the importance of tensor calculus is to consider geometric complications when drawing right angles on a spherical surface when the Pythagorean Theorem no longer works. It’s here that the metric tensor comes to the rescue. It generalizes coordinates and geometries so that distance can be measured in any given space. The magic of tensors comes from their special transformational properties that enable them to describe the same physics in all reference frames.

Tensors are inherently related to vector spaces and their dual spaces, and can take several different forms – for example: a scalar, a vector, a dual vector at a point, or a multi-linear map between vector spaces. Euclidean vectors and scalars (which are often used in elementary physics and engineering applications where general relativity is irrelevant) are the simplest tensors.

Tensors are important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, …), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, …), or general relativity (stress–energy tensor, curvature tensor, … ) and others.

In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are simply called “tensors”.

A tensor decomposition is unique whenever components are linearly independent, where a decomposition is a schema for expressing a tensor as elementary operations between simpler tensors. In contrast, a matrix decomposition is unique only when components are orthogonal. Compared to traditional matrix-based code, tensor-based modeling is faster and requires less memory space.

Objects that tensors may map between include, but are not limited to, vectors and scalars, and, recursively, even other tensors. For example, a matrix is a map between vectors, and is thus a tensor.

Tensor functions fall into one of four main categories: reshaping, element-wise operations, reduction, and access. Some of the tensor reshaping operations includes squeeze, unsqueeze, flatten, and reshape. Combining with another tensor will also reshape a tensor.

### Tensors applications in Warfare

Since long military organizations have invested considerable effort into developing theories of war and battle. The purpose of such theories is ultimately to inform decisions: what equipment to acquire, what processes to institute, what training and education to develop, what research to conduct, how to conduct operations, and what operations to conduct in the first place. Perhaps unsurprisingly, many such efforts at the development of theories of war and battle have been oriented around the idea of achieving something like a complete and correct theory by which future outcomes might be predicted and thereby the means of determining the means to guarantee, or at least maximize the chances of, obtaining the outcome one desires. Therefore numerous theories have been proposed such as Network Centric Warfare (NCW) and Effects-Based Operations (EBO) and distributed C4ISR system (Command, Control, Communications, Computing, Intelligence, Surveillance and Reconnaissance).

Yet as predicted even from the outset by Clausewitz, a programme aimed at developing theories predictive of outcomes is ultimately not possible, because―to use modern mathematical language―the conditions of war and battle are simply not ergodic. That is, the inherent complex nonlinearity of such systems yield strong limitations on what can be predicted about their outcomes; conditions are not static and distributions are not nicely behaved, making it not possible to sample from the unrealized future by collecting data about the past.

Recent years have seen a growing knowledge about, and interest in, the burgeoning knowledge across the sciences about complexity and uncertainty, among defence and military thinkers. The general emerging view is that defence and military matters feature burgeoning complexity of technological, social, economic, cultural and political varieties.

In a paper authors proposed the basis for a rigorous approach to modeling combat, specifically under conditions of complexity and uncertainty. According to them, the theories of war and battle intended as explanatory and predictive bases for guiding force development decision-making such as NCW and EBO, none has proven satisfactory.

Their concept of tensor-centric warfare (TCW) represents complex warfighting dynamics on such battle-manifolds is naturally defined as an interplay of spatiotemporal vector and tensor fields flowing on them. The proposed basis is a tensorial generalization of earlier Lanchester-type equations, inspired by the contemporary debate in defence and military circles around how to best utilize information and communications systems in military operations, including the distributed C4ISR system (Command, Control, Communications, Computing, Intelligence, Surveillance and Reconnaissance).

“Despite attracting considerable interest and spawning several efforts to develop sound theoretical frameworks for informing force design decision-making, the development of good frameworks for analytically modeling combat remains anything but decided. Using a simple combat scenario, we first develop a tensor generalization of the Lanchester square law, and then extend it to also include the Lanchester linear law, which represents the effect of suppressive fire.

The emphasis in our formalization lies in the possibility of better addressing the complexity and uncertainty inherent in war and battle, which, despite having been studied since the Military Enlightenment period, have continued to prove challenging to military thinking.

### Tensors are the most fundamental data type used in all major AI frameworks.

An example of a 3D tensor (or cube) is time series data used with radar processing, which has three parameters (time, frequency, and channel). Described by width, height, and depth (color), a two-dimensional JPG image can be expressed with a 3D tensor. Adding the number of pictures to process increases it to a 4D tensor. A collection of videos would be stored as a 5D tensor (number of videos, number of frames per video, width, height, and depth). As these images process through the deep learning layers, they can be broken down into hundreds of features, thus expanding the number of dimensions.

Serving where the compute engines, the algorithms, and the data all intersect, tensors are at the heart of deep learning, and as demonstrated, they easily represent high-order relationships. Tensors will often discover hidden relationships that a human did not see in the data and could not program as a feature. And like linear algebra, tensor algebra is parallelizable.

### Google’s TensorFlow

TensorFlow is a free and open-source software library for dataflow and differentiable programming across a range of tasks. TensorFlow was developed by the Google Brain team for internal Google use. It is a symbolic math library, and is also used for machine learning applications such as neural networks. Google also uses this AI engine to recognize spoken words, translate from one language to another, improve Internet search results, and more. It was released under the Apache License 2.0 on November 9, 2015.

Deep learning relies on neural networks—systems that approximate the web of neurons in the human brain. TensorFlow is a way of building and running these neural networks—both at the training stage and the execution stage. It’s a set of software libraries—a bunch of code—that you can slip into any application so that it too can learn tasks like image recognition, speech recognition, and language translation.

TensorFlow computations are expressed as stateful dataflow graphs. The name TensorFlow derives from the operations that such neural networks perform on multidimensional data arrays, which are referred to as tensors. Its flexible architecture allows for the easy deployment of computation across a variety of platforms (CPUs, GPUs, TPUs), and from desktops to clusters of servers to mobile and edge devices.

### Google’s AI is being used by US military drone programme

Google’s TensorFlow AI systems are being used by the US Department of Defense’s (DoD) Project Maven, which was established in July last year to use machine learning and artificial intelligence to analyse the vast amount of footage shot by US drones. The initial intention is to have AI analyse the video, detect objects of interest and flag them for a human analyst to review.

Drew Cukor, chief of the DoD’s Algorithmic Warfare Cross-Function Team, said in July: “People and computers will work symbiotically to increase the ability of weapon systems to detect objects. Eventually we hope that one analyst will be able to do twice as much work, potentially three times as much, as they’re doing now. That’s our goal.”

A Google spokesperson said: “This specific project is a pilot with the Department of Defense, to provide open source TensorFlow APIs that can assist in object recognition on unclassified data. The technology flags images for human review, and is for non-offensive uses only.”

### References and Resources also include:

http://mil-embedded.com/guest-blogs/what-is-a-tensor-and-why-should-i-care/

https://www.scirp.org/journal/PaperInformation.aspx?paperID=84904