Tensors, the geometric tool that solved Einstein's relativity problem
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Tensors, the geometric tool that solved Einstein's relativity problem
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Re: Tensors, the geometric tool that solved Einstein's relativity problem
#2If you have any linear algebra background, then the definition of a tensor is straightforward: given a vector space V over a field K (in physics, K = R or C), a tensor T is a multilinear (i.e. linear in each argument) function from vectors and dual vectors in V to numbers in K. That's it! A type (p, q) tensor T takes p vectors and q dual vectors as arguments (p+q is often called the rank of T but is ambiguous compared to the type).
(If you're unfamiliar with the definition of dual vector, it's even simpler: it's just a linear function from V to K.)
Re: Tensors, the geometric tool that solved Einstein's relativity problem
#3I've always thought the use of "Tensor" in the "TensorFlow" library is a misnomer. I'm not too familiar with ML/theory, is there a deeper geometric meaning to the multi-dimensional array of numbers we are multiplying or is "MatrixFlow" a more appropriate name?
Re: Tensors, the geometric tool that solved Einstein's relativity problem
#4> Talk to a computer scientist, and they might tell you that a tensor is an array of numbers that stores important data
The conflicting definitions of tensors have precedent in lower dimensions: vectors were already being used in computer science to mean something different than in mathematics / physics, long before the current tensormania.
Its not clear if that ambiguity will ever be a practical problem though. For as long as such structures are containers of numerical data with no implied transformation properties we are really talking about two different universes.
Things might get interesting though in the overlap between information technology and geometry [1] :-)
Re: Tensors, the geometric tool that solved Einstein's relativity problem
#5I would argue that today, geometric algebra/Clifford calculus and space time algebra are more intuitive and useful.
Gibbs/Heavysides vectors were more popular at the time.
At least for me.
Re: Tensors, the geometric tool that solved Einstein's relativity problem
#6If you have any linear algebra background, then the definition of a tensor is straightforward: given a vector space V over a field K (in physics, K = R or C ), a tensor T is a multilinear (i.e. linear in each argument) function from vectors and dual vectors in V to numbers in K . That's it! A type (p, q) tensor T takes p vectors and q dual vectors as arguments ( p+q is often called the rank of T but is ambiguous comp…
The definition may be simple, but it's not very concrete and I'd argue that makes it not strait forward. While examples of vector spaces can be very concrete (think R, R^2, R^30), I struggle to think of a concrete example of a multilinear function from vectors and dual vectors in V to numbers in K. On top of that when working with tensors, you don't usually use the definition os a multilinear function at least as far as I remember.
Re: Tensors, the geometric tool that solved Einstein's relativity problem
#7I've always thought the use of "Tensor" in the "TensorFlow" library is a misnomer. I'm not too familiar with ML/theory, is there a deeper geometric meaning to the multi-dimensional array of numbers we are multiplying or is "MatrixFlow" a more appropriate name?
The tensors in tensorflow are often higher dimensional. Is a 3d block of numbers (say 1920x1080x3) still a matrix? I would argue it's not. Are there transformation rules for matrices?
You're totally correct that the tensors in tensorflow do drop the geometric meaning, but there's precedence there from how CS vs math folk use vectors.
Re: Tensors, the geometric tool that solved Einstein's relativity problem
#8If you have any linear algebra background, then the definition of a tensor is straightforward: given a vector space V over a field K (in physics, K = R or C ), a tensor T is a multilinear (i.e. linear in each argument) function from vectors and dual vectors in V to numbers in K . That's it! A type (p, q) tensor T takes p vectors and q dual vectors as arguments ( p+q is often called the rank of T but is ambiguous comp…
The definition may be simple, but it's not very concrete and I'd argue that makes it not strait forward. While examples of vector spaces can be very concrete (think R, R^2, R^30), I struggle to think of a concrete example of a multilinear function from vectors and dual vectors in V to numbers in K. On top of that when working with tensors, you don't usually use the definition os a multilinear function at least as far…
Not really to push back as I do agree that this is a bit trickier to get an intuition for than the OP suggests, but the most trivial concrete example of a (1, 1) tensor would just be the evaluation function (v, f) |-> f(v), which, given a metric, corresponds to the inner product.
Re: Tensors, the geometric tool that solved Einstein's relativity problem
#9If you have any linear algebra background, then the definition of a tensor is straightforward: given a vector space V over a field K (in physics, K = R or C ), a tensor T is a multilinear (i.e. linear in each argument) function from vectors and dual vectors in V to numbers in K . That's it! A type (p, q) tensor T takes p vectors and q dual vectors as arguments ( p+q is often called the rank of T but is ambiguous comp…
For those without a strong math background but more of a programmers background;
You know the matrices you work with in 2D or 3D graphics environments that you can apply to vectors or even other matrices to more easily transform (rotate, translate, scale)?
Well tensors are the generalisation of this concept. If you’ve noticed 2D games transformation matrices seem similar (although much simpler) to 3D games transformation mateices you’ve probably wondered what it’d look like for a 4D spacetime or even more complex scenarios. Well you’ve now started thinking about tensors.
Re: Tensors, the geometric tool that solved Einstein's relativity problem
#10If you have any linear algebra background, then the definition of a tensor is straightforward: given a vector space V over a field K (in physics, K = R or C ), a tensor T is a multilinear (i.e. linear in each argument) function from vectors and dual vectors in V to numbers in K . That's it! A type (p, q) tensor T takes p vectors and q dual vectors as arguments ( p+q is often called the rank of T but is ambiguous comp…
For those without a strong math background but more of a programmers background; You know the matrices you work with in 2D or 3D graphics environments that you can apply to vectors or even other matrices to more easily transform (rotate, translate, scale)? Well tensors are the generalisation of this concept. If you’ve noticed 2D games transformation matrices seem similar (although much simpler) to 3D games transforma…
Slight disagree here -- matrices are enough for transformations in 2, 3, 4, and 100 dimensions. Tensors are not arrays with more rows and columns; they are higher dimensional objects -- more indices, not greater range of indices.