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It's true though that vectors are basically 1D lists if we're speaking about elements from the vector space R^n. Is it true that tensors are multidimensional ar

It's true though that vectors are basically 1D lists if we're speaking about elements from the vector space R^n. Is it true that tensors are multidimensional arrays if they come from the space R^{m1 x m2 x ... x mn} ? I'm also confused: do you think about a tensor as a vector from vector space? as a object that maps between vector spaces? or something completely different? I realize there's a LOT of detail here - just trying to flesh out my 500,000,000 mile view a little more.

tel 11y ago
It's not true, though. A list of numbers in R is not a vector until you imbue the position of each of those numbers with a geometric basis, or, alternatively, relate that list of numbers to the set of all lists of such numbers along with some algebraic operations which turn it into a vector space. This is similar with tensors. It's even harder to talk about them, though, without the geometric/algebraic bits.
im3w1l 11y ago
They use tensors here to mean simply multidimensional array. And as this discussion shows, that choice of words is unfortunate and confusing.
- rspeer 11y ago
  What I'm seeing is that physicists and geometers disagree with programmers about what "tensor" means -- not in that their views are inconsistent, but in that the programmers' view is too limited for the things that the physicists and geometers want to do with tensors. They also disagree about what "vector" and "matrix" mean for the same reason, but they can't really put that horse back in the barn. But "tensors" remain unfamiliar enough that this disagreement can come up whenever anyone says "tensor". Here's a possible truce: - Programmers should default to talking about "arrays" - Geometers and physicists should default to talking about "vectors", "matrices", and "tensors" - We all generally recognize that some (vectors|matrices|tensors) are arrays, and some arrays are (vectors|matrices|tensors), and it's not the worst thing ever if someone uses one to mean the other - We all generally recognize that you can sometimes do damn cool things by using an array as a tensor or a tensor as an array, so there isn't that much benefit in shouting "NO THEY'RE TOTALLY DIFFERENT".
  - tel 11y ago
    Eh, I think as you stack more and more "structure" atop something it becomes less and less appropriate to confuse it with its representation. There is afterall already a completely appropriate, cross-disciplinary term for what "programmers" here call "tensors": multidimensional arrays, or even nd-arrays. No mathematician will bat an eye if you say that a tensor can be represented by an nd-array---but if you equate them, then you're clearly missing something.
    - dvanduzer 11y ago
      So... Do "programmers" think of "matrix" and "tensor" as "synonymous"? edit: that was actually a serious question, because i really don't understand this idea that programmers have a common notion of what a matrix is.
      - tel 11y ago
        I didn't mention the term "matrix". I think it has a much more complex story with respect to common use by "programmers" along with vector. Since linear algebra is such a commonly taught subject there are lots of people with practical working knowledge of matrices. If you don't have the geometric or abstract algebraic perspective here then matrix is almost certainly "2d-array" at first blush, but you'll also admit that there's a sort of non-trivial relationship between these "matrices" and size-conformant "vectors" (1d-arrays) and another one between two size-conformant matrices. This distinction is made from time to time [0][1]. Finally, of course, a matrix is a special case of a tensor. Tensor is such a broader topic that it becomes more and more difficult to talk about it without its geometric underpinnings. This is what I was writing about above. [0] Is a black-and-white image a matrix? I'd imagine a lot of people would have a bit of a tough time saying yes directly---and for good reason. But there's also a reasonable argument for pretending like it is, and even reasonable arguments for turning it into one! [1] Numpy distinguishes between nd-arrays and matrices exactly right in that matrices are 2d-arrays only and are imbued with multiplication as composition of maps. In practice, many people I know who are Numpy users just think of matrix as a convenient way to overload *, though.
  - 11y ago
- musgravepeter 11y ago
  I agree it's unfortunate. I think this is really a flimsy use of a technical term which does have a precise meaning in the context of mathematical physics to make the idea of multi-dimensional array seem "deeper" than it really is. There are enough hard problems in data science - I don't think it needs to burdened with terms like this.
  - tel 11y ago
    Especially if information geometry ever grows and data scientists start to actually have to use real tensors.
musgravepeter 11y ago
A 1D list of numbers is not necessarily a "vector" in the Differential Geometry sense. The definition of a vector or tensor is based on how it transforms as the co-ordinate system transforms. See e.g. http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors http://en.wikipedia.org/wiki/Covariance_and_contravariance_o... One weird consequence of this is that if A and B are vectors, the cross product (AxB) is NOT a vector in the tensorial sense. [Aside: Cross products are really weird since they are defined in the familiar way 3 and 7 dimensions only! (see http://math.stackexchange.com/questions/185991/is-the-vector-cross-product-only-defined-for-3d) http://math.stackexchange.com/questions/185991/is-the-vector...]
evanpw 11y ago
The word "vector" just means an element of a vector space: a collection of "things" that you can add together and multiply by numbers. The vector space R^n is defined as the collection of all lists of n real numbers, so you are right in that sense. But usually when people say that a vector "is" a list of numbers, they mean that if you choose a coordinate system, then the coordinates of a vector uniquely identify it. This is exactly analogous to locations on the earth being uniquely identified by their latitude and longitude: we first have to make a (somewhat arbitrary) choice of coordinates. You wouldn't say that a point on the earth "is" a pair of numbers, though. The case of R^n is confusing, because if you choose the usual coordinate system, then the coordinates of a vector are the same list of numbers as the vector itself. I like to think of tensors as functions that map some number of vectors to some other number of vectors. But you can add together two tensors (with the same number of inputs and outputs) or multiply a tensor by a number, so you can also think of them as vectors in some other vector space.
- platz 11y ago
  Almost seems like tensors are natural transformations in the category theoretic sense.

tel 11y ago

It's not true, though. A list of numbers in R is not a vector until you imbue the position of each of those numbers with a geometric basis, or, alternatively, relate that list of numbers to the set of all lists of such numbers along with some algebraic operations which turn it into a vector space. This is similar with tensors. It's even harder to talk about them, though, without the geometric/algebraic bits.

im3w1l 11y ago

They use tensors here to mean simply multidimensional array. And as this discussion shows, that choice of words is unfortunate and confusing.

rspeer 11y ago
What I'm seeing is that physicists and geometers disagree with programmers about what "tensor" means -- not in that their views are inconsistent, but in that the programmers' view is too limited for the things that the physicists and geometers want to do with tensors. They also disagree about what "vector" and "matrix" mean for the same reason, but they can't really put that horse back in the barn. But "tensors" remain unfamiliar enough that this disagreement can come up whenever anyone says "tensor". Here's a possible truce: - Programmers should default to talking about "arrays" - Geometers and physicists should default to talking about "vectors", "matrices", and "tensors" - We all generally recognize that some (vectors|matrices|tensors) are arrays, and some arrays are (vectors|matrices|tensors), and it's not the worst thing ever if someone uses one to mean the other - We all generally recognize that you can sometimes do damn cool things by using an array as a tensor or a tensor as an array, so there isn't that much benefit in shouting "NO THEY'RE TOTALLY DIFFERENT".
- tel 11y ago
  Eh, I think as you stack more and more "structure" atop something it becomes less and less appropriate to confuse it with its representation. There is afterall already a completely appropriate, cross-disciplinary term for what "programmers" here call "tensors": multidimensional arrays, or even nd-arrays. No mathematician will bat an eye if you say that a tensor can be represented by an nd-array---but if you equate them, then you're clearly missing something.
  - dvanduzer 11y ago
    So... Do "programmers" think of "matrix" and "tensor" as "synonymous"? edit: that was actually a serious question, because i really don't understand this idea that programmers have a common notion of what a matrix is.
    - tel 11y ago
      I didn't mention the term "matrix". I think it has a much more complex story with respect to common use by "programmers" along with vector. Since linear algebra is such a commonly taught subject there are lots of people with practical working knowledge of matrices. If you don't have the geometric or abstract algebraic perspective here then matrix is almost certainly "2d-array" at first blush, but you'll also admit that there's a sort of non-trivial relationship between these "matrices" and size-conformant "vectors" (1d-arrays) and another one between two size-conformant matrices. This distinction is made from time to time [0][1]. Finally, of course, a matrix is a special case of a tensor. Tensor is such a broader topic that it becomes more and more difficult to talk about it without its geometric underpinnings. This is what I was writing about above. [0] Is a black-and-white image a matrix? I'd imagine a lot of people would have a bit of a tough time saying yes directly---and for good reason. But there's also a reasonable argument for pretending like it is, and even reasonable arguments for turning it into one! [1] Numpy distinguishes between nd-arrays and matrices exactly right in that matrices are 2d-arrays only and are imbued with multiplication as composition of maps. In practice, many people I know who are Numpy users just think of matrix as a convenient way to overload *, though.
- 11y ago
musgravepeter 11y ago
I agree it's unfortunate. I think this is really a flimsy use of a technical term which does have a precise meaning in the context of mathematical physics to make the idea of multi-dimensional array seem "deeper" than it really is. There are enough hard problems in data science - I don't think it needs to burdened with terms like this.
- tel 11y ago
  Especially if information geometry ever grows and data scientists start to actually have to use real tensors.

musgravepeter 11y ago

A 1D list of numbers is not necessarily a "vector" in the Differential Geometry sense. The definition of a vector or tensor is based on how it transforms as the co-ordinate system transforms. See e.g. http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors http://en.wikipedia.org/wiki/Covariance_and_contravariance_o... One weird consequence of this is that if A and B are vectors, the cross product (AxB) is NOT a vector in the tensorial sense. [Aside: Cross products are really weird since they are defined in the familiar way 3 and 7 dimensions only! (see http://math.stackexchange.com/questions/185991/is-the-vector-cross-product-only-defined-for-3d) http://math.stackexchange.com/questions/185991/is-the-vector...]

evanpw 11y ago

The word "vector" just means an element of a vector space: a collection of "things" that you can add together and multiply by numbers. The vector space R^n is defined as the collection of all lists of n real numbers, so you are right in that sense. But usually when people say that a vector "is" a list of numbers, they mean that if you choose a coordinate system, then the coordinates of a vector uniquely identify it. This is exactly analogous to locations on the earth being uniquely identified by their latitude and longitude: we first have to make a (somewhat arbitrary) choice of coordinates. You wouldn't say that a point on the earth "is" a pair of numbers, though. The case of R^n is confusing, because if you choose the usual coordinate system, then the coordinates of a vector are the same list of numbers as the vector itself. I like to think of tensors as functions that map some number of vectors to some other number of vectors. But you can add together two tensors (with the same number of inputs and outputs) or multiply a tensor by a number, so you can also think of them as vectors in some other vector space.

platz 11y ago
Almost seems like tensors are natural transformations in the category theoretic sense.