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Eigenvectors and Eigenvalues (2015)

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Re: Eigenvectors and Eigenvalues (2015)

#11

Post by aje403 »

xevb3k wrote: post #3

Whenever this kind of stuff comes up I feel like a bit of a fraud... I’ve written a bunch of scientific data analysis code. I have a science PhD. Written large image analysis pipelines that worked as well as the state of the art... been published etc. For the most part I’ve found basic math and heuristics to be good enough. Every so often I go relearn calculus. But honestly, none of this stuff ever seems to come in h…

This is frightening but believable. I've worked with a few "quants" who stared at me doe eyed explaining eigen* and basic calculus concepts to them in the context of why their calculations don't add up. You mention you've used fourier transforms before - if you don't understand an eigenbasis then you don't have a fundamental understanding the math you're deploying.

Re: Eigenvectors and Eigenvalues (2015)

#12

Post by cowboysauce »

xevb3k wrote: post #10
jdonaldson wrote: post #7

Earlier quoted context omitted.

Eigenvectors and Eigenvalues show up everywhere, although sometimes it's in the form of an iterative estimate (PageRank is basically the power method estimation of the first eigenvector of a connected graph of web pages). They're in the same class as logarithms and Fourier transforms IMHO. You won't need to calculate them by hand, but you should know what they do and why they're important.

Interesting, perhaps that’s it... I’ve used FFTs a few times, but even FFTs have never been make or break in terms of getting a pipeline or analysis working well.

The Fourier transform is actually linked with linear algebra. You can think of it as taking a vector in an infinite dimensional Hilbert space (your signal) and decomposing it into its components (the amplitudes of the frequencies).

Re: Eigenvectors and Eigenvalues (2015)

#13

Post by stephencanon »

cowboysauce wrote: post #12
xevb3k wrote: post #10

Earlier quoted context omitted.

Interesting, perhaps that’s it... I’ve used FFTs a few times, but even FFTs have never been make or break in terms of getting a pipeline or analysis working well.

The Fourier transform is actually linked with linear algebra. You can think of it as taking a vector in an infinite dimensional Hilbert space (your signal) and decomposing it into its components (the amplitudes of the frequencies).

The Fourier transform as used in practice in signal processing is ~always discrete, hence finite-dimensional. It's literally "just" a rotation with an especially nice decomposition that allows efficient multiplication.

Re: Eigenvectors and Eigenvalues (2015)

#14

Post by danharaj »

stephencanon wrote: post #13
cowboysauce wrote: post #12

Earlier quoted context omitted.

The Fourier transform is actually linked with linear algebra. You can think of it as taking a vector in an infinite dimensional Hilbert space (your signal) and decomposing it into its components (the amplitudes of the frequencies).

The Fourier transform as used in practice in signal processing is ~always discrete, hence finite-dimensional. It's literally "just" a rotation with an especially nice decomposition that allows efficient multiplication.

A particular fft in practice may be discrete but the relation between fft's of different resolutions of the "same" signal hints at the infinite structure which bundles up all the finite subspaces into one conceptual object.

Re: Eigenvectors and Eigenvalues (2015)

#15

Post by WhompingWindows »

jdonaldson wrote: post #7
xevb3k wrote: post #3

Whenever this kind of stuff comes up I feel like a bit of a fraud... I’ve written a bunch of scientific data analysis code. I have a science PhD. Written large image analysis pipelines that worked as well as the state of the art... been published etc. For the most part I’ve found basic math and heuristics to be good enough. Every so often I go relearn calculus. But honestly, none of this stuff ever seems to come in h…

Eigenvectors and Eigenvalues show up everywhere, although sometimes it's in the form of an iterative estimate (PageRank is basically the power method estimation of the first eigenvector of a connected graph of web pages). They're in the same class as logarithms and Fourier transforms IMHO. You won't need to calculate them by hand, but you should know what they do and why they're important.

Why should you know what they do and why they're important? How does that practically change my R code?

Re: Eigenvectors and Eigenvalues (2015)

#16

Post by haskellandchill »

The visual explanation movement falls flat for me. It's like trying to understand Monads through blog posts. It's great if you already understand the concept to develop your intuition, or if you've never heard of the concept to pique your interest, but it won't help in the intermediate area where you know what you want to know but don't understand it fully. I need to build proofs through incremental exercises to grasp these concepts.

Re: Eigenvectors and Eigenvalues (2015)

#19

Post by whatshisface »

haskellandchill wrote: post #16

The visual explanation movement falls flat for me. It's like trying to understand Monads through blog posts. It's great if you already understand the concept to develop your intuition, or if you've never heard of the concept to pique your interest, but it won't help in the intermediate area where you know what you want to know but don't understand it fully. I need to build proofs through incremental exercises to gras…

As someone who understands eigenfunctions already, I don't understand the pictures either. Here is the best way to think about it: a matrix is a transformation, a composition of rotation, scaling, etc. Eigensets are lines going through the origin that the matrix moves points along. So a rotation would have no eigenvectors because none of the points move in a straight line, while a scaling along the x axis would have an eigenset that was also along the x axis, consisting of the points that were moved straight up or down.

To imagine finding the eigenset, just ask, could I draw a line through 0,0 such that any point I put on it would stay on it after the matrix acted?

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