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A tensor is a multilinear map that is invariant under coordinate change? What do those have to do with machine learning?
by Konohamaru 6y ago
A tensor is a multilinear map that is invariant under coordinate change? What do those have to do with machine learning?
- the_svd_doctor 6y agoIt basically means a multidimensional array in ML.
- Konohamaru 6y agoThat's a severe abuse of terminology, almost to the point of ignorance. A tensor is an algebraic object (not limited to linear algebra) with a very specific definition.
- 7402 6y agoWhat's going on here is a collision between the way certain terms are used in two different fields. People who learn about tensors and vectors in the culture of Mechanical Engineering, Applied Mathematics, or Physics, are strenuously trained that a "vector" is not just any list of numbers, but a specific mathematical object with specific properties. (Similarly with tensors.) In that culture, the use of the term "vector" the way it is used in Software Engineering looks quite odd. An undergraduate might even have a problem set that shows various number triples and be asked to identify which ones are vectors and which ones aren't. I was educated under that system, but spent years as a software engineer, so I am comfortable switching between the two contexts. When I wear my Software Engineer hat, I am quite comfortable with the generalized use of vector and tensor as done in software and ML, but for someone who has only grown up in one culture, it looks very wrong.
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- ajtulloch 6y agocrudely, linear transformations : matrices :: multilinear transformations : tensors.
- Konohamaru 6y agoIt has to behave in a certain way to be a tensor.
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- ogogmad 6y agoGiven a vector space V over a field F, a tensor is a multilinear map T: V x V x V x ... x V^* x V^* x V^* -> F You're mixing up geometric objects (multilinear transformations) with their coordinate representations. A geometric object, such as a multilinear map, is automatically invariant to the coordinate system. An array of numbers, like a matrix, is not.
- Konohamaru 6y agoActually you're right. I was mixing up covariant and contravariant vectors with tensors. Although tensors do decompose into covariant and contravariant components, as your definition states.
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