[HN Gopher] Higher Order Derivatives of Transforms
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Higher Order Derivatives of Transforms
Author : nosferalatu123
Score : 72 points
Date : 2024-01-29 06:44 UTC (16 hours ago)
(HTM) web link (nosferalatu.com)
(TXT) w3m dump (nosferalatu.com)
| shiandow wrote:
| Don't think I'll ever get used to stars in LaTeX when someone
| means simple multiplication.
|
| The preceding blog post[1] seems to contain the more interesting
| parts though. This is just (d/dt)^n e^At x = A^n e^At x, which is
| kind of obvious from the definition of e^At.
|
| [1]: https://nosferalatu.com/DerivativesLogarithmsTransforms.html
| cycomanic wrote:
| The article (even more the previous article) really misses a
| reference to matrix exponentials [1]. Everything about X(t) =
| X^t becomes a bit clearer. Also I think using the term
| transform is a bit loose here. IIRC Matrix exponentials only
| work with linear transforms that are represented by a square
| matrix.
|
| [1] https://en.m.wikipedia.org/wiki/Matrix_exponential
| magicalhippo wrote:
| The previous post in the series[1], helpfully linked to in the
| introduction, has a lot more details of the underlying concepts.
|
| It also set my mind wandering to the not-technically-related
| functional derivatives[2], where you vary the function slightly
| rather than the argument value.
|
| I'm not great at math, but I do love this what-if exploration you
| can do in math. Due to the various proofs underlying it all it
| seems sometimes more fruitful than similar exploration in
| programming, where one might quickly stumble upon obscure
| compiler errors or similar obstacles.
|
| [1]: https://nosferalatu.com/DerivativesLogarithmsTransforms.html
|
| [2]: https://en.wikipedia.org/wiki/Functional_derivative
| abhgh wrote:
| If you're looking for interesting derivative-adjacent ideas, I
| would also recommend Clarke derivatives [1]. They occasionally
| show up in ML papers, e.g., [2], [3]. Unrelated bu tangential,
| another place where you need derivatives but don't have access
| to them (standard or otherwise) is in the area of black-box
| optimization. Within this area, Bayesian Optimization
| (BayesOpt) has picked up quite a bit, which I've successfully
| used quite a bit in my work - I've an introduction here [4].
| There is also a good book available online for free on the
| topic [5].
|
| [1] https://en.wikipedia.org/wiki/Clarke_generalized_derivative
|
| [2]
| https://proceedings.neurips.cc/paper/2021/file/70afbf2259b44...
|
| [3] http://proceedings.mlr.press/v202/lee23p/lee23p.pdf
|
| [4] https://blog.quipu-strands.com/bayesopt_1_key_ideas_GPs
|
| [5] https://bayesoptbook.com/
| c32c33429009ed6 wrote:
| What is meant by a Transform in this context? The author doesn't
| seem to give a definition.
| magicalhippo wrote:
| They give a brief statement in their previous post (see link at
| start of post), essentially it's a linear transformation also
| known as linear map[1].
|
| [1]: https://en.wikipedia.org/wiki/Linear_map
| c32c33429009ed6 wrote:
| I read their previous post, and nowhere do they explicitly
| say "a transform is a...". One might assume that it is indeed
| a linear transformation, as you suggest, but it shouldn't be
| up to the reader to do detective work just to understand the
| objects the author is talking about.
| magicalhippo wrote:
| What I meant was that based on what they said in their
| previous post ("[g]iven a transform T and a point x, we can
| find the transformed point with T*x") and the interactive
| graphics, I felt certain they meant a linear map.
|
| I agree it's sloppy, at least a reference or something
| should be given if one doesn't want to spend time on the
| full definition.
| makerdiety wrote:
| Maybe the author could have used a more general notion,
| then, if omission and brevity were going to be present?
| Like, instead of a linear map or transform, he could have
| said an operator or something.
|
| I don't know what is the general form of a transform or
| linear map. I think it's something like operator, though.
| magicalhippo wrote:
| True, however I don't know how well versed the author is.
| Back in my late teens when I was deep into 3D graphics
| and ray tracing, I knew a lot about that specific math
| but not much beyond it. To me, "transform" was crystal
| clear to mean some kind of linear transform, and I hadn't
| yet learned of the more general operator notion[1].
|
| So I can see myself writing something similar thinking it
| was clear.
|
| [1]: https://en.wikipedia.org/wiki/Operator_(mathematics)
| dimatura wrote:
| In robotics, it's pretty widely used to refer to a 6-degree of
| freedom pose or relative pose in space, for example it's widely
| used in ROS, a de facto standard https://docs.ros.org/en/melodi
| c/api/geometry_msgs/html/msg/T.... Not sure if there they're
| just using it as an example in the graphics.
| nosferalatu123 wrote:
| I meant a "transform" as a linear map. I'm using the word as it
| is used in computer graphics (my background), so it's something
| that translates, rotates, scales, etc. other things (such as
| points). That is often a 3x3 or 4x4 matrix, although it can
| also be a vec3 translation and a quaternion, or just a
| quaternion. I think "Transform" is clear in the context of
| computer graphics, but I see what you mean about it being
| vaguely defined in my blog post.
| dimatura wrote:
| I like this summary of Lie theory in the context of robotics:
| https://arxiv.org/pdf/1812.01537.pdf
| tnecniv wrote:
| I like this tutorial because it doesn't get too bogged down in
| abstractions and has numerous examples. When I've tried to
| learn differential geometry in the past, standard texts get
| very abstract very quickly and it's hard for me to envision
| what the generalization is doing for me.
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