This is mostly a semantic argument, but I find this to be a very annoying perspective. Given a basis, there is a natural isomorphism between tensors of a certain type and multidimensional arrays of certain dimensions.
Of course there is, but if you perform an operation on a multidimensional array, there is no guarantee it corresponds to an operation on tensors, ie. the resulting tensor may depend on the basis.
Sure, if you perform an arbitrary operation on a multidimensional array. But the same is true of any representation of any mathematical object. It makes no physical sense to take the sine of a mass, or two to the power of a length. But that doesn't mean that whenever someone says "oh, the mass of an object is a real number" I need to nitpick them.
I'm not familiar with tensor contraction as practiced by a machine learning package, but summation convention is just that, it's not a fundamental property of tensors.
As a way of describing physics or geometry they have additional structure which I'm not seeing.
Comments
This is mostly a semantic argument, but I find this to be a very annoying perspective. Given a basis, there is a natural isomorphism between tensors of a certain type and multidimensional arrays of certain dimensions.
Of course there is, but if you perform an operation on a multidimensional array, there is no guarantee it corresponds to an operation on tensors, ie. the resulting tensor may depend on the basis.
Sure, if you perform an arbitrary operation on a multidimensional array. But the same is true of any representation of any mathematical object. It makes no physical sense to take the sine of a mass, or two to the power of a length. But that doesn't mean that whenever someone says "oh, the mass of an object is a real number" I need to nitpick them.
But it's like calling my table a cat because they both have four legs.
If your table is furry and has a tail it may actually be a cat (from a machine learning perspective).
The machine learning packages have an einsum function / tensor contraction, etc. What more do you need for it to be called a tensor?
I'm not familiar with tensor contraction as practiced by a machine learning package, but summation convention is just that, it's not a fundamental property of tensors.
As a way of describing physics or geometry they have additional structure which I'm not seeing.