5 ms·

Basically, order of a tensor is the number of dimensions of an array of data. Vector - one dimensional array or a tensor of order 1 Matrix - two dimensional arr

Basically, order of a tensor is the number of dimensions of an array of data. Vector - one dimensional array or a tensor of order 1 Matrix - two dimensional array of data or a tensor of order 2 Three and more dimensional arrays of data are tensors of higher order.

ska 8y ago
This isn't quite right, but it's a common misconception. Any rank-2 tensor can be represented in matrix form, but not all matrices are tensors, similarly with rank-1 tensors and vectors. The distinction is important because thinking about the way you have presented leads to confusion about what tensors are...
- IlyaKisil 8y ago
  Was just explaining the meaning of “higher-order” within the context of this library. But thanks, now I have a clearer picture of what to include in the main README file and notebooks with tutorials, so to avoid misleading.
- Lxr 8y ago
  Can you explain this distinction in more detail?
  - ska 8y ago
    Sure! There are several different ways to define them, but fundamentally tensors ar geometric objects that define linear relationships with respect to particular vector spaces. They are often defined somewhat loosly by their behaviour under transformations. In context we often refer to scalars, vectors,matrices as order/rank 0,1,2 tensors (higher order tensors don’t have the same sort of common shorthand). This works fine when you have the context of the underlying vector space, and nderstand the “rules”. Physicists do this a lot, and they often love shortcuts :) However, there is a growing use/abuse of the terminology (see machine learning) to just mean n-dimensional arrays. The analogy is drawn that a matrix is a 2d version but you can have 3D, 4d, etc. While it’s true that a NxN matrix can represent a tensor (given the context as previously) that misses most of the structure... as such, it’s an unfortunate use of the name.
- dbranes 8y ago
  Sorry this is just wrong. Maybe you’re trying to get at the “co/contravariant” properties of tensors, in which case your statement can be more clearly stated as, e.g., the space rank (2, 0), and rank (1,1) vectors, admit different interpretations as internal hom spaces of vector spaces. But in any interpretation of your statement the distinction is never important because all spaces distinguished by this distinction are isomorphic via cononical isomorphisms.
  - ska 8y ago
    You might want to think that through a bit more, leaving aside the issue of needing the underlying vector space, are you sure you are comfortable with the statement that all matrices are tensors?
    - dbranes 8y ago
      Yes absolutely. Given a matrix as an array of numbers there's a number of natural ways to interpret it as a tensor. What are you uncomfortable about?
ataspinar 8y ago
As was stated by others before, I find the name very misleading. I know "Higher Order Tensors Toolbox" sounds way cooler than "ndarray toolbox", but wouldn't the second one be closer to the truth? Furthermore, if 1-D Arrays are called vectors, and 2-D Arrays are called Matrices and 3-D (or higher) Arrays are called Tensors, does that not automatically mean that every Tensor is higher order? ...

ska 8y ago

This isn't quite right, but it's a common misconception. Any rank-2 tensor can be represented in matrix form, but not all matrices are tensors, similarly with rank-1 tensors and vectors. The distinction is important because thinking about the way you have presented leads to confusion about what tensors are...

IlyaKisil 8y ago
Was just explaining the meaning of “higher-order” within the context of this library. But thanks, now I have a clearer picture of what to include in the main README file and notebooks with tutorials, so to avoid misleading.
Lxr 8y ago
Can you explain this distinction in more detail?
- ska 8y ago
  Sure! There are several different ways to define them, but fundamentally tensors ar geometric objects that define linear relationships with respect to particular vector spaces. They are often defined somewhat loosly by their behaviour under transformations. In context we often refer to scalars, vectors,matrices as order/rank 0,1,2 tensors (higher order tensors don’t have the same sort of common shorthand). This works fine when you have the context of the underlying vector space, and nderstand the “rules”. Physicists do this a lot, and they often love shortcuts :) However, there is a growing use/abuse of the terminology (see machine learning) to just mean n-dimensional arrays. The analogy is drawn that a matrix is a 2d version but you can have 3D, 4d, etc. While it’s true that a NxN matrix can represent a tensor (given the context as previously) that misses most of the structure... as such, it’s an unfortunate use of the name.
dbranes 8y ago
Sorry this is just wrong. Maybe you’re trying to get at the “co/contravariant” properties of tensors, in which case your statement can be more clearly stated as, e.g., the space rank (2, 0), and rank (1,1) vectors, admit different interpretations as internal hom spaces of vector spaces. But in any interpretation of your statement the distinction is never important because all spaces distinguished by this distinction are isomorphic via cononical isomorphisms.
- ska 8y ago
  You might want to think that through a bit more, leaving aside the issue of needing the underlying vector space, are you sure you are comfortable with the statement that all matrices are tensors?
  - dbranes 8y ago
    Yes absolutely. Given a matrix as an array of numbers there's a number of natural ways to interpret it as a tensor. What are you uncomfortable about?

ataspinar 8y ago

As was stated by others before, I find the name very misleading. I know "Higher Order Tensors Toolbox" sounds way cooler than "ndarray toolbox", but wouldn't the second one be closer to the truth? Furthermore, if 1-D Arrays are called vectors, and 2-D Arrays are called Matrices and 3-D (or higher) Arrays are called Tensors, does that not automatically mean that every Tensor is higher order? ...