Tensor software is a class of mathematical software designed for manipulation and calculation with tensors. Maxima [24] is a free open source general purpose computer algebra system which includes several packages for tensor algebra calculations in its core distribution. It is particularly useful for calculations with abstract tensors, i.

It comes with three tensor packages: [25]. From Wikipedia, the free encyclopedia. This article has multiple issues. Please help improve it or discuss these issues on the talk page.

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Namespaces Article Talk. Views Read Edit View history. Languages Add links. By using this site, you agree to the Terms of Use and Privacy Policy.There has been a lot of recent interest and innovation in compiler technology for high performance sparse tensor computation.

This workshop will bring together relevant parties from academia and industry to discuss individual approaches, how they relate, and whether the ideas can be combined. Program There will be seven session of three speakers. The sessions will consist of three ten minute talks followed by a twenty minute speaker and room discussion.

Liu, C. Wen, A. Sarwate, and M. Mehri Dehnavi Cluster Sparse tensor optimizations for tensor decompisitons used for sparse count data Memory-efficient parallel tensor decompositions Baskaran, M. Hammond, John F. Chesmi, S. Kamil, M. Strout, and M. Strout, M. Hall and C. Olschanowsky Proc.

Blanco, B. Liu, and M. Signal Processing Mehri Dehnavi Cluster. Sparse tensor optimizations for tensor decompisitons used for sparse count data Memory-efficient parallel tensor decompositions Baskaran, M.

Cyclops tensor framework distributed memroy tensor algebra A massively parallel tensor contraction framework for coupled-cluster computations Edgar Solomonik, Devin Matthews, Jeff R.

Mehri Dehnavi SC PPoPP JavaScript is disabled for your browser. Some features of this site may not work without it.

Toggle navigation. Additional downloads Corrected text of report restores missing figure captions Other Contributors Computer Architecture. Advisor Saman Amarasinghe. Terms of use Creative Commons Attribution 4. Metadata Show full item record. Abstract Tensor and linear algebra is pervasive in data analytics and the physical sciences. Often the tensors, matrices or even vectors are sparse.

Computing expressions involving a mix of sparse and dense tensors, matrices and vectors requires writing kernels for every operation and combination of formats of interest. The number of possibilities is infinite, which makes it impossible to write library code for all.

This problem cries out for a compiler approach. This paper presents a new technique that compiles compound tensor algebra expressions combined with descriptions of tensor formats into efficient loops. The technique is evaluated in a prototype compiler called taco, demonstrating competitive performance to best-in-class hand-written codes for tensor and matrix operations. Date issued Search DSpace. This Collection.Login or Subscribe Newsletter. Abby Abazorius Email: abbya mit.

Media can only be downloaded from the desktop version of this website. The table would be mostly zeroes. With sparse data, analytic algorithms end up doing a lot of addition and multiplication by zero, which is wasted computation. Programmers get around this by writing custom code to avoid zero entries, but that code is complex, and it generally applies only to a narrow range of problems.

That code offers a fold speedup over existing, non-optimized software packages. The system is called Taco, for tensor algebra compiler. People figured out a few very specific operations — sparse matrix-vector multiply, sparse matrix-vector multiply plus a vector, sparse matrix-matrix multiply, sparse matrix-matrix-matrix multiply.

The biggest contribution we make is the ability to generate code for any tensor-algebra expression when the matrices are sparse. In recent years, the mathematical manipulation of tensors — tensor algebra — has become crucial to not only big-data analysis but machine learning, too. Traditionally, to handle tensor algebra, mathematics software has decomposed tensor operations into their constituent parts.

So, for instance, if a computation required two tensors to be multiplied and then added to a third, the software would run its standard tensor multiplication routine on the first two tensors, store the result, and then run its standard tensor addition routine. In the age of big data, however, this approach is too time-consuming. Computer science researchers have developed kernels for some of the tensor operations most common in machine learning and big-data analytics, such as those enumerated by Amarasinghe.

But the number of possible kernels is infinite: The kernel for adding together three tensors, for instance, is different from the kernel for adding together four, and the kernel for adding three three-dimensional tensors is different from the kernel for adding three four-dimensional tensors.

Many tensor operations involve multiplying an entry from one tensor with one from another. If either entry is zero, so is their product, and programs for manipulating large, sparse matrices can waste a huge amount of time adding and multiplying zeroes. Hand-optimized code for sparse tensors identifies zero entries and streamlines operations involving them — either carrying forward the nonzero entries in additions or omitting multiplications entirely.

This makes tensor manipulations much faster, but it requires the programmer to do a lot more work. The code for multiplying two matrices — a simple type of tensor, with only two dimensions, like a table — might, for instance, take 12 lines if the matrix is full meaning that none of the entries can be omitted. But if the matrix is sparse, the same operation can require lines of code or more, to track omissions and elisions. Taco adds all that extra code automatically.

For any given operation on two tensors, Taco builds a hierarchical map that indicates, first, which paired entries from both tensors are nonzero and, then, which entries from each tensor are paired with zeroes. All pairs of zeroes it simply discards. Taco also uses an efficient indexing scheme to store only the nonzero values of sparse tensors. But using the Taco compression scheme, it takes up only 13 gigabytes — small enough to fit on a smartphone.

This has the potential to be a real game-changer.

It is one of the most exciting advances in recent times in the area of compiler optimization. MIT News Office. Browse or. Browse Most Popular.Download the video from iTunes U or the Internet Archive. We didn't get terribly far, but I'd like to start with the Cartesian coordinate system that we set up. Rather than using x, y, and z, I'm labeling the axes x1, x2, and x3. And we'll see that the subscripts play a very useful role in the formalism we're about to develop. Now, the first thing we might want to specify in this coordinate is the orientation of a vector and its components.

So let's suppose that this is some vector P. And what I will do to define its orientation is to use the three angles that the vector makes, or the direction makes with respect to x1, x2, x3. And we could define these angles as theta1, that's the angle between the direction and x1, theta2, the angle between our direction or our vector and x2, and finally, not surprisingly, I'll call this one theta3.

So the three components of the vector could be written as P1, the component along x1 is going to be the magnitude of P times the cosine of theta1. The x2 component of P would be the magnitude of P times the cosine of theta2. And P3, the third component, would be the magnitude of P times the cosine of theta3. Now, we will have so many relations that involve the cosine of the angle between a direction and one of our reference axes that it is convenient to define a special term for the cosines of these angles.

So I'll define this as magnitude of P times the quantity l1, magnitude of P times l2, magnitude of P times l3, which is a lot easier to write. And we will define these things as the direction cosines. With these equations it's easy to attach some meaning to the direction cosines. Suppose we had a vector of magnitude 1, something that we will refer to as a unit vector.

And if we put in magnitude of P equal to 1, it follows that l1m l2m l3 are simply the components of a unit vector in a particular direction along, obviously, x1, x2, and x3, respectively.

Trivial piece of algebra, but it attaches a physical and geometric significance to the direction cosines. Now, the vector is something that could represent a physical quantity.

In any case, it is something that is absolute. And it sits embedded majestically, relative to some absolute coordinate system. The magnitudes of the components P1, P2, and P3 will change their values if we would decide to change the coordinate system that we're using as our reference system. So the next question we might ask is, suppose we change the coordinate system to some new values, x1 prime, x2 prime, and x3 prime? And I'll illustrate my point with just a two dimensional analog of this.Contact: Julian J.

Shun, jshun csail. Reminders to: seminars csail.

Abstract: Tensor and Linear Algebra are powerful tools with applications in data analytics, machine learning, science, and engineering.

The massive growth of data in these applications makes performance critical. For applications that use sparse tensors, where most components are zeros, programmers must choose between libraries with hand-optimized implementations of select operations and generalized software systems with poor performance.

In this talk, I will present compiler abstractions and techniques that combine tensor expressions with specifications of sparse irregular tensor data structures to produce efficient parallel source code.

I will show solutions to the three main problems of sparse tensor algebra compilation: how to represent tensor data structures, how to characterize sparse iteration spaces, and how to generate code to coiterate over irregular data structures. I will also show how to optimize sparse tensor algebra code in a compiler and how to programmatically map sparse data to tensors.

We have implemented these techniques in the TACO sparse tensor algebra compiler. It is the first compiler to generate sparse code for any basic tensor expression on many sparse tensor representations.

The generated code matches or exceeds the performance of hand-optimized libraries while generalizing to any expression and many user-specified irregular data structures. He will join Stanford as an Assistant Professor in He has received the Eureka and Rosing prizes for his bachelor project, the Adobe Fellowship, a best poster award, and two best paper awards. Created by Julian J. Shun at Wednesday, July 03, at PM. Seminar Series List. The Sparse Tensor Algebra CIt is the free algebra on Vin the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing Vin the sense of the corresponding universal property see below.

## Invited Workshop on Compiler Techniques for Sparse Tensor Algebra

The tensor algebra is important because many other algebras arise as quotient algebras of T V. These include the exterior algebrathe symmetric algebraClifford algebrasthe Weyl algebra and universal enveloping algebras.

The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebraand a more complicated one, which yields a bialgebraand can be extended by giving an antipode to create a Hopf algebra structure.

Note : In this article, all algebras are assumed to be unital and associative. The unit is explicitly required to define the coproduct.

Let V be a vector space over a field K. For any nonnegative integer kwe define the k th tensor power of V to be the tensor product of V with itself k times:.

That is, T k V consists of all tensors on V of order k.

### Tensor algebra

By convention T 0 V is the ground field K as a one-dimensional vector space over itself. This multiplication rule implies that the tensor algebra T V is naturally a graded algebra with T k V serving as the grade- k subspace. The construction generalizes in a straightforward manner to the tensor algebra of any module M over a commutative ring. If R is a non-commutative ringone can still perform the construction for any R - R bimodule M.

It does not work for ordinary R -modules because the iterated tensor products cannot be formed. The tensor algebra T V is also called the free algebra on the vector space Vand is functorial.

**Determinants and Volume - MIT 18.06SC Linear Algebra, Fall 2011**

As with other free constructionsthe functor T is left adjoint to some forgetful functor. In this case, it's the functor that sends each K -algebra to its underlying vector space.

Explicitly, the tensor algebra satisfies the following universal propertywhich formally expresses the statement that it is the most general algebra containing V :. Here i is the canonical inclusion of V into T V the unit of the adjunction. One can, in fact, define the tensor algebra T V as the unique algebra satisfying this property specifically, it is unique up to a unique isomorphismbut one must still prove that an object satisfying this property exists.

The above universal property shows that the construction of the tensor algebra is functorial in nature. If V has finite dimension nanother way of looking at the tensor algebra is as the "algebra of polynomials over K in n non-commuting variables".

If we take basis vectors for Vthose become non-commuting variables or indeterminates in T Vsubject to no constraints beyond associativitythe distributive law and K -linearity. Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain relations on the generators, i. Examples of this are the exterior algebrathe symmetric algebraClifford algebrasthe Weyl algebra and universal enveloping algebras.

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