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Functions are vectors

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Re: Functions are vectors

#31

Post by formally »

noduerme wrote: post #24
formally wrote: post #18

Earlier quoted context omitted.

You should look into the pigeonhole principle.

Looked it up. How does that apply to this, or am I lacking the imagination to see it?

If the data set is large enough then there is no way to represent it as a functional relationship between finite dimensional vector spaces. In fact, this problem already is visible in existing large neural networks because they can only work with data that conforms to the dimensional constraints of the input space. It's why image transformers trained on NxM images don't work on any other grid size.

Re: Functions are vectors

#32

Post by lying4fun »

thaumasiotes wrote: post #13
adammarples wrote: post #7

Earlier quoted context omitted.

That's not the definition of a function, what you're describing would be called a bijective function. A simple function that is not bijective and maps to two distinct values would be sqrt(x)

That is the definition of a function, but it's purely a formalism. In reality, and in theoretical math, we are frequently interested in functions that generate multiple values, and it's very easy to represent those in terms of the formal definition: 1. When we want to consider both positive and negative square roots, we can just say that instead of the function sqrt(x): ℝ ⟶ ℝ which always gives a nonnegative real num…

> That is the definition of a function, but it's purely a formalism...

Isn’t the function definition meant to be interpreted as “maps consistently to exactly one element of the codomain”? So an ordered pair of R^2 is still one element of R^2

It seems the parent has just mixed up the domain and codomain, because under that assumption he would be right about both the definition and bijectivity

Re: Functions are vectors

#33

Post by HeavyStorm »

opportune wrote: post #28

I’d probably have titled this something including either the term “linear” or “functional analysis”. Because submitted here, we will first interpret “functions” in the context of a function in computer programming, where the statement is more provocative and thus clickbaity. The problem is many real world functions and problems are nonlinear. But they may have linear components. For example, a dog can be recognized b…

Yep, went there to learn how to use vectors to replace methods...

Re: Functions are vectors

#34

Post by chongli »

noduerme wrote: post #29
chongli wrote: post #23

Earlier quoted context omitted.

You may be thinking of a vector as a concrete collection of values, like a vector in R^3: [x y z]. This piece is about abstract vector spaces, their properties (vector addition, scalar multiplication, etc.) and specifically how functions meet the definition, giving you vector spaces of functions (function spaces). So the idea is that if you two functions, f and g, and a scalar b, then you can do stuff like: f + g = g…

Isn't this just describing a tautology? I was reading here earlier today about the naming of the constant for light as c , and I had a question which I was too embarrassed to ask. It is this: In e=mc^2 , what are the units, and if the units aren't defined and it's just a relationship, then why specify c^2? What's the point of squaring a constant, since it's just another constant? Not that I understand a damn thing ab…

But the idea that two functions can be thought of operating additively on a vector - or space - seems... trivial.

No, the functions are not operating on a vector, the functions themselves are vectors, which means you can do things like define a linear transformation L and then you have things like this:

L(af + bg) = aL(f) + bL(g)

You may also define a norm on a function space (such as an L^P space [1]), ||f||_p, which maps functions to non-negative real numbers and obeys the properties of norms that we expect, such as the triangle inequality:

||f + g||_p [1] https://en.wikipedia.org/wiki/Lp_space

Re: Functions are vectors

#35

Post by nine_k »

noduerme wrote: post #24
formally wrote: post #18

Earlier quoted context omitted.

You should look into the pigeonhole principle.

Looked it up. How does that apply to this, or am I lacking the imagination to see it?

I suspect the idea is that there can be many functions that fit a particular vector, because there are more functions than vectors, but I nay be wrong.

Re: Functions are vectors

#36

Post by boulos »

noduerme wrote: post #29
chongli wrote: post #23

Earlier quoted context omitted.

You may be thinking of a vector as a concrete collection of values, like a vector in R^3: [x y z]. This piece is about abstract vector spaces, their properties (vector addition, scalar multiplication, etc.) and specifically how functions meet the definition, giving you vector spaces of functions (function spaces). So the idea is that if you two functions, f and g, and a scalar b, then you can do stuff like: f + g = g…

Isn't this just describing a tautology? I was reading here earlier today about the naming of the constant for light as c , and I had a question which I was too embarrassed to ask. It is this: In e=mc^2 , what are the units, and if the units aren't defined and it's just a relationship, then why specify c^2? What's the point of squaring a constant, since it's just another constant? Not that I understand a damn thing ab…

Assuming this is in good faith, the units are:

- c: speed of light in meters per second

- m: mass in kilograms

- energy: joules

Re: Functions are vectors

#37

Post by hayasaki »

thorel wrote: post #15

The realization that functions can be treated as elements in an abstract vector space (with infinitely many dimensions) is a turning point in the history of mathematics that led to the emergence of the sub-field known as functional analysis . The significance of this paradigm shift is that it allowed mathematicians to apply some of the geometric intuition developed from the study of finite-dimensional spaces (such as…

My friend, you don't even need it to be in vector space for functional analysis. Truly what is needed is just an inner product. I will grant you the inner product must be linear and hence in a vector space.

Re: Functions are vectors

#38

Post by noduerme »

chongli wrote: post #34
noduerme wrote: post #29

Earlier quoted context omitted.

Isn't this just describing a tautology? I was reading here earlier today about the naming of the constant for light as c , and I had a question which I was too embarrassed to ask. It is this: In e=mc^2 , what are the units, and if the units aren't defined and it's just a relationship, then why specify c^2? What's the point of squaring a constant, since it's just another constant? Not that I understand a damn thing ab…

But the idea that two functions can be thought of operating additively on a vector - or space - seems... trivial. No, the functions are not operating on a vector, the functions themselves are vectors , which means you can do things like define a linear transformation L and then you have things like this: L(af + bg) = aL(f) + bL(g) You may also define a norm on a function space (such as an L^P space [1]), ||f||_p, whi…

Thanks. I'm trying to understand this. So take a function that returns a random shuffled deck of cards with 52! possible real number vector outputs. What is the generalized insight from treating this function itself as a vector space... is there a shortcut to monte carlo-ing a million random shuffles if you can "divide" that function by one which produces a straight flush?

Re: Functions are vectors

#39

Post by noduerme »

formally wrote: post #31
noduerme wrote: post #24

Earlier quoted context omitted.

Looked it up. How does that apply to this, or am I lacking the imagination to see it?

If the data set is large enough then there is no way to represent it as a functional relationship between finite dimensional vector spaces. In fact, this problem already is visible in existing large neural networks because they can only work with data that conforms to the dimensional constraints of the input space. It's why image transformers trained on NxM images don't work on any other grid size.

ah. I mean, the dataset can be infinitely large and still be covered perfectly by a function, if it was generated from a function. That's why absurd functions can be found that overfit for some subset of basic real world results. (Ask anyone who built a funky physics system for a videogame in the 90s). I think what's more interesting is the question of what that essential function is for a stop sign or a pedestrian, as opposed to the function for finding it in a 512 grid or something.

Re: Functions are vectors

#40

Post by trostaft »

hayasaki wrote: post #37
thorel wrote: post #15

The realization that functions can be treated as elements in an abstract vector space (with infinitely many dimensions) is a turning point in the history of mathematics that led to the emergence of the sub-field known as functional analysis . The significance of this paradigm shift is that it allowed mathematicians to apply some of the geometric intuition developed from the study of finite-dimensional spaces (such as…

My friend, you don't even need it to be in vector space for functional analysis. Truly what is needed is just an inner product. I will grant you the inner product must be linear and hence in a vector space.

Why even require an inner product! You can get away with a lot just sitting in an Banach space (only a norm required).
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