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I realized that derivatives are linear

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Re: I realized that derivatives are linear

#51

Post by viraptor »

fwdpropaganda wrote: post #50
viraptor wrote: post #49

Earlier quoted context omitted.

You may be overestimating the banality. I studied CS with many math courses and some of the article goes over my head. (It would be clearer 10y ago) I don't expect many people here actually studied math as their main goal.

That aside I think there's something bigger here. I think learning is really hard work, and so most people's first reaction to hard work is to say No, and then go and construct an a posteriori rationale for why actually they shouldn't do that hard work (it's not that useful, you're never gonna use it, you're an expert at something else, etc). Similar story for why asking data structures in job interviews is a bad ide…

Or... you know, this is actually new information to some people. Why does there have to be something bigger behind it?

Re: I realized that derivatives are linear

#52

Post by fwdpropaganda »

viraptor wrote: post #51
fwdpropaganda wrote: post #50

Earlier quoted context omitted.

That aside I think there's something bigger here. I think learning is really hard work, and so most people's first reaction to hard work is to say No, and then go and construct an a posteriori rationale for why actually they shouldn't do that hard work (it's not that useful, you're never gonna use it, you're an expert at something else, etc). Similar story for why asking data structures in job interviews is a bad ide…

Or... you know, this is actually new information to some people. Why does there have to be something bigger behind it?

Or... both can be true.

Re: I realized that derivatives are linear

#53

Post by geezerjay »

mduerksen wrote: post #8
anonytrary wrote: post #2

This is why you take linear algebra and calculus before doing machine learning.

On the contrary - ML is a great motivator to finally grapple the "prerequisites". During school I never understood what the math was for, so my unconscious brain never saw the necessity to actually learn it. Now I want to learn - with hugely better results. This mechanism should be utilized much more often instead of shoving seemingly unrelated knowledge into peoples ears without letting them feel the need for it fir…

> On the contrary - ML is a great motivator to finally grapple the "prerequisites".

Your comment doesn't make sense, considering that you acknowledge math is in fact a prior requirement. Just because you didn't made heads or tails out of math during school that doesn't mean math ceased to be a prerequisite to learn applied math.

Re: I realized that derivatives are linear

#54

Post by geezerjay »

enriquto wrote: post #48
tzahola wrote: post #11

Better later than never, I guess. I’ve heard that the next big thing will be this “chain rule”, you should check that out too.

hey, it is better to be nice in that case! Because you don't know anything when your are born, everyday thousands of people learn about well-known cool things that are new for them: https://www.xkcd.com/1053/ This guy even explained his rediscovery in a beautiful form.

> Because you don't know anything when your are born, everyday thousands of people learn about well-known cool things that are new for them: https://www.xkcd.com/1053/

That doesn't mean that we should praise everyone who announces they discovered or invented basic math properties which for some reasom were ignored although they are basic prerequisites for a specific task.

Should we also praise carpenters who discover the pythagorean theorem?

Re: I realized that derivatives are linear

#55

Post by chombier »

4ad wrote: post #45
chombier wrote: post #28

Earlier quoted context omitted.

Of course you are correct, even though one could argue the linearity of differentiation is a property you can obtain by differentiating in coordinate charts , where the reasoning is still valid. In any case, it is probably a good thing to get a good intuition of what differentiation and derivatives are in the vector space setting before digging into differential geometry.

Not all vector spaces are manifolds (in fact most aren't), and you don't need charts to define differential operators (just see functional analysis).

I most definitely agree with you, however this brings us even further away from the author's setting.

Re: I realized that derivatives are linear

#56

Post by steamer25 »

fwdpropaganda wrote: post #50
viraptor wrote: post #49

Earlier quoted context omitted.

You may be overestimating the banality. I studied CS with many math courses and some of the article goes over my head. (It would be clearer 10y ago) I don't expect many people here actually studied math as their main goal.

That aside I think there's something bigger here. I think learning is really hard work, and so most people's first reaction to hard work is to say No, and then go and construct an a posteriori rationale for why actually they shouldn't do that hard work (it's not that useful, you're never gonna use it, you're an expert at something else, etc). Similar story for why asking data structures in job interviews is a bad ide…

Yeah it seems analogous to bike shedding or Paul Graham's 'blub' paradox. I.e., "we don't understand that and haven't [realized that we've] needed it to date therefore it's probably not important and certainly not of interest".

I like your conservation-of-mental-energy interpretation.

Re: I realized that derivatives are linear

#57

Post by vole »

>Most of the other non-polynomial functions have an equivalent Taylor polynomial

_analytic_ functions have a Taylor _series_, but it would be incorrect to say that "most" functions have a taylor series, and a taylor series is not a polynomial.

Re: I realized that derivatives are linear

#58

Post by acheron »

bbeonx wrote: post #20
mduerksen wrote: post #8

Earlier quoted context omitted.

On the contrary - ML is a great motivator to finally grapple the "prerequisites". During school I never understood what the math was for, so my unconscious brain never saw the necessity to actually learn it. Now I want to learn - with hugely better results. This mechanism should be utilized much more often instead of shoving seemingly unrelated knowledge into peoples ears without letting them feel the need for it fir…

Right. There are a lot of comments along the lines of "Duhhh, that's the point of calculus" here. But you know what? Screw that. People learn when people learn, and discovering it for yourself is a lot cooler than someone telling you about it, especially if it's related to investigating something you care about. I can't tell you the number of 'trivial' math facts that I have (re)discovered because they were in the co…

> The point isn't to remember D_x is a linear operator--math isn't about memorization. It's about understanding the context where this is a useful fact and knowing how to figure it out.

Totally agree with this part.

> Post it to the internet and you get ridiculed and mocked for it so that you wish you could forget it.

Telling everybody you meet about a basic fact that you just learned is mildly cute when a 6-year-old does it.

It's great that this person learned something that was new to them, sure. But that doesn't mean that they need to shout about it to the tens of thousands of their closest friends who read the front page of HN.

Re: I realized that derivatives are linear

#59

Post by MaxBarraclough »

mduerksen wrote: post #8
anonytrary wrote: post #2

This is why you take linear algebra and calculus before doing machine learning.

On the contrary - ML is a great motivator to finally grapple the "prerequisites". During school I never understood what the math was for, so my unconscious brain never saw the necessity to actually learn it. Now I want to learn - with hugely better results. This mechanism should be utilized much more often instead of shoving seemingly unrelated knowledge into peoples ears without letting them feel the need for it fir…

Agreed. This is the approach taken by the generally brilliant '3Blue1Brown' YouTube channel.

His video on determinants (for example) teaches what they actually do, rather than emphasising a seemingly arbitrary algorithm for you to blindly follow (which is how I was taught determinants).

Re: I realized that derivatives are linear

#60

Post by msie »

sampo wrote: post #5

The derivative is a linear operator, but it's not a bounded operator. That is, for example, the vector norm of f(x) = k·sin(x/k) → 0 when k→0, but the norm of d/dx f(x) does not. This also means that it's not continuous. Of the mappings between vector spaces, the most well behaving are the bounded linear operators, and the derivative doesn't belong to these. But yes, it's linear. Edit: Originally wrote f(x) = k·sin(k…

So what level of mathematics is needed to understand the following thread? Undergrad real analysis? Grad math courses?
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