The jump from spherical harmonics to eigenfunctions on a general mesh, and the specific example mesh chosen, might be the finest mathematical joke I've seen this decade.
sixo · 2h ago
Would you explain the joke for the rest of us?
MalbertKerman · 1h ago
It's quietly reversing the traditional "We approximate the cow to be a sphere" and showing how the spherical math can, in fact, be generalized to solutions on the cow.
sixo · 1h ago
oh. I did not interpret that blob as a cow. Thanks.
xeonmc · 2h ago
Spherical Haromics approximating Spherical Cows?
dark__paladin · 1h ago
assume spherical cow
sixo · 1h ago
A few questions occur to me while reading this, which I am far from qualified to answer:
- How much of this structure survives if you work on "fuzzy" real numbers? Can you make it work? Where I don't necessarily mean "fuzzy" in the specific technical sense, but in any sense in which a number is defined only up to a margin of error/length scale, which in my mind is similar to "finitism", or "automatic differentiation" in ML, or a "UV cutoff" in physics. I imagine the exact definition will determine how much vectorial structure survives. The obvious answer is that it works like a regular Fourier transform but with a low-pass filter applied, but I imagine this might not be the only answer.
- Then if this is possible, can you carry it across the analogy in the other direction? What would be the equivalent of "fuzzy vectors"?
- If it isn't possible, what similar construction on the fuzzy numbers would get you to the obvious endpoint of a "fourier analysis with a low pass filter pre-applied?"
- The argument arrives at fourier analysis by considering an orthonormal diagonalization of the Laplacian. In linear algebra, SVD applies more generally than diagonalizations—is there an "SVD" for functions?
xeonmc · 1h ago
I’d guess that it would be factored as “nonlinearity”, which might be characterized as some form of harmonic distortion, analogous to clipping nonlinearity of finite-ranged systems?
Perhaps some conjugate relation could be established between finite-range in one domain and finite-resolution in another, in terms of the effect such nonlinearities have on the spectral response.
sitkack · 40m ago
A fuzzy vector is a Gaussian? Thinking of what it would be in 1, 2, 3 and n dimensions.
jschveibinz · 54m ago
An engineering, signal processing extension/perspective:
An infinite sequence approximates a general function, as described in the article (see the slider bar example). In signal processing applications, functions can be considered (or forced) to be bandlimited so a much lower-order representation (i.e. vector) suffices:
- The subspace of bandlimited functions is much smaller than the full L^2 space
- It has a countable orthonormal basis (e.g., shifted sinc functions)
- The function can be written as (with sinc functions):
While that video obviously is not fancy, it will help with building an intuition about fixed points.
Explaining how the dimensions are points needed to describe a functions in a plane and not as much about orthogonal dimensions.
Specifically with fixed points and non-expansive mappings.
Hopefully this helps someone build intuitions.
chongli · 2h ago
I see this a lot with math concepts as they begin to get more abstract: strange visualizations to try to build intuition. I think this is ultimately a dead-end approach which misleads rather than enlightens.
To me, the proper way of continuing to develop intuition is to abandon visualization entirely and start thinking about the math in a linguistic mode. Thus, continuous functions (perhaps on the closed interval [0,1] for example) are vectors precisely because this space of functions meet the criteria for a vector space:
* (+) vector addition where adding two continuous functions on a domain yields another continuous function on that domain
* (.) scalar multiplication where multiplying a continuous function by a real number yields another continuous function with the same domain
* (0) the existence of the zero vector which is simply the function that maps its entire domain of [0,1] to 0 (and we can easily verify that this function is continuous)
We can further verify the other properties of this vector space which are:
* associativity of vector addition
* commutativity of vector addition
* identity element for vector addition (just the zero vector)
* additive inverse elements (just multiply f by -1 to get -f)
* compatibility of scalar multiplication with field multiplication (i.e a(bf) = (ab)f, where a and b are real numbers and f is a function)
* identity element for scalar multiplication (just the number 1)
* distributivity of scalar multiplication over vector addition (so a(f + g) = af + ag)
* distributivity of scalar multiplication over scalar addition (so (a + b)f = af + bf)
So in other words, instead of trying to visualize an infinite-dimensional space, we’re just doing high school algebra with which we should already be familiar. We’re just manipulating symbols on paper and seeing how far the rules take us. This approach can take us much further when we continue on to the ideas of normed vector spaces (abstracting the idea of length), sequences of vectors (a sequence of functions), and Banach spaces (giving us convergence and the existence of limits of sequences of functions).
ajkjk · 2h ago
Funny, I agree that visualizations aren't that useful after a point, but when you said "start thinking about the math in a linguistic mode" I thought you were going to describe what I do, but then you described an entirely different thing! I can't learn math the way you described at all: when things are described by definitions, my eyes glaze over, and nothing is retained. I think the way you are describing filters out a large percentage of people who would enjoy knowing the concepts, leaving only the people whose minds work in that certain way, a fairly small subset of the interested population.
My third way is that I learn math by learning to "talk" in the concepts, which is I think much more common in physics than pure mathematics (and I gravitated to physics because I loved math but can't stand learning it the way math classes wanted me to). For example, thinking of functions as vectors went kinda like this:
* first I learned about vectors in physics and multivariable calculus, where they were arrows in space
* at some point in a differential equations class (while calculating inner products of orthogonal hermite polynomials, iirc) I realized that integrals were like giant dot products of infinite-dimensional vectors, and I was annoyed that nobody had just told me that because I would have gotten it instantly.
* then I had to repair my understanding of the word "vector" (and grumble about the people who had overloaded it). I began to think of vectors as the N=3 case and functions as the N=infinity case of the same concept. Around this time I also learned quantum mechanics where thinking about a list of binary values as a vector ( |000> + |001> + |010> + etc, for example) was common, which made this easier. It also helped that in mechanics we created larger vectors out of tuples of smaller ones: spatial vector always has N=3 dimensions, a pair of spatial vectors is a single 2N = 6-dimensional vector (albeit with different properties under transformations), and that is much easier to think about than a single vector in R^6. It was also easy to compare it to programming, where there was little difference between an array with 3 elements, an array with 100 elements, and a function that computed a value on every positive integer on request.
* once this is the case, the Fourier transform, Laplace transform, etc are trivial consequences of the model. Give me a basis of orthogonal functions and of course I'll write a function in that basis, no problem, no proofs necessary. I'm vaguely aware there are analytic limitations on when it works but they seem like failures of the formalism, not failures of the technique (as evidenced by how most of them fall away when you switch to doing everything on distributions).
* eventually I learned some differential geometry and Lie theory and learned that addition is actually a pretty weird concept; in most geometries you can't "add" vectors that are far apart; only things that are locally linear can be added. So I had to repair my intuition again: a vector is a local linearization of something that might be macroscopically, and the linearity is what makes it possible to add and scalar-multiply it. And also that there is functionally no difference between composing vectors with addition or multiplication, they're just notations.
At no point in this were the axioms of vector spaces (or normed vector spaces, Banach spaces, etc) useful at all for understanding. I still find them completely unhelpful and would love to read books on higher mathematics that omit all of the axiomatizations in favor of intuition. Unfortunately the more advanced the mathematics, the more formalized the texts on it get, which makes me very sad. It seems very clear that there are two (or more) distinct ways of thinking that are at odds here; the mathematical tradition heavily favors one (especially since Bourbaki, in my impression) and physics is where everyone who can't stand it ends up.
chongli · 1h ago
I can't learn math the way you described at all: when things are described by definitions, my eyes glaze over, and nothing is retained. I think the way you are describing filters out a large percentage of people who would enjoy knowing the concepts, leaving only the people whose minds work in that certain way, a fairly small subset of the interested population.
If you told me this in the first year of my math degree I would have included myself in that group. I think you’re right that a lot of people are filtered out by higher math’s focus on definitions and theorems, although I think there’s an argument to be made that many people filter themselves out before really giving themselves the chance to learn it. It took me another year or two to begin to get comfortable working that way. Then at some point it started to click.
I think it’s similar to learning to program. When I’m trying to write a proof, I think of the definitions and theorems as my standard library. I look at the conclusion of the theorem to prove as the result I need to obtain and then think about how to build it using my library.
So for me it’s a linguistic approach but not a natural language one. It’s like a programming language and the proofs are programs. Believe it or not, this isn’t a hand-wavey concept either, it’s a rigorous one [1].
> When I’m trying to write a proof, I think of the definitions and theorems as my standard library. I look at the conclusion of the theorem to prove as the result I need to obtain and then think about how to build it using my library.
fwiw, this is exactly the thing that you when you're trying to formally prove some theorem in a language like Lean.
MalbertKerman · 1h ago
> and I was annoyed that nobody had just told me that because I would have gotten it instantly.
Right?! In my path through the physics curriculum, this whole area was presented in one of two ways. It went straight from "You don't need to worry about the details of this yet, so we'll just present a few conclusions that you will take on faith for now" to "You've already deeply and thoroughly learned the details of this, so we trust that you can trivially extend it to new problems." More time in the math department would have been awfully useful, but somehow that was never suggested by the prerequisites or advisors.
ajkjk · 47m ago
oh, my point was the opposite of that. The math department was totally useless for learning how anything made sense. I only understood linear algebra when I took quantum mechanics for instance. The math department couldn't be bothered to explain anything in any sort of useful way; you were supposed to prove pointless theorems about things you didn't understand.
olddustytrail · 3h ago
> infinite dimensions for functions is often more effective when thought of as around 8:00
I guess it works if you look at it sideways.
ttoinou · 2h ago
Isn't this the opposite way ? Vectors are functions whose input space are discrete dimensions. Let's not pretend going from natural numbers to real is "simple", reals numbers are a fascinating non-obvious math discovery. And also the passage from a few numbers to all natural numbers (aleph0) is non obvious. So basically we have two alephs passages to transforms N-D vectors as functions over reals.
xeonmc · 1h ago
Vectors are not necessarily discrete-domained. Anything that satisfies the vector space properties is a vector.
ttoinou · 1h ago
I agree but I'm operating under the assumption of the article
Conceptualizing functions as infinite-dimensional vectors lets us apply the tools of linear algebra to a vast landscape of new problems
layer8 · 1h ago
Linear algebra isn’t limited to discrete-dimensional vector spaces. Or what do you mean?
gizmo686 · 1h ago
Vectors are an abstract notion. If you have two sets and two operations that satisfy the definition of a vector space, then you have a vector space; and we refer to elements of the vector set as "vectors" within that vector space.
The observation here is that set of real value functions, combined with the set of real numbers, and the natural notion of function addition and multiplication by a real number satisfies the definition of a vector space. As a result all the results of linear algebra can be applied to real valued functions.
It is true that any vector space is isomorphic to a vector space whose vectors are functions. Linear algebra does make a lot of usage of that result, but it is different from what the article is discussing.
It seems like mentioning some of the applications at the beginning would motivate learning all these definitions.
ethan_smith · 2h ago
This perspective is crucial for understanding signal processing, machine learning, and quantum mechanics. Viewing functions as vectors enables practical techniques like Fourier transforms and kernel methods that underlie many modern technologies.
sixo · 2h ago
The genre of this article is not pedagogical, really. One usually learns these techniques in the course of a particular field like physics, electrical engineering, or theoretical chemistry. This article is best thought of as "a story you've seen before, but told from the beginning / ground up, with a lot of the connections to other topics and examples laid out for you". For that purpose, it's excellent, perhaps the best I've ever seen. It might also whet the appetite of a novice, but it's not really for that.
gizmo686 · 1h ago
The first paragraph and table of context both mention applications.
skybrian · 1h ago
Yes, so it does. Perhaps I read too quickly.
almostgotcaught · 3h ago
> "The material is not motivated." Not motivated? Judas just stick a dagger in my heart. This material needs no motivation. Just do it. Faith will come. He's teaching you analysis. Not selling you a used car. By the time you are ready to read this book you should not need motivation from the author as to why you need to know analysis. You should just feel a burning in you chest that can only be quenched by arguments involving an arbitrary sequence {x_n} that converges to x in X.
Not sure what I'm supposed to get from that. I guess some people care a little too much about math and have trouble relating to others?
TheRealPomax · 2h ago
If you need "practical applications" for some part of math to have value to you, then large parts of math will not be for you. That's fine, but that's also something you should accept and internalize: math is already its own application, we dig through it in order to better understand it, and that understanding will (with rather advanced higher education) be applicable to other fields, which in turn may have practical uses.
Those practical uses are someone else's problem to solve (even if they rely on math to solve them), and they can write their own web pages on how functions as vectors help solve specific problems in a way that's more insightful than using "traditional" calculus, and get those upvoted on HN.
But this link has a "you must be this math to ride" gate, it's not for everyone, and that's fine. It's a world wide web, there's room for all levels of information. You need to already appreciate the problems that you encountered in non-trivial calculus to appreciate this interpretation of what a function even is and how to exploit the new power that gives you.
skybrian · 2h ago
I don't see any such "math gate" on this link. Also, this math does have practical applications, but they're not mentioned until very late in the article.
My suggestion is that briefly mentioning them up front might be nice. I didn't mean to start a big argument about it.
almostgotcaught · 2h ago
i'll never fathom why people on hn treat a post as an auto-invite for unsolicited feedback.
LegionMammal978 · 2h ago
Yet some parts of math are 'preferred' over others, in that most 'serious' mathematicians would rather read 100 pages about functional analysis than 100 pages of meandering definitions from some rando trying to solve the Collatz conjecture.
Some people would like to have a filter for what to spend their time on, better than "your elders before you have deemed these ideas deeply important". One such filter is "Can these ideas tell us nontrivial things about other areas of math?" That is, "Do they have applications?"
Short of the strawman of immediate economic value, I don't think it's wrong to view a subject with light skepticism if it seemingly ventures off into its own ivory tower without relating back to anything else. A few well-designed examples can defuse this skepticism.
almostgotcaught · 3h ago
you're supposed to get that the cynical lens you're applying here doesn't fit - if you aren't intrinsically motivated to read this stuff then it's not for you. which is fine btw because (functional) analysis isn't a required class.
simpaticoder · 1h ago
The author asserts vectors are functions, specifically a function that takes an index and returns a value. He notes that as you increase the number of indices, a vector can contain an arbitary function (he focuses on continuous, real-valued functions).
It's fun to simulate one thing with another, but there is a deeper and more profound sense in which vectors are functions in Clifford Algebra, or Geometric Algebra. In that system, vectors (and bi-vectors...k-vectors) are themselves meaningful operators on other k-vectors. Even better, the entire system generalizes to n-dimensions, and decribes complex numbers, 2-d vectors, quaternions, and more, essentially for free. (Interestingly, the primary operation in GA is "reflection", the same operation you get in quantum computing with the Hadamard gate)
- How much of this structure survives if you work on "fuzzy" real numbers? Can you make it work? Where I don't necessarily mean "fuzzy" in the specific technical sense, but in any sense in which a number is defined only up to a margin of error/length scale, which in my mind is similar to "finitism", or "automatic differentiation" in ML, or a "UV cutoff" in physics. I imagine the exact definition will determine how much vectorial structure survives. The obvious answer is that it works like a regular Fourier transform but with a low-pass filter applied, but I imagine this might not be the only answer.
- Then if this is possible, can you carry it across the analogy in the other direction? What would be the equivalent of "fuzzy vectors"?
- If it isn't possible, what similar construction on the fuzzy numbers would get you to the obvious endpoint of a "fourier analysis with a low pass filter pre-applied?"
- The argument arrives at fourier analysis by considering an orthonormal diagonalization of the Laplacian. In linear algebra, SVD applies more generally than diagonalizations—is there an "SVD" for functions?
Perhaps some conjugate relation could be established between finite-range in one domain and finite-resolution in another, in terms of the effect such nonlinearities have on the spectral response.
An infinite sequence approximates a general function, as described in the article (see the slider bar example). In signal processing applications, functions can be considered (or forced) to be bandlimited so a much lower-order representation (i.e. vector) suffices:
- The subspace of bandlimited functions is much smaller than the full L^2 space - It has a countable orthonormal basis (e.g., shifted sinc functions) - The function can be written as (with sinc functions):
x(t) = \sum_{n=-\infty}^{\infty} f(nT) \cdot \text{sinc}\left( \frac{t - nT}{T} \right)
- This is analogous to expressing a vector in a finite-dimensional subspace using a basis (e.g. sinc)
Discrete-time signal processing is useful for comp-sci applications like audio, SDR, trading data, etc.
The popular lens is the porcupine concept when infinite dimensions for functions is often more effective when thought of as around 8:00 in this video.
https://youtu.be/q8gng_2gn70
While that video obviously is not fancy, it will help with building an intuition about fixed points.
Explaining how the dimensions are points needed to describe a functions in a plane and not as much about orthogonal dimensions.
Specifically with fixed points and non-expansive mappings.
Hopefully this helps someone build intuitions.
To me, the proper way of continuing to develop intuition is to abandon visualization entirely and start thinking about the math in a linguistic mode. Thus, continuous functions (perhaps on the closed interval [0,1] for example) are vectors precisely because this space of functions meet the criteria for a vector space:
* (+) vector addition where adding two continuous functions on a domain yields another continuous function on that domain
* (.) scalar multiplication where multiplying a continuous function by a real number yields another continuous function with the same domain
* (0) the existence of the zero vector which is simply the function that maps its entire domain of [0,1] to 0 (and we can easily verify that this function is continuous)
We can further verify the other properties of this vector space which are:
* associativity of vector addition
* commutativity of vector addition
* identity element for vector addition (just the zero vector)
* additive inverse elements (just multiply f by -1 to get -f)
* compatibility of scalar multiplication with field multiplication (i.e a(bf) = (ab)f, where a and b are real numbers and f is a function)
* identity element for scalar multiplication (just the number 1)
* distributivity of scalar multiplication over vector addition (so a(f + g) = af + ag)
* distributivity of scalar multiplication over scalar addition (so (a + b)f = af + bf)
So in other words, instead of trying to visualize an infinite-dimensional space, we’re just doing high school algebra with which we should already be familiar. We’re just manipulating symbols on paper and seeing how far the rules take us. This approach can take us much further when we continue on to the ideas of normed vector spaces (abstracting the idea of length), sequences of vectors (a sequence of functions), and Banach spaces (giving us convergence and the existence of limits of sequences of functions).
My third way is that I learn math by learning to "talk" in the concepts, which is I think much more common in physics than pure mathematics (and I gravitated to physics because I loved math but can't stand learning it the way math classes wanted me to). For example, thinking of functions as vectors went kinda like this:
* first I learned about vectors in physics and multivariable calculus, where they were arrows in space
* at some point in a differential equations class (while calculating inner products of orthogonal hermite polynomials, iirc) I realized that integrals were like giant dot products of infinite-dimensional vectors, and I was annoyed that nobody had just told me that because I would have gotten it instantly.
* then I had to repair my understanding of the word "vector" (and grumble about the people who had overloaded it). I began to think of vectors as the N=3 case and functions as the N=infinity case of the same concept. Around this time I also learned quantum mechanics where thinking about a list of binary values as a vector ( |000> + |001> + |010> + etc, for example) was common, which made this easier. It also helped that in mechanics we created larger vectors out of tuples of smaller ones: spatial vector always has N=3 dimensions, a pair of spatial vectors is a single 2N = 6-dimensional vector (albeit with different properties under transformations), and that is much easier to think about than a single vector in R^6. It was also easy to compare it to programming, where there was little difference between an array with 3 elements, an array with 100 elements, and a function that computed a value on every positive integer on request.
* once this is the case, the Fourier transform, Laplace transform, etc are trivial consequences of the model. Give me a basis of orthogonal functions and of course I'll write a function in that basis, no problem, no proofs necessary. I'm vaguely aware there are analytic limitations on when it works but they seem like failures of the formalism, not failures of the technique (as evidenced by how most of them fall away when you switch to doing everything on distributions).
* eventually I learned some differential geometry and Lie theory and learned that addition is actually a pretty weird concept; in most geometries you can't "add" vectors that are far apart; only things that are locally linear can be added. So I had to repair my intuition again: a vector is a local linearization of something that might be macroscopically, and the linearity is what makes it possible to add and scalar-multiply it. And also that there is functionally no difference between composing vectors with addition or multiplication, they're just notations.
At no point in this were the axioms of vector spaces (or normed vector spaces, Banach spaces, etc) useful at all for understanding. I still find them completely unhelpful and would love to read books on higher mathematics that omit all of the axiomatizations in favor of intuition. Unfortunately the more advanced the mathematics, the more formalized the texts on it get, which makes me very sad. It seems very clear that there are two (or more) distinct ways of thinking that are at odds here; the mathematical tradition heavily favors one (especially since Bourbaki, in my impression) and physics is where everyone who can't stand it ends up.
If you told me this in the first year of my math degree I would have included myself in that group. I think you’re right that a lot of people are filtered out by higher math’s focus on definitions and theorems, although I think there’s an argument to be made that many people filter themselves out before really giving themselves the chance to learn it. It took me another year or two to begin to get comfortable working that way. Then at some point it started to click.
I think it’s similar to learning to program. When I’m trying to write a proof, I think of the definitions and theorems as my standard library. I look at the conclusion of the theorem to prove as the result I need to obtain and then think about how to build it using my library.
So for me it’s a linguistic approach but not a natural language one. It’s like a programming language and the proofs are programs. Believe it or not, this isn’t a hand-wavey concept either, it’s a rigorous one [1].
[1] https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
fwiw, this is exactly the thing that you when you're trying to formally prove some theorem in a language like Lean.
Right?! In my path through the physics curriculum, this whole area was presented in one of two ways. It went straight from "You don't need to worry about the details of this yet, so we'll just present a few conclusions that you will take on faith for now" to "You've already deeply and thoroughly learned the details of this, so we trust that you can trivially extend it to new problems." More time in the math department would have been awfully useful, but somehow that was never suggested by the prerequisites or advisors.
I guess it works if you look at it sideways.
The observation here is that set of real value functions, combined with the set of real numbers, and the natural notion of function addition and multiplication by a real number satisfies the definition of a vector space. As a result all the results of linear algebra can be applied to real valued functions.
It is true that any vector space is isomorphic to a vector space whose vectors are functions. Linear algebra does make a lot of usage of that result, but it is different from what the article is discussing.
(This is where I learned at least half of the math on this page: theoretical chemistry.)
Polynomials come to mind.
Functions on a countable domain are sequences.
No comments yet
https://www.amazon.com/review/R23MC2PCAJYHCB
Those practical uses are someone else's problem to solve (even if they rely on math to solve them), and they can write their own web pages on how functions as vectors help solve specific problems in a way that's more insightful than using "traditional" calculus, and get those upvoted on HN.
But this link has a "you must be this math to ride" gate, it's not for everyone, and that's fine. It's a world wide web, there's room for all levels of information. You need to already appreciate the problems that you encountered in non-trivial calculus to appreciate this interpretation of what a function even is and how to exploit the new power that gives you.
My suggestion is that briefly mentioning them up front might be nice. I didn't mean to start a big argument about it.
Some people would like to have a filter for what to spend their time on, better than "your elders before you have deemed these ideas deeply important". One such filter is "Can these ideas tell us nontrivial things about other areas of math?" That is, "Do they have applications?"
Short of the strawman of immediate economic value, I don't think it's wrong to view a subject with light skepticism if it seemingly ventures off into its own ivory tower without relating back to anything else. A few well-designed examples can defuse this skepticism.
It's fun to simulate one thing with another, but there is a deeper and more profound sense in which vectors are functions in Clifford Algebra, or Geometric Algebra. In that system, vectors (and bi-vectors...k-vectors) are themselves meaningful operators on other k-vectors. Even better, the entire system generalizes to n-dimensions, and decribes complex numbers, 2-d vectors, quaternions, and more, essentially for free. (Interestingly, the primary operation in GA is "reflection", the same operation you get in quantum computing with the Hadamard gate)