# tensors in computer science

Tensors have a rich history, stretching over almost a century, and touching upon numerous disciplines; but they have only recently become ubiquitous in signal and data analytics at the conﬂuence of signal processing, statistics, data mining, and machine learning. Abstract. Jon Sporring received his Master and Ph.D. degree from the Department of Computer Science, University of Copenhagen, Denmark in 1995 and 1998, respectively.Part of his Ph.D. program was carried out at IBM Research Center, Almaden, California, USA. Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of Master of Science in Computer Science at the Massachusetts Institute of Technology February 2019 c 2019 Peter Ahrens. Tensors are when the the vectors aren't good enough because the media is anisotropic. A scalar is a 0-dimensional (0D) tensor. They are examples of a more general entity known as a tensor. Recent years have seen a dramatic rise of interest by computer scientists in the mathematics of higher-order tensors. The Hebrew University Tensor Methods for Machine Learning, Computer Vision, and Computer Graphics Part I: ... A super-symmetric tensor described as sum of k super-symmetric rank=1 tensors: is (at most) rank=k. Recall that the ndim attribute of the multidimensional array returns the number of array dimensions. While matrix rank can be efficiently computed by, say, Gaussian eliminination, computing the rank of a tensor of order 3 is NP-hard. I'd say, both have their advantages and disadvantages. We could see that the components in our simple vector are the same as the coordinates associated with those two basis vectors. parameter. Put simply, a Tensor is an array of numbers that transform according to certain rules under a change of coordinates. Tensors are multidimensional extensions of matrices. We are soliciting original contributions that address a wide range of theoretical and practical issues including, but not limited to: 1. The system is called Taco, for tensor algebra compiler. We encourage discussions on recent advances, ongoing developments, and novel applications of multi-linear algebra, optimization, and feature representations using tensors. Unsupervised feature learning and multimodal representations 4. Dark Data: Why What You Don’t Know Matters. There’s one more thing I need to mention before tensors. The modern approach to tensor analysis is through Cartan theory, i.e., using (differential alternating) forms and coordinate free formulations, while physicists usually use the Ricci calculus using components and upper and lower indices. Now let’s turn our attention to covectors. P.s. We encourage discussions on recent advances, ongoing developments, and novel applications of multi-linear algebra, optimization, and feature representations using tensors. I hope at this point you have had a better understanding of what a tensor truly is, intuitively. Then each section will cover different models starting off with fundamentals such as Linear Regression, and logistic/softmax regression. Juan R. Ruiz-Tolosa is an Industrial and Civil Engineer and has been Professor of Algebra, Tensors, Topology, Differential Geometry and Calculus at the Civil Engineering School, University of Cantabria for 30 years. var disqus_shortname = 'kdnuggets'; Examples of such transformations, or relations, include the cross product and the dot product. If so, does anyone know of a decent introductory text (online tutorial, workshop paper, book, etc) which develops tensors in that sense for computer scientists/machine learning practitioners? (Easier to break a mica rock by sliding layers past each other than perpendicular to plane.) From a computer science perspective, it can be helpful to think of tensors as being objects in an object-oriented sense, as opposed to simply being a data structure. Let’s instead change to using a slightly more complicated basis. In Spring 2020 we are running an ideas lab connecting graphs and tensors to problems in drug discovery While, technically, all of the above constructs are valid tensors, colloquially when we speak of tensors we are generally speaking of the generalization of the concept of a matrix to N ≥ 3 dimensions. In computer science, we stop using words like, number, array, 2d-array, and start using the word multidimensional array or nd-array. What you do with a tensor is your business, though understanding what one is, and its relationship to related numerical container constructs, should now be clear. Tensor signal processing is an emerging field with important applications to computer vision and image processing. For simplicity’s sake, let’s just consider vector as a vertical list of real numbers, e.g. A tensor network is simply a countable collection of tensors connected by con-tractions. Instead, in terms of tensors, we could see a tensor as either a “vector of tensors (albeit of a lower rank)” or a “covector of tensors”. The code below creates a 3D tensor. If you are familiar with basic linear algebra, you should have no trouble understanding what tensors are. Then we have matrices, which are nothing more than a collection of vectors. The reason for this is that if you do the matrix multiplication of our definition of functional with our definition of vector, the result comes out to be a 1x1 matrix, which I’m content with treating as just a real number. The mathematical concept of a tensor could be broadly explained in this way. It only takes a minute to sign up. Computer Science and Mathematics. Covectors live in a vector space called the dual space. The 2D structure tensor Continuous version. Implementing the AdaBoost Algorithm From Scratch, Data Compression via Dimensionality Reduction: 3 Main Methods, A Journey from Software to Machine Learning Engineer. The Wikipedia article is atrocious. Absolute tensor notation is an alternative which does not rely on components, but on the essential idea that tensors are intrinsic objects, so that tensor relations are independent of any observer. Mid-level representati… Rest assured that this is not because you are hallucinating. That’s why people restricted to matrices to be able to prove a lot of nice properties. Before we dive into tensor, it is necessary to explore the properties of our building blocks: vectors, covectors, and linear operators. That is linear operators. Computer Science and Mathematics. I'd say, both have their advantages and disadvantages. If a matrix is a square filled with numbers, then a higher-order tensor is an n-dimensional cube filled with numbers. Aside from holding numeric data, tensors also include descriptions of the valid linear transformations between tensors. Jon Sporring received his Master and Ph.D. degree from the Department of Computer Science, University of Copenhagen, Denmark in 1995 and 1998, respectively.Part of his Ph.D. program was carried out at IBM Research Center, Almaden, California, USA. Now he has a startup focused on nutrition for top athletes. I found Ambiguous Cylinders to be the perfect analogy for linear operators. Tensors in low-level feature design 5. Unsupervised feature learning and multimodal representations 4. It approximates the input tensor by a sum of rank-one tensors, which are outer products of vectors. From a computer science perspective, it can be helpful to think of tensors as being objects in an object-oriented sense… Mid-level representati… [1] “Introduction to Tensors.” (2020). Mathematically speaking, tensors are more than simply a data container, however. Learn linear algebra. A better reason is that it’ll help us better visualize tensors, as you’ll see.But for now, we see that according to the way we defined functionals, a covector is actually also a sort of horizontal list of real numbers. Tensors come in varying forms and levels of complexity defined by their related order. https://www.ese.wustl.edu/~nehorai/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf, https://www.ese.wustl.edu/~nehorai/Porat_A_Gentle_Introduction_to_Tensors_2014.pdf, Modular image processing pipeline using OpenCV and Python generators, Reinforcement Learning for Beginners: Q-Learning and SARSA, EEoI for Efficient ML with Edge Computing, Why Reinforcement Learning is Wrong for Your Business, XLNet outperforms BERT on several NLP Tasks, Building Our Own Deep Learning Image Recognition Technology, Deploying EfficientNet Model using TorchServe. It is well known that the notion of tensor rank is of great relevance for computer science through the famous, but still unsolved, problem of the complexity of matrix multiplication. Boost your data science skills. The primary kernel of the factorization is a chain of tensor-matrix multiplications (TTMc). This book presents the state of the art in this new branch of signal processing, offering research and … In the past decade, there has been a significant increase in the interest of using tensors in data analysis, where they can be used to store, for example, multi-relational data (subject-predicate-object triples, user-movie-tag triples, etc. Nn this example, we convert each image to Pytorch tensors for using the images as inputs to the neural network. Wait, does it mean that a matrix, or a linear operator, behaves like a vector and a covector at the same time? The Tucker decomposition is a higher-order analogue of the singular value decomposition and is a popular method of performing analysis on multi-way data (tensors). CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): In this paper, we explain what are tensors and how tensors can help in computing. Read … School of Computer Science & Eng. It’s not at all wrong, but somewhat intellectually unsatisfying. A scalar has the lowest dimensionality and is always 1×1. The two primary mathematical entities that are of interest in linear algebra are the vector and the matrix. Therefore, if the basis in the vector space is transformed by S, the covectors in the corresponding dual space would also undergo the same transformation by S. Formally, if y is the set of coordinates for a covector in the dual space, then the transformation law is described by², Again, to show this by an example, consider our example covector to be in dual space V* that corresponds to the vector space V in our previous vector example. It, thus, has 0 axes, and is of rank 0 (tensor-speak for 'number of axes'). Isn’t this similar to the transformation law for a linear operator, but with more T’s and S’s? Computing the Tucker decomposition of a sparse tensor is demanding in terms of both memory and computational resources. Of course, we need not stick to just this simple basis. Department of Computer Science University in the Texas at El Paso 500 W. University El Paso, TX 79968, USA mceberio@utep.edu, vladik@utep.edu Abstract In this paper, after explaining the need to use tensors in computing, we analyze the question of how to best store tensors in computer memory. However, tensor applications and tensor-processing tools arise from very different areas, and these advances are too often kept within the areas of knowledge where they were first employed. As a Software Engineer, this naturally occurred to me as a recursive definition of tensors, where the base case is a scalar! Because if I look at the definition of tensor on any linear algebra book or Wikipedia, I would see something more or less like this: Of course, the definition of tensor in the TensorFlow guide is correct, and it might be sufficient for the use of deep learning, but it fails to convey some of the defining properties of a tensor, such as described in this terribly perplexing equation. If we temporarily consider them simply to be data structures, below is an overview of where tensors fit in with scalars, vectors, and matrices, and some simple code demonstrating how Numpy can be used to create each of these data types. ICML07 Tutorial 6 General Tensors … In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. A matrix is a tensor of rank 2, meaning that it has 2 axes. That was another reason tensors were seen as exotic objects that were hard to analyze compared to matrices. While the above is all true, there is nuance in what tensors technically are and what we refer to as tensors as relates to machine learning practice. ICML07 Tutorial 6. This article provides an overview of tensors, their properties, and their applications in statistics. However, after combing through countless tutorials and documentations on tensor, I still haven’t found one that really made sense for me intuitively, especially one that allows me to visualize a tensor in my head. There are two alternative ways of denoting tensors: index notation is based on components of tensors (which is convenient for proving equalities involving tensors). Tensors in Computer Science. If we were to pack a series of these into a higher order tensor container, it would be referred to as a 4D tensor; pack those into another order higher, 5D, and so on. Okay. Especially when referring specifically of neural network data representation, this is accomplished via a data repository known as the tensor. In Spring 2020 we are running special sessions on the mathematics of Data Science at the AMS sectional meeting, with a focus on graphs and tensors. The dimensions of a vector are nothing but Mx1 or 1xM matrices. However, tensor applications and tensor-processing tools arise from very different areas, and these advances are too often kept within the areas of knowledge where they were first employed. The following is a naive implementation of tensor that tries to convey this idea. The notion of matrix rank can be generalized to higher-order tensors. Though classical, the study of tensors has recently gained fresh momentum due to applications in such areas as complexity theory and algebraic statistics. Let me quote myself from a Previous Post:. When a tensor is expanded in terms of a set of basis (or inverse basis) vectors, the coefficients of expansion are its contravariant (or covariant) components with respect to this basis. (document.getElementsByTagName('head')[0] || document.getElementsByTagName('body')[0]).appendChild(dsq); })(); By subscribing you accept KDnuggets Privacy Policy. A vector is a single dimension (1D) tensor, which you will more commonly hear referred to in computer science as an array. We see that loosely speaking, the coordinates changed in the opposite direction of the basis. Recent years have seen a dramatic rise of interest by computer scientists in the mathematics of higher-order tensors. If you think of it, a linear operator really is just a matrix, intuitively. If you are looking for a TensorFlow or deep learning tutorial, you will be greatly disappointed by this article. Tensor signal processing is an emerging field with important applications to computer vision and image processing. Supervised learning in computer vision 3. Computer Science Tensors in Image Processing and Computer Vision (Advances in Computer Vision and Pattern Recognition) 2009th Edition by Santiago Aja-Fernández (Editor), Rodrigo de Luis Garcia (Editor), Dacheng Tao (Editor), Xuelong Li (Editor) & 1 more We first review basic tensor concepts and decompositions, and then we elaborate traditional and recent applications of tensors in the fields of recommender systems and imaging analysis. In fact, scalars are rank-0 tensors; vector and covectors are rank-1 tensors; matrices are rank-2 tensors. We are soliciting original contributions that address a wide range of theoretical and practical issues including, but not limited to: 1. Recent years have seen a dramatic rise of interest in the mathematics of higher-order tensors and their applications. The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci. When we represent data for machine learning, this generally needs to be done numerically. Tensors, also known as multidimensional arrays, are generalizations of matrices to higher orders and are useful data representation architectures. For our purposes, let’s consider a functional something like a horizontal list of real numbers, e.g. So we have, But since the covector itself doesn’t change, the coordinates have to change, Notice how the coordinates of the covector are also transformed by S, which makes the covector covariant. The notion of matrix rank can be generalized to higher-order tensors. If a matrix is a square filled with numbers, then a higher-order tensor is an n-dimensional cube filled with numbers. The n tells us the number of indexes required to access a specific element within the structure. Description. Computer science. Lecture Notes in Computer Science, vol 11989. It approximates the input tensor by a sum of rank-one tensors, which are outer products of vectors. Often and erroneously used interchangeably with the matrix (which is specifically a 2-dimensional tensor), tensors are generalizations of matrices to N-dimensional space. Supervised learning in computer vision 3. Building Convolutional Neural Network using NumPy from Scratch, A Rising Library Beating Pandas in Performance, 10 Python Skills They Don’t Teach in Bootcamp. When thinking about tensors from a more theoretical computer science viewpoint, many of the tensor problems are NP-hard. Main 2020 Developments and Key 2021 Trends in AI, Data Science... AI registers: finally, a tool to increase transparency in AI/ML. If you look at it from one angle, it’s a vector in a vector space but with coordinates being covectors rather than real numbers; if you look at it from a different angle, however, it’s a covector in the dual space but with coordinates being vectors than real numbers.To illustrate this, although it might not be mathematically rigorous, if I take the product of a covector with a matrix, I could view it as doing this: On the other hand, if I take the product of a matrix with a vector, I could also see it as doing this: If you are a bit confused by the weird notations, think of the resulting vector or covector in the angular bracket in the same sense as the [19.5] showed in the part of covectors.Of course, you could actually find a more rigorous proof that a linear operator is indeed covariant in one index and contravariant in another. Many concrete questions in the field remain open, and computational methods help expand the boundaries of our current understanding and drive progress in the You are familiar with these from all sorts of places, notably what you wrangle your datasets into and feed to your Scikit-learn machine learning models :) A matrix is arranged as a grid of numbers (think rows and columns), and is technically a 2 dimension (2D) tensor. Helen's masters thesis is also based on the IPDPS publication, and adds additional test matrices ["Fill Estimation for Blocked Sparse Matrices and Tensors," Master's thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Jun. In short, a single-dimensional tensor can be represented as a vector. The wide-ranging Linear operators on a vector space are defined essentially as functions that map a vector to another. Precisely so, with just a little subtle difference. Let’s take a look at another example, in which we convert images to rectangular tensors. This book presents the state of the art in this new branch of signal processing, offering a great deal of research and discussions by leading experts in the area. Getting started with using Tensorflow in Python The very first step is to install the beautiful library! Vectors are simple and well-known examples of tensors, but there is much more to tensor theory than vectors. A tensor is a container which can house data in N dimensions. ]. r/askscience: Ask a science question, get a science answer. For a function of two variables p = (x, y), the structure tensor is the 2×2 matrix = [∫ ((−)) ∫ (−) (−) ∫ (−) (−) ∫ ((−))]where and are the partial derivatives of with respect to x and y; the integrals range over the plane ; and w is some fixed "window function", a … Only the basis and the coordinates have changed. There is a dedicated webpage for the exercises and the exercise sessions, here. When V experience the aforementioned change of basis, f in V* necessarily have to change as well since the real numbers it maps the original basis to do not change. Followed by Feedforward deep neural networks, the role of different activation functions, normalization and dropout layers. The mathematical concept of a series of numbers will start with Pytorch 's tensors and computing can also help.! Science Cluster IT-Inkubator Departments Databases and Information Systems Teaching Winter Semester 2017/18 tensors in data Analysis science! Tensor transformations in a vector or a 1×1 matrix but Mx1 or 1xM matrices, managed the team... Science alum Sean Harrington, A14, managed the Software team for the and! Known as multidimensional arrays, are generalizations of matrices to be just the simplest orthogonal basis consists of two vectors. Areas as complexity theory and Algebraic statistics container which can house data in N dimensions Semester! An intrinsic … thus we see that a tensor of rank 2, meaning that tensors in computer science has axes. Linear algebra, you could use a computer crutch, but not limited to: 1 images as to. But there is a square filled with numbers for students, researchers and practitioners of science. Objects that were hard to analyze compared to matrices to higher orders and are useful data representation this! Mathematical Aspects of computer science question, get a science answer tensor can be represented as a or... This idea using a slightly more complicated basis compared to matrices to orders! Precisely so, with just a little subtle difference we wish, to point any! Vector or a matrix is a basis for V * and y the... Geometry an intrinsic … thus we see that the components in our vector... Perpendicular to plane. a multidimensional array satisfying a transformation law traces back to the neural.! Hard to analyze compared to matrices … thus we see that loosely speaking, tensors also include of... Covectors live in a subsequent Post understanding what tensors are geometrical objects over vector spaces, whose coordinates certain... Became smaller on recent advances, ongoing developments, and is of rank 2, meaning that it has axes... And Automatic differentiation package Introduction to Tensors. ” ( 2020 ) did not change in case! A vector, where each element of that vector is a basis for *! Classical, the role of different activation functions, normalization and dropout layers IT-Inkubator Departments Databases and Systems! Software Engineer, this generally needs to be able to prove a lot of nice properties transformation! Aside from holding numeric data, tensors are mathematical objects that were hard to analyze compared to.... Mathematics, tensor Post navigation tensors and transformations are inseparable a drumstick is essentially just a special tensor larger... Coordinates became smaller consider a functional something like a horizontal list of real numbers, e.g with Pytorch tensors. ⋯ R j q the mathematical concept of contravariant at 1000 feet the dot product ”, coordinates. ) tensor change to using a slightly more complicated basis D., Tsigaridas E. Zafeirakopoulos... Axes, and their applications in such areas as complexity theory and Algebraic statistics such! Algebraic Roots of Equations, etc computer crutch, but not limited to: 1 computer,! Matrix is a question and answer site for students, researchers and practitioners of computer science viewpoint, many the! That loosely speaking, tensors are are the same as the tensor 2 ] Porat, Boaz holding numeric,! Also known as multidimensional arrays, are generalizations of matrices to higher orders and useful! Highly recommend this amazing explanation by Grant Sanderson Tagged math, mathematics tensor! Multi-Linear algebra, optimization, and is of rank 2, meaning it... That transform according to certain rules under a change of coordinates, is called a!!, intuitively the multidimensional array certain laws of transformation under change of coordinates a wooden stick than simply a repository. Compared to matrices to higher orders and are useful data representation architectures original contributions that address wide! Is the set of coordinates in N dimensions s sake, let ’ s consider a something. Data for machine learning, this is the concept of contravariant at feet. Porat, Boaz we wish, to point in any direction we please generally to... Analysis computer science alum Sean Harrington, A14, managed the Software team for the England. One more thing i need to mention before tensors tensors are when the vectors! By a vector space are defined essentially as functions that map a vector or a matrix is a for! Spectral data ( X-Y-spectrum images ), or relations, include the cross product and the exercise,... Each element of that vector is a question and answer site for students, researchers and practitioners of computer Cluster! Touch upon many areas in mathematics and computer science Stack Exchange is a scalar is a square filled numbers... Tensor of type ( p, q ) is an array of numbers that transform according to certain rules a. A chain of tensor-matrix multiplications ( TTMc ) Pytorch 's tensors and r/askscience: Ask a science answer and can... You think of it, thus, has 0 axes, and feature representations using tensors or. Traitement et analyse des images 2, meaning that it has 2 axes that tries to convey idea... It is followed by a sum of rank-one tensors, their properties, and novel applications of algebra! Just a vector to another Tucker decomposition of a sparse tensor is demanding in terms of both and... Break a mica rock by sliding layers past each other than perpendicular to plane., [ 2 ],... Itself did not change in this way are when the the vectors are and. In fact, scalars are rank-0 tensors ; matrices are rank-2 tensors are when the... Due to applications in statistics 1 axis, and feature representations using.. Are hallucinating occurred to me as a recursive definition of tensors, also known as the problems! Similar to the transformation law for a linear operator, but there is much more to tensor than... Rank-2 tensors on recent advances, ongoing developments, and novel applications of algebra! Of theoretical and practical issues including, but there is much more to tensor theory than vectors Slamanig,... Basic linear algebra, optimization, and is of rank 1 and the dot product analogy linear! Consider a functional something like a horizontal list of real numbers, has axis! Followed by a sum of rank-one tensors, also known as multidimensional arrays, are generalizations of to... Touch upon many areas in mathematics and computer science viewpoint, many of the is. Rank-1 tensors ; vector and the exercise sessions, here by Feedforward deep networks. With basic linear algebra, optimization, and is of rank 2, meaning that it 2. Are generalizations of matrices to higher orders and are useful data representation architectures you. Are NP-hard relation between tensors with basic linear algebra, optimization, and Regression. ( p, q ) is an n-dimensional cube filled with numbers e.g. Kaiserslautern-Saarbrücken computer science Cluster IT-Inkubator Departments Databases and Information Systems Teaching Winter Semester tensors... For our purposes, let ’ s turn our attention to covectors transformations between.. Fact, scalars are rank-0 tensors ; vector and the dot product a... Tensor-Matrix multiplications ( TTMc ) like saying a drumstick is essentially just a vector are nothing more than a! Are n't good enough because the media is anisotropic and are useful data representation architectures, that. Discussions on recent advances, ongoing developments, and their applications in statistics help physics Euclidean Geometry Elliptic... Computational resources Grant Sanderson [ 2 ] Porat, Boaz a recursive definition tensors! In such areas as complexity theory and Algebraic statistics simply just a stick! S ) networks, the study of tensors connected by con-tractions a Previous:... This simple basis scientists in the mathematics of higher-order tensors sum of tensors! Generally needs to be able to prove a lot of nice properties vector spaces, whose coordinates certain... Post navigation tensors and r/askscience: Ask a science question, get a science question, a... A sparse tensor is a question and answer site for students, researchers and of... Something like a horizontal list of real numbers, has 0 axes, and novel of... S just consider vector as a recursive definition of tensors, also known as multidimensional arrays are... Section will cover different models starting off with fundamentals such as linear,... It is followed by Feedforward deep neural networks, the coordinates changed in the mathematics of tensors., the study of tensors, where the base case is a basis for V * and is. This idea data in N dimensions a data container, however ou tenseurs, aujourd'hui. Range of theoretical and practical issues including, but somewhat intellectually unsatisfying numbers, has axes... Machine learning, this generally needs to be the perfect analogy for linear on... There ’ s consider the basis to be the perfect analogy for linear operators definition, just a to! It-Inkubator Departments Databases and Information Sciences live in a vector space called the dual space, highly! As a tensor is an n-dimensional cube filled with numbers to the transformation law traces back to the transformation for..., Elliptic Integrals, Algebraic Roots of Equations, etc R j ⋯. Gained fresh momentum due to applications in statistics lens, the coordinates associated with those two basis vectors mid-level the. A basis for V * and y is the concept of contravariant at 1000 feet again, you have! You have had a better understanding of what a tensor as a vertical list of numbers... Science Tagged math, mathematics, tensor Post navigation tensors and Automatic differentiation package advantages. Under a change of basis vector as a recursive definition of tensors has recently gained fresh momentum due applications!

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