Why tensors? A beginner's perspective(mfaizan.github.io) |
Why tensors? A beginner's perspective(mfaizan.github.io) |
For example, in the beginning, author describes tensors as things behaving according to tensor transformation formula. This is already very much a physicist kind of thinking: it assumes that there is some object out there, and we’re trying to understand what it is in terms of how it behaves. It also uses the summation notation which is rather foreign to non-physicist mathematicians. Then, when it finally reaches the point where it is all related to tensors in TensorFlow sense, we find that there is no reference made to the transformation formula, purportedly so crucial to understanding tensors. How comes?
The solution here is quite simple: what author (and physicists) call tensors is not what TensorFlow (and mathematicians) call tensors. Instead, author describes what mathematicians call “a tensor bundle”, which is a correspondence that assigns each point of space a unique tensor. That’s where the transformation rule comes from: if we describe this mapping in terms of some coordinate system (as physicist universally do), the transformation rule tells you how to this description changes in terms of change of the coordinates. This setup, of course, has little to do with TensorFlow, because there is no space that its tensors are attached to, they are just standalone entities.
So what are the mathematician’s (and TensorFlow) tensors? They’re actually basically what the author says, after very confusing and irrelevant introduction talking about change of coordinates of underlying space — irrelevant, because TensorFlow tensors are not attached as a bundle to some space (manifold) as they are on physics, so no change of space coordinates ever happens. Roughly, tensors are a sort of universal objects representing multi linear maps: bilinear maps V x W -> R correspond canonically one-to-one to regular linear maps V (x) W -> R, where V (x) W is a vector space called tensor product of V and W, and tensors are simply vectors in this tensor product space.
Basically, the idea is to replace weird multi linear objects with normal linear objects (vectors), that we know how to deal with, using matrix multiplication and stuff. That’s all there is to it.
The author is perfectly clear in the first sentence that the piece's focus is about the usefulness of tensors in a physics context.
Technically, that’s a tensor field which is a section of the tensor bundle. Similarly, a vector field is a section of the tangent bundle (the collection of all the tangent spaces of the points on the manifold). A vector field is just a choice of a tangent vector for each point from that point’s tangent space.
In grade school it drove me nuts when the homework required us to describe a word without using the word (or it’s Latinate siblings). And yet as an adult there are few enough weeks that go by where some grownup doesn’t try to pull that same trick.
If you think developers are guilty of circular logic, check out some of the math pages on Wikipedia. You can get lost in moments.
Is it just me or are we horrible at teaching advanced math? Where are the examples (with actual numbers)? Where is the motivation? Where are the pictures?
Eg "Numbers (formal) are those objects which behave like numbers (informal)."
It is still a work in progress, but does it help address some of the problems you see in learning mathematics? Any feedback is greatly appreciated. Thanks.
TensorFlow "tensor"(and most other use of "tensor" in programmer jargon) is not a tensor at all, it's just a multidimensional array.
Matrices (or linear transformations in general) are important examples of tensors. There's a nice adjunction between tensor spaces A(x)B and the space of linear transformations B=>C given by:
Hom(A(x)B, C) = Hom(A, B=>C)
In the case of Tensorflow I think they do actually still talk about linear transformations of some kind so it's perfectly fine to call them tensors.
Wald's approach in General Relativity is much better - he treats Tensors as a multilinear map from vectors and dual vectors to scalars.
He then derives the underlying coordinate transformaton rules, for the vector spaces used in differential geometry. But
Most of the article tries to provide some intuition behind why multilinear maps, which sound like a fairly abstract concept, might be relevant in physics. The key link being the importance of coordinate invariance.
I didn’t go into deriving the coordinate transforms from the multilinear map definition as I didn’t feel that it’d provide much better intuition, but I did mention the equivalence near the end.
Yeah, the idea that there are pre-existing things that we're trying to describe is somewhat weird to me when we're trying to come up with a definition of a tensor. The whole point of mathematics is that you come up with the definitions and theorems fall out.
In particular, this comment is funny and speaks to some difference in how I and the author view what we're doing when defining a tensor:
> But why that specific transformation law - why must tensors transform in that way in order to preserve whatever object the tensor represents?
Because we defined it like that! When you make the definition "a tensor is a thing that follows X laws", you don't get to ask why, you just defined it!
Just a funny bit of phrasing, I get what is meant :)
That's just how it's presented in textbooks. It's obviously not math is actually done.
>[the explanation in the OP] will make sense to physicist, but no sense to most everyone else, including mathematicians
And then went on to describe tensors in a way that is unfriendly to non mathematicians by saying
> tensors are a sort of universal objects representing multi linear maps: bilinear maps V x W -> R correspond canonically one-to-one to regular linear maps V (x) W -> R, where V (x) W is a vector space called tensor product of V and W, and tensors are simply vectors in this tensor product space.
For example the stress and strain calculations which are used for calculating Deformation (Say if you were rolling a sheet of steel in a mill) makes use of tensors and also something called an "Invariant" I assume this also comes from Physics/Mathematics world.
Even as a physicist I found it highly confusing when I got told in physics classes that a tensor is "just a thing (or object) that behaves like so under coordinate transformation". Like, what do you mean by "thing"? I have no intuition to this yet, I need it concise definitions! Fortunately I took a differential geometry class at the same time, which was really helpful.
https://www.youtube.com/watch?v=f5liqUk0ZTw
Very simple and basic.
Edit: incorrectly wrote vectors instead of tensors.
Definitely not beginner level.
Normally, you first study the distinction between vectors (which can be expanded to tensors) and scalars in second-year Analytical Mechanics class. You also get a taste of tensors toward the later material in Electromagnetism (which is also probably second-year). And you finally arrive at a rigorous definition of tensors when you take Mathematical Physics (second-year or third-year depending on your skills).
https://grinfeld.org/books/An-Introduction-To-Tensor-Calculu...
The kicker? All of them are tensors. Tensor is just a generalisation of the concept.
I am no licensed mathematician, so this could be off. However, every time I dive into this topic, I have to wade through way too complex mathnobabble to arrive at that notion. So let's keep it simple: tensors are a mathematician's template for arrays of any dimension.
(I kid, but I think this is true, right?)
For example, if I have a vector x in V and a map T from V to W, then I would like the truth of T(x)=y to be independent of how I represent T and x.
Whether or not the concrete block breaks under that stress obviously does not depend on your choice of basis or units, so your transformation rules had better reflect that reality.
I'm sure this has confused a lot of people, especially beginners. Clashing terminology is one of the main difficulties in interdisciplinary work, in my experience. I don't think it's good to shrug it off like that.
(Sort of like how vectors kind of got going via a list of numbers and then they found the right axioms for vector spaces and then linear algebra shifted from a lot of computation to a sort of spare and elegant set of theorems on linearity).
You're getting annoyed that people are confusing the map with the territory [1]. Multidimensional arrays with certain properties can be used to represent tensors, but aren't tensors. In the same way a diagram of torus isn't a topological space, or a multiplication table isn't a group, or a matrix is not a linear map. Isomorphic but not literally the thing.
Or you're annoyed that people forget an array representing a tensor needs to satisfy some transformation law and can't just be any big array with some numbers in it.
Or maybe you're a fan of basis-free linear algebra!
Which one is it?
1: https://en.wikipedia.org/wiki/Map%E2%80%93territory_relation
More importantly though, “tensors” as commonly used in machine learning seem to rely on a single special basis, so they really are just multidimensional arrays. A machine learning algorithm isn’t really invariant under a change of basis. For example, the ReLU activation function is not independent of a change of basis.
One of my old physics professors taught us to think of tensors as "arrays with units." If it's a vector/matrix/higher dimensional array but has physical units, it's probably a tensor. The fact that it has units means it represents something physical which must obey additional constraints (like the coordinate system transformation rule).
Words can have multiple meanings, but I think we can all agree it's preferable if they don't have multiple slightly different depending on context meanings. That's just confusing.
The machine learning packages have an einsum function / tensor contraction, etc. What more do you need for it to be called a tensor?
As a way of describing physics or geometry they have additional structure which I'm not seeing.
Dot product: https://en.wikipedia.org/wiki/Dot_product
Matrix multiplication > Dot product, bilinear form and inner product: https://en.wikipedia.org/wiki/Matrix_multiplication#Dot_prod...
> The dot product of two column vectors is the matrix product
Tensor > Geometric objects https://en.wikipedia.org/wiki/Tensor :
> The transformation law for a tensor behaves as a functor on the category of admissible coordinate systems, under general linear transformations (or, other transformations within some class, such as local diffeomorphisms.) This makes a tensor a special case of a geometrical object, in the technical sense that it is a function of the coordinate system transforming functorially under coordinate changes.[24] Examples of objects obeying more general kinds of transformation laws are jets and, more generally still, natural bundles.[25][26]
Tensors require some algebraic structure, usually a vector space.
The most "modern" treatment of classical geometry that I know of is Geometry by Brannan, Esplen, Gray. It might be worth a look.
That’s a pretty good bar and I wonder if we could just cut to that chase earlier. But I also believe that people need enough math to see when they’re being cheated, and I feel like you could just tell middle schoolers that and they would pay attention. Maybe even primary school.
You told Billy he could have three apples, and now there are two left. Did Billy take more apples than he should have?
It’s always how do you share your cookies fairly with your friends and if they’re my cookies why do I have to share them at all? Screw “fairly” I’m keeping the extras at least. That sort of sharing is a socially advanced concept they don’t entirely get just yet.
Actually being able to do stuff with Bayes law by hand is going to be not only hard to teach, but probably impossible to remember for those of us who don't actually do math in real life. People forget stuff after a few months or years.
I highly doubt the average person is interested in checking the math on a science paper, so if you want the general public to understand statistics you... have to show us all a reason to, and also teach us all of the related skills needed to make it useful. Or else.... we will all just forget, even with the best teacher in the world.
Most of us aren't doing random game engines as a hobby project or testing things on bacteria cultures.
Maybe they should teach it in context of how to understand a scientific paper, since that's one of the more relevant things for non-pros. If you just teach statistics alone people will say
"Ok, now I know that it's easy to lie to yourself if you don't use any numbers but I don't have collections of large numbers of data points in my life to actually analyze"
If you have a way to fix this, you would be set for life, going around playing a sort of corduroy jacketed Robin Hood, keeping the rich from stealing from the poor.
In upper-level undergraduate math, I made a game of seeing how many pages I would go before seeing 7 printed anywhere. It was usually 10 pages, if I included the page numbers.
Here's an example:
https://en.wikipedia.org/wiki/Homology_(mathematics)
It has motivation, examples, and even actual numbers (though they're really just 0 and 1. most of the time). In my opinion, it's very good and clear exposition, for an encyclopedic article. However, I strongly suspect that people without enough mathematical knowledge (and "enough" in this case is something in the neighborhood of "enough to obtain an undergraduate degree in Mathematics") will simply not get anything about it beyond "it's about number of holes" (and that's not even remotely close to the whole picture: homology theories are important and useful in context of things with no "holes" to speak of). If you think otherwise, but not know what a quotient group is, you're just fooling yourself.
This is something I observe on HN a lot: people don't understand advanced mathematics, and are dumbfounded by the fact, trying to blame weird notation mathematicians insist on, or lack of motivation/examples/pictures etc. I never see people here do the same with advanced physics ("if the Standard Model is so standard, why can't they briefly and clearly describe what it is" is not something I ever see), molecular biology, or material science. People seem to know their limits and understand that really grokking these fields requires many years of deep study.
I think it's because many people on HN have good experience learning mathematics at school: it was something they always grasped really easily, and were easily able to figure out how to calculate derivatives, integrals, get matrices into normal forms etc. I don't want to rain on anyone's parade, because these things are still relatively difficult, and it does require more intellectual ability and effort that probably 3/4ths of the population aren't capable of. However, relative to advanced mathematics, undergraduate calculus is really rather trivial stuff.
Point is, if you don't understand modern advanced mathematics, you shouldn't get any more disappointed than you are about not being able to play violin. These things just don't come easy.
Realistically, it is (somewhat) fixable in small contexts. I've worked (and continue to work) on teams that are somewhat decent at risks and probabilities, but it's definitely an exceptional experience.
I don't know how to widely teach that. But I'm not yet ready to give up and say "it can't be taught", because those folks on my team are the counterexamples.
Obviously they are not. One is a linear operator, the other is a data structure for implementing computations using that operator. This description extends to all tensors.
It's like saying "queues are not just lists". That is true and also neither insightful nor helpful.
I don't see it as mystifying or complicated, what am I missing?
The coordinate change stuff that physicists talk about stems from observing that a matrix can be used to represent some tensors, but the rule for changing basis changes along with the kind of tensor. So if M is a matrix which represents a linear map and P is a matrix whose columns are basis vectors, then PMP^{-1} is the same linear map as M but in basis P; if on the other hand the matrix M represents a bilinear form as opposed to a linear map, then the basis change formula is actually PMP^T, where we use the matrix transpose. Sylvester's Law Of Inertia is then a non-trivial observation about matrix representations of bilinear forms.
Physicists conflate a tensor with its representation in some coordinate system. Then they show how changing the coordinate system changes the coordinates. This point of view does provide some concrete intuition, though, so it's not all bad. By a coordinate system, I mean a linear basis.
Hope that helps.
The OP also emphasizes the abstract interpretation as providing more intuition than the coordinate transformation rule.
Rather, mathematicians that complain about the physicist's approach just haven't advanced far enough in their studies to understand how vector bundles are associated to the frame bundle ;)
Physicists' tensors = generalization of arrays with units; have to transform according to certain coordinate laws.
Mathematicians' tensors = generalization of arrays, transformation rules don't matter.
That's definitely inaccurate, at least it doesn't match what I think of as tensors in mathematics.
In mathematics, tensors are the most general result of a bilinear opteration. This does imply that they transform according to certain laws: if you represent the tensor using some particular basis, that basis can be expressed in the original vector spaces you multiplied, and choosing a different basis for your vector spaces results in a different basis for your tensors.
By "most general bilinear operation" I am talking about what is expressed in category theory as a universal property... with a morphism that preserves bilinear maps.
Tensors can be over multiple vector spaces or a single vector space (in which case it's typically implied that it's over the vector space and its dual). When you use a vector space and its dual, I believe you get the kind of tensor that physicists deal with, and all of the same properties. Note that while vector spaces and their dual may seem to be equivalent at first glance (and they are isomorphic in finite-dimensional cases), both mathematicians and physicists must know that they have different structure and transform differently.
Something that will throw you off is that mathematicians often like to use category theory and "point free" reasoning where you talk about vector spaces and tensor products in terms of things like objects and morphisms, and often avoid talking about actual vectors and tensors. Physicists talk about tensors using much more concrete terms and specify coordinate systems for them. It can require some insight in order to figure out that mathematicians and physicists are actually talking about the same thing, and figure out how to translate what a physicist says about a tensor to what a mathematician says.
Tensors are introduced by physicists to ensure various physical quantities (which involve coordinates and their derivatives) do not depend on the arbitrarily chosen coordinate system. This is ensured through the transformation properties of tensors.
The name tensor itself comes from the theory of elasticity, Cauchy stress tensor, which BTW is uniform in many practical cases, and obeys the following tensor transformation rule:
https://en.wikipedia.org/wiki/Cauchy_stress_tensor#Transform...
like any other (contravariant) tensor must.
Matrices are not examples of tensors. Matrices can be used for representation of tensors, in which case tensor product becomes Kronecker product, but matrices in general don't have to represent tensors. You can put anything, including your favorite colors or a list of random numbers, in a matrix, and it won't be a tensor in general, not unless it must transform like a tensor under coordinate system changes.
Similarly, TensorFlow "tensor" is just a multidimensional data array, with no transformation rules enforced on it, and therefore is not a tensor.
it’s like some people invented a new word and won’t tell you what it actually means in sufficient detail to differentiate it from all the other words you know. so you keep using it with others in the hopes that contextual information will finally make it clear. one day.
Thankfully, there is a great historical example of this. The electric field vector \vec{E} = (E_x, E_y, E_z), is not a tensor. It doesn't obey the tensor-transformation law. Similarly, the magnetic field vector is not a tensor. These are matrices, but not tensors.
As you know the Electromagnetic tensor [1] is the tensor that correctly transforms under coordinate transformation, and hence allows different observers to agree with each other.
But not any multidimensional data is a valid tensor.
Sure there are: Any basis of the underlying vector space(s) induces a basis of the tensor space. Components respective to some basis are coordinates. You can then investigate what happens to the induced basis (or rather, the respective components) under a basis transformation of the underlying vector space(s), which is where the "physicist's" definition of tensors originates.
A change of coordinates does indeed induce a change of basis, but a change of basis isn't really a change of coordinates. And strictly speaking some vector spaces don't really have an obvious basis (without invoking choice), so having a basis be a prerequisite for the definition is not ideal.
The whole requirement that a tensor is 'something that transforms like [...] under a coordinate transformation' is just how physicists have chosen to phrase that a vector bundle is only well defined if it's definition isn't dependent on some arbitrary choice of coordinates. In my opinion this requirement is more easily apparent in the mathematical definition where there is no choice of coordinates in the first place, rather than the physicists way of working with some choice of coordinates and checking how things transform.
However, physicists get introduced to tensors far earlier than any excursions into differential geometry when discussing rigid bodies.
ps - one thing that always annoyed me was the limitation of linearity in so many of these models (which i totally understand why, but still). all the interesting real-world stuff happens non-linearly...
In the words of Dirac, "I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it." This usually only works if your equations are linear.
[0]: but at this point, not really. even in grad school, i only did linear modeling, and relatively rudimentary ones, at that.
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FYI, matrices transform between vectors within a vector space.
You bring up a good point though, if this were meant to be a transformation, then we're talking about modules (Z/2^8Z being the underlying ring) and not vector spaces, which is fine. I was needlessly narrow when I said "vector spaces" earlier.
Square ones do, but m x n ones represent linear maps from an n-dimensional to m-dimensional vector space (over the/a field containing the elements of the matrix).
Why do you mention things like data structures, enums and labels? This is math, not C++.
Anyway, the point is, any arbitrary 2D arrangement of numbers can be a valid matrix, whatever those numbers may represent.
I'm genuinely claiming they aren't numbers due to the explicit context. The entries are colors with numerals as labels.
Now, you've said something slightly different where I think we can agree. If the entries are from a set closed under a notion of multiplication and addition, then it represents a matrix even if it isn't being used in that way. But I still say that if you're merely using the array as a place to keep some data then I won't be using the word matrix, I'll just call it an array.
Anyway. I started this by saying it was interesting that we use the word matrix slightly differently. I still think that's interesting and think it's totally fine if you want to call an array a matrix even if the entries don't come from something where a transformation makes sense. I wouldn't "correct" someone's usage. I just think it's interesting how we use it different. Anyway, I feel like you just think I'm dumb, so I'm ending this here. You might not, but it's hard to read the vibe via text. If you have something to add you think will get me to change my word choices, feel free to respond and I'll read it, but I'm not replying on this thread anymore.