Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Functions are vectors

thenumb.at

Post reply on HN
Print view View original

101–110 of 124 posts

Re: Functions are vectors

#101

Post by zmgsabst »

GolDDranks wrote: post #78

I haven't read the article yet, but I've known that functions are (infinite) vectors for some years. However, there's something that has been bothering me: most of my understanding of linear algebra comes from 2D and 3D spaces, and then in different context of machine learning, datasets that have from tens to even millions of dimensions. In the former, geometric context, the connection between the dimensions are clea…

> But surely there is a lot of interesting math that is concerned with the connectedness of the input and output spaces themselves, in context of there still existing a mapping between the input and output.

Topology.

Topology studies neighborhoods in your space and how they relate to things like functions, limits, etc.

Since you brought up images:

https://en.wikipedia.org/wiki/Digital_topology

And data science:

https://en.wikipedia.org/wiki/Topological_data_analysis

Re: Functions are vectors

#102

Post by [deleted] »

LudwigNagasena wrote: post #89

I think the article has it backwards and provides bad intuition. It is not input that makes functions form a vector space, it is the output. Functions from any set X to a field F can form a vector space, even if X is unordered.

[deleted]

Re: Functions are vectors

#103

Post by bigbillheck »

cykotic wrote: post #95
bigbillheck wrote: post #91

Earlier quoted context omitted.

> You will not find this in the literature. It would have cost you nothing except a minute or so of your time to have refuted yourself before posting: https://arxiv.org/abs/1912.08274 https://arxiv.org/abs/1703.01700 (See also: https://en.wikipedia.org/wiki/Multivalued_function )

A multivalued function is not a function. Hence the need to call it something else. In complex analysis you have branch cuts and things like that. We don’t call something a function that doesn’t have a unique output for a given input. EDIT: Complex analysis was developed well before the formal definition of functions, relations, set theory. As such there are legacy terms in use. Mathematics has a lot of abuse of nota…

A set-valued function is a perfectly valid kind of function, and a multivalued function in complex analysis is just a set-valued function with certain continuity and other restrictions.

Re: Functions are vectors

#104

Post by cykotic »

bigbillheck wrote: post #103
cykotic wrote: post #95

Earlier quoted context omitted.

A multivalued function is not a function. Hence the need to call it something else. In complex analysis you have branch cuts and things like that. We don’t call something a function that doesn’t have a unique output for a given input. EDIT: Complex analysis was developed well before the formal definition of functions, relations, set theory. As such there are legacy terms in use. Mathematics has a lot of abuse of nota…

A set-valued function is a perfectly valid kind of function, and a multivalued function in complex analysis is just a set-valued function with certain continuity and other restrictions.

If set up that way then yes it meets the definition of a function. I’m not a complex analyst but I imagine they think of a function from C to C on a given branch rather than multi-function being a function from C to the countable product of C (or however one wants to formally define the set value). I think this because they tend to care about analytic properties and saying function from C to countable product of C includes far too many functions. Saying multifunction will immediately make one think of branch cuts.

They don’t think of Log(z) as a set of values. At least I don’t. I assume one means the principal branch when writing Log(z) unless specifically stated otherwise. Maybe in their mind they do think of an infinite set when seeing Log(z). I’m skeptical of this possibility though.

Re: Functions are vectors

#105

Post by cmehdy »

ubj wrote: post #41

I wish I could upvote this twice. This is the best basic introduction to concepts in functional analysis that I've seen. Another great overview that goes deeper into the math is [1]. Another fantastic application that the website doesn't mention is the composition / Koopman operator. In control theory (e.g. autonomous drones, cars, robot arms, etc.), most real-world systems are described by nonlinear dynamics which a…

I am so thankful for Steve Brunton's content. Had I been able to access his content a decade ago during my M.Sc, I would have probably pursued a PhD out of the passion and quality he brings to this domain - instead I just felt done with academia, strugglign to find grants, reading yet again terse books all alone, and just moved on.

Solid YouTube educators are creating immense future opportunities and we'll all benefit from that. Control theory connects the dots between all sorts of things and can be a joy for those of us who love seeing patterns and structures everywhere (iirc Steve has a recent video about control theory for social models also).

Re: Functions are vectors

#106

Post by lanstin »

rrobukef wrote: post #71
lanstin wrote: post #30

I have never seen these index functions used as a transfinite basis for a vector space. And it seems like the function is not a limit point of finite sequences of basis functions, but some weird transfinite sum with mostly zero entries? Clearly there is no Fourier transformation possible on all functions? I think diagonalization methods would be easy to disprove any useful result. Even Hilbert spaces are usually just…

The article is some summary of a book with chapters. At some point they limit the space to the subspace of functions periodic over (b-a) and change the basis (with proof) from dirac delta to sines of frequency 2pi*k/(b-a) [with k in N]. In this subspace all functions have Fourier transformations.

Big changes there.

Re: Functions are vectors

#107

Post by Someone »

smokel wrote: post #92
GolDDranks wrote: post #78

I haven't read the article yet, but I've known that functions are (infinite) vectors for some years. However, there's something that has been bothering me: most of my understanding of linear algebra comes from 2D and 3D spaces, and then in different context of machine learning, datasets that have from tens to even millions of dimensions. In the former, geometric context, the connection between the dimensions are clea…

This is bothering me a little as well. Perhaps someone more knowledgeable could guide us into the light. As a concrete example, I am toying with a reinforcement learning to solve the game of Yahtzee. When trying to formulate the state space, I have to lump in together the states of the dice (1..6), the current state of the score card (13 boolean values, and 13 integer scores in the range 0..50), and the current turn…

In general, you want to do one-hot encoding whenever your number doesn’t really represent a quantity.

For example, for the current state of a rolled die in Yahtzee, ‘2’ mostly (see below) isn’t smaller than ‘3’, isn’t between ‘1’ and ‘3’, and isn’t closer to ‘3’ than it is to ‘4’.

So, ‘2’ isn’t a quantity there; it’s just one of the markings on the dice.

About that ‘mostly’: the straights require the ability to order the values on the dice, but even then, it’s just an order, not anything about them being equidistant, or about (2,6) showing more eyes than (3,4) (again, there’s an exception there, with the ‘Chance’ score)

Re: Functions are vectors

#108

Post by AmpsterMan »

thaumasiotes wrote: post #64
eternityforest wrote: post #63

Earlier quoted context omitted.

Seems like a useful formalism though, at least coming from a programming background. Maybe some kind of "it's all of them at once" mental model is useful in math, but object oriented programming gives you a hierarchal mindset, everything is in a container of some sort, so multiple return values having a container like a set makes perfect sense and just a bunch of loose values isn't a very familiar concept. But perhap…

It's useful in many places and less useful in others. Over in the programming world, Common Lisp allows you to return multiple values without wrapping them in a container. If you want the primary value, you just treat the function as if it returned one value. If you want additional return values, you use multiple-value-bind. (I believe the general idea is along these lines: your function does a certain amount of comp…

I can think of two simple, practical answers to your query:

The expressive power of single value functions is very powerful and the constraint is not necessarily restrictive but may even drive a stronger analysis. (Where does this function have multiple returns? Is it for the whole domain? Etc)

By contrast the expressive power of negation and implication are relatively low given very intuitive and well defined alternatives exist.

Second, there already exist good enough paradigms for dealing with multi-valued functions. Splitting the function up into multiple functions, mapping to an ordered pair, etc.

Defining a function the traditional way is more than just notational convenience. The single value constraint allows for many simplifying assumptions, enough that is worth to pay the cost when dealing with relations that you want to talk about in functional ways

Re: Functions are vectors

#109

Post by xigoi »

lanstin wrote: post #30

I have never seen these index functions used as a transfinite basis for a vector space. And it seems like the function is not a limit point of finite sequences of basis functions, but some weird transfinite sum with mostly zero entries? Clearly there is no Fourier transformation possible on all functions? I think diagonalization methods would be easy to disprove any useful result. Even Hilbert spaces are usually just…

Mathematicians HATE this weird trick! Learn how he constructed a basis for the vector space of real functions without the axiom of choice.

Re: Functions are vectors

#110

Post by xigoi »

eigenket wrote: post #82
HeavyStorm wrote: post #33

Earlier quoted context omitted.

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

For some reason computer programmers took the mathematical definition of "functions", used it for their things which are emphatically not mathematical functions, and are now complaining that they get confused by people talking about the old math definition.

Same with “vector”.
Post reply on HN