If you take your table of contents from 0:19 and move it to your video description, UA-cam will turn these into chapters. You have to start with "0:00" for the first section for UA-cam to pick it up. ua-cam.com/video/MUqeoo5Tp2M/v-deo.html. Good luck!
One of the things I noticed in some videos in general is, for some reason extremely basic concepts are still explained. Like if linear algebra is prerequisite, then euclidean distance should be required too
To be honest, when I was learning this topic I had forgotten what the Euclidean distance is and kept on mistaking it for the something else. I just wanted it to be clear.
"mu = n^-1 * Sigma..." is peak math notation optimized to obscure meaning. This "shorthand" notation takes as much space as avg = (x1 + ... + xn) / n with the added benefits of: - 2 greek letters - obfuscated division via negative exponent (when the video says DIVIDE by n) - implied iterator i from 1? 0? to n - cursive x as a variable why, as a reader, do I have to go through those ideas to decrypt the avg = (x1 + ... + xn) / n The only reason I'm saying this is because I like the ideas in math and love the some1/2 initiative to make math accessible, but the way you use math notation is like this jarring inaccessible way like it's found in many textbooks. Having to decode and track notation transforms this video from lunchtime break to dedicated study session. Another example would be @3:58, that does not! look like a dot product it takes me a good 20 secs just to parse out the cursive non-colored indices (which look to me like mmunvnnuvun) to see what goes where and how they get grouped/iterated These are obviously the bad cases, I really liked the intro so far with the dots moving around in 2D become a 2D matrix then the distances as lines then those becoming a matrix, and going along vectors to find distances then shown as column vectors becoming the norm.
The point, though, is to learn a new way of looking at things and a notation that allows you to look at things in the new way. Such explanations will often use examples that overlap with concepts you already understand as a prop to help you with the transition, but that does _not_ mean the new perspective is just a funny way of writing the old concept. For instance, when you learned multiplication you were probably given an explanation in terms of "repeated addition" -- from which you might have incorrectly concluded that there was no value to the multiplication since it was just an obsfucated rehash of something you already knew. I trust you can see how this would have been an incorrect conclusion. In the present example, there are good reasons for the notation to work differently; for example, in fields where multiplication isn't commutative (meaning we can't assume that A*B = B*A) , we write division as multiplication by an inverse so that we can distinguish A * B^-1 from B^1 * A ... something we can not do if we write it as a fraction A over B. Likewise, the notation used to find the centroid happens in this case to produce the average, but it is far more general. And (this is the whole point of it) it allows us to use algebra to combine and manipulate all the different things it can represent. Giving a special name (average) to one of them eliminates these advantages. Sure, we can then talk about the average in a comfortable and familiar way, but we can't _do_ anything with it, and we can''t go anywhere with it.
@@markusroberts2703 My point was not about math, but communication, mainly the two following aspects: When teaching, you want to lower the computation overhead for the listener to let them focus on what's important in that lesson, as you say in some fields it is important, but here it is not. Specificity changes meaning: have a nice day vs have a nice next 86400 seconds. If that extra notation carried crucial information which was then used, it'd be totally fine, but it wasn't. Regardless of my or your arguments, it was distracting enough for me to stop the video and leave that comment so either I'm not the target audience or there's something wrong with the presentation. For reference, I've watched the whole 3b1b channel and never had this problem with notation.
Very interesting! So if I'm understanding correctly, if you have n points in n-dimensional space, then any set of pairwise distances that satisfies every triangle inequality is achievable? (Heuristically this makes sense because you can set your axes so that the first point is at the origin, the second point is on the x-axis, the third point is on the xy-plane, etc.; so the number of coordinates you have available is 0+1+2+...+(n-1), which is also the number of equations you have from your given distances.) So finding these n points in n-dimensional space is what the 'square root of B' part of the algorithm is doing. And then the eigenvalues part is just seeing whether those n points are actually in a lower dimension; e.g. if you have a 5x5 matrix but the column vectors all lie roughly in a plane, then there will be 3 eigenvalues near zero because the subspace of R^5 complement to the plane, which has dimension 3, would be mapped to basically zero. The two other eigenvectors would be a basis for that plane and you can then use those to express your five 5-d points with 2-d coordinates. I hope my understanding is correct. The applications you mentioned are really fascinating. Great topic and great video!
Thanks. Sometimes orientation and translation information is not desired. E.g. Advent of Code 2021 Day 19: You need to stitch pointclouds, that may be offset and rotated. Learning about MDS from your video confirms my hunch that only considering the distance matrix is the right choice.
Something I think is funny is that the python library supports the Minkowski Metric, which actually violates positive definiteness (and possibly the triangle inequality, but I'm less certain about that). Meaning it's not _really_ a "metric" by that definition. From what I can tell, it's considered a "pseudo-euclidean metric."
I've never heard it called a "pseudo-euclidean" metric. What it is is the generalization of Euclidean distance with arbitrary exponents/roots. It also generalizes Manhattan distance. Wikipedia shows a case where for exponents
@@ilonachan By positive-definite, I mean d(x, y) ≥ 0. With the Minkowski distance, you have a "null cone" and any vectors along it effectively have 0 distance from the origin, and vectors on one side of the null cone can even have negative squared distances. In special relativity, the null cone is also called the light cone.
@@angeldude101 oh I see. Yea I'm aware of that, but I didn't know it was ALSO called Minkowski metric! According to wikipedia we're talking about two different things that have the same name. What I mean is under "Minkowski Distance" (that's also probably the one that library uses) and yours is a subsection of "Minkowski Space". There's no way THAT could ever be confusing amirite I don't think "positive definite" is the right term in this case though. I'd just call that metric a non-negative function. However, inner products (e.g. the scalar product) ARE positive definite! If that is the case, you can define lengths of vectors by applying the inner product to the same vector on both sides (the result is non-negative by definition), and distances as the length of the vector between two points. Doing it that way is actually the basics of "Riemannian Metrics", which we studied in a course on "(Elementary) Differential Geometry". Allows you to define distances and angles on abstract manifolds, so it's pretty cool. It's not exactly the math behind general relativity, but it's REALLY close (basically a minus sign away)
19:15 "This would probably be better than not knowing what France looks like at all" I disagree. It's better to have never known what France look like. #ObligatoryFranceSlander
"(DO NOT DO BY HAND UNLESS YOU ARE THE LAST TURING COMPLETE THING IN EXISTENCE)"
omg i love it man. i'm going to steal this for sure
If you take your table of contents from 0:19 and move it to your video description, UA-cam will turn these into chapters. You have to start with "0:00" for the first section for UA-cam to pick it up. ua-cam.com/video/MUqeoo5Tp2M/v-deo.html. Good luck!
"eigen-" means "of the self"/"own" in german, which I don't believe has a translation in "special"
Ah that would make more sense, at least in my mind.
it can sometimes mean "weird" or "special", as in "das ist schon ein recht eigenes System" or something (German native speaker here)
In French, we call EigenVectors "Vecteur Propres", which is the same as in german, since in that context "Propres" also refers to own/self
@@TheDrax72 or we simply put the in the washing machine
@@JayDee-b5u "Characteristic" is the way I've heard it translated.
One of the things I noticed in some videos in general is, for some reason extremely basic concepts are still explained. Like if linear algebra is prerequisite, then euclidean distance should be required too
some being summer of math exposition. Never knew this acronym was so bad haha
To be honest, when I was learning this topic I had forgotten what the Euclidean distance is and kept on mistaking it for the something else. I just wanted it to be clear.
This is an extremely informative video and I learned a lot!
"mu = n^-1 * Sigma..." is peak math notation optimized to obscure meaning. This "shorthand" notation takes as much space as avg = (x1 + ... + xn) / n with the added benefits of:
- 2 greek letters
- obfuscated division via negative exponent (when the video says DIVIDE by n)
- implied iterator i from 1? 0? to n
- cursive x as a variable
why, as a reader, do I have to go through those ideas to decrypt the avg = (x1 + ... + xn) / n
The only reason I'm saying this is because I like the ideas in math and love the some1/2 initiative to make math accessible, but the way you use math notation is like this jarring inaccessible way like it's found in many textbooks. Having to decode and track notation transforms this video from lunchtime break to dedicated study session.
Another example would be @3:58, that does not! look like a dot product it takes me a good 20 secs just to parse out the cursive non-colored indices (which look to me like mmunvnnuvun) to see what goes where and how they get grouped/iterated
These are obviously the bad cases, I really liked the intro so far with the dots moving around in 2D become a 2D matrix then the distances as lines then those becoming a matrix, and going along vectors to find distances then shown as column vectors becoming the norm.
The point, though, is to learn a new way of looking at things and a notation that allows you to look at things in the new way. Such explanations will often use examples that overlap with concepts you already understand as a prop to help you with the transition, but that does _not_ mean the new perspective is just a funny way of writing the old concept.
For instance, when you learned multiplication you were probably given an explanation in terms of "repeated addition" -- from which you might have incorrectly concluded that there was no value to the multiplication since it was just an obsfucated rehash of something you already knew. I trust you can see how this would have been an incorrect conclusion.
In the present example, there are good reasons for the notation to work differently; for example, in fields where multiplication isn't commutative (meaning we can't assume that A*B = B*A) , we write division as multiplication by an inverse so that we can distinguish A * B^-1 from B^1 * A ... something we can not do if we write it as a fraction A over B.
Likewise, the notation used to find the centroid happens in this case to produce the average, but it is far more general. And (this is the whole point of it) it allows us to use algebra to combine and manipulate all the different things it can represent. Giving a special name (average) to one of them eliminates these advantages. Sure, we can then talk about the average in a comfortable and familiar way, but we can't _do_ anything with it, and we can''t go anywhere with it.
@@markusroberts2703 My point was not about math, but communication, mainly the two following aspects:
When teaching, you want to lower the computation overhead for the listener to let them focus on what's important in that lesson, as you say in some fields it is important, but here it is not.
Specificity changes meaning: have a nice day vs have a nice next 86400 seconds. If that extra notation carried crucial information which was then used, it'd be totally fine, but it wasn't.
Regardless of my or your arguments, it was distracting enough for me to stop the video and leave that comment so either I'm not the target audience or there's something wrong with the presentation.
For reference, I've watched the whole 3b1b channel and never had this problem with notation.
okay, there were plenty of reasons to upvote, but that MONTAGE MUSIC
Wow! Thank you for beautiful explanations!
Very interesting! So if I'm understanding correctly, if you have n points in n-dimensional space, then any set of pairwise distances that satisfies every triangle inequality is achievable? (Heuristically this makes sense because you can set your axes so that the first point is at the origin, the second point is on the x-axis, the third point is on the xy-plane, etc.; so the number of coordinates you have available is 0+1+2+...+(n-1), which is also the number of equations you have from your given distances.) So finding these n points in n-dimensional space is what the 'square root of B' part of the algorithm is doing. And then the eigenvalues part is just seeing whether those n points are actually in a lower dimension; e.g. if you have a 5x5 matrix but the column vectors all lie roughly in a plane, then there will be 3 eigenvalues near zero because the subspace of R^5 complement to the plane, which has dimension 3, would be mapped to basically zero. The two other eigenvectors would be a basis for that plane and you can then use those to express your five 5-d points with 2-d coordinates. I hope my understanding is correct. The applications you mentioned are really fascinating. Great topic and great video!
thanks a lot!!!! You saved my life!!!
Thanks. Sometimes orientation and translation information is not desired. E.g. Advent of Code 2021 Day 19: You need to stitch pointclouds, that may be offset and rotated. Learning about MDS from your video confirms my hunch that only considering the distance matrix is the right choice.
The summation variable `n` at 3:53 is incorrect, it should be `v`
THIS EXPLANATION IS NOT BRILLIANT,
IT'S BEYOND EXTRAORDINARY!🤯🤯🤯
Dot producting the vector to itslef does not produce that it rather produces that squared
Something I think is funny is that the python library supports the Minkowski Metric, which actually violates positive definiteness (and possibly the triangle inequality, but I'm less certain about that). Meaning it's not _really_ a "metric" by that definition. From what I can tell, it's considered a "pseudo-euclidean metric."
I've never heard it called a "pseudo-euclidean" metric. What it is is the generalization of Euclidean distance with arbitrary exponents/roots. It also generalizes Manhattan distance.
Wikipedia shows a case where for exponents
@@ilonachan By positive-definite, I mean d(x, y) ≥ 0. With the Minkowski distance, you have a "null cone" and any vectors along it effectively have 0 distance from the origin, and vectors on one side of the null cone can even have negative squared distances. In special relativity, the null cone is also called the light cone.
@@angeldude101 oh I see. Yea I'm aware of that, but I didn't know it was ALSO called Minkowski metric! According to wikipedia we're talking about two different things that have the same name. What I mean is under "Minkowski Distance" (that's also probably the one that library uses) and yours is a subsection of "Minkowski Space". There's no way THAT could ever be confusing amirite
I don't think "positive definite" is the right term in this case though. I'd just call that metric a non-negative function. However, inner products (e.g. the scalar product) ARE positive definite! If that is the case, you can define lengths of vectors by applying the inner product to the same vector on both sides (the result is non-negative by definition), and distances as the length of the vector between two points.
Doing it that way is actually the basics of "Riemannian Metrics", which we studied in a course on "(Elementary) Differential Geometry". Allows you to define distances and angles on abstract manifolds, so it's pretty cool. It's not exactly the math behind general relativity, but it's REALLY close (basically a minus sign away)
Flock of birds and the doughnut!
19:15 "This would probably be better than not knowing what France looks like at all" I disagree. It's better to have never known what France look like. #ObligatoryFranceSlander
give me another example
Could do without the “music”