Great video! I want to add a couple of references to what you mentioned in the video related to neural networks: 1. Ali Rahimi got the Neurips 2017 "test of time" award for a method called - Random kitchen sinks (kernel method with random features). 2. Choromansky (from Google) made a variation of this idea to alleviate the quadratic memory cost of self-attention in transformers (which also works like a charm - I tried it myself, and I'm still perplexed how it didn't become one of the main efficiency improvements for transformers.). Check "retrinking attention with performers". Thank you for the great work on the video - keep them coming please! :)

reminds me of that episode of veggie tales when larry was like "in the future, linear algebra will be randomly generated!"

As a mathematician specializing in probability and random processes, I approve this message. N thumbs up where N ranges between 2.01 and 1.99 with 99% confidence!

3:03 Linear algebra professor, here. Please stop teaching that it's the rows of matrices which are vectors. Yes, both rows and columns of matrices correspond to vectors in separate vector spaces, but when they don't have the full picture yet, beginning students should be thinking of the columns of the matrix as 'the' vectors. I've had to spend so much work fixing the perspective of engineers in their masters program who only think of the rows as vectors. It's much easier to broaden a student's perspective from columns to also rows, than it is to broaden their perspective from rows to also columns.

As a developer at AMD I feel somewhat obligated to note we have an equivalent to cuBLAS called rocBLAS, as well as an interface layer hipBLAS designed to compile code to make use of either AMD or NVIDIA GPUs.

The part about matrix multiplication reminded me of studying cache hit and miss patterns in university. Interesting video.

Another tidbit about LinPack: One of its major strengths at the time it was written was that all of its double precision algorithms were truly double precision. At that time other packages often had double precision calculations hidden within the single precision routines where as their double precision counter parts did not have quad-precision parts anywhere inside. The LinPack folks were extraordinarily concerned about numerical precision in all routines. It was a great package. It also provided the basis for Matlab

Brunton, Kutz et al. in the paper you mentioned here "Randomized Matrix Decompositions using R," recommended in their paper using Nathan Halko's algo, developed at the CU Math department. B&K give some timing data, but the time and memory complexity were already computed by Halko, and he had implemented it in MATLAB for his paper - B&K ported it to R. Halko's paper from 2009 "FINDING STRUCTURE WITH RANDOMNESS: STOCHASTIC ALGORITHMS FOR CONSTRUCTING APPROXIMATE MATRIX DECOMPOSITIONS" laid this all out 7 years before the first draft of the B&K paper you referenced. Halko's office was a mile down the road from me at that time, and I implemented Python and R code based on his work (it was used in medical products, and my employer didn't let us publish). It does work quite well.

I'm writing a paper on a related topic. Didn't know about many of these papers, thanks for sharing! I really enjoyed your video

I have seen a very similar idea in compressed sensing. In compressed sensing we also use a randomized sampling matrix, because the errors can be considered as white noise. We can then use a denoising algorithm to recover the original data. In fact I know Philips MRI machines use this technique to speed up scans, because you have to take less pictures. Fascinating

Golub and Van Loan’s textbook is goated. I loved studying and learning numerical linear algebra for the first time in undergrad.

I'm finally far enough in education to see how well made your stuff is. Super excited to see a new one from you. Thanks for expanding people's horizons!

Dang, I absolutely love videos and articles that summarize the latest in a field of research and explain the concepts well!

As always this is BRILLIANT. I started following your videos since I saw the GP regression video. Great content! Thank you very much.

The nice thing is, with continuous systems (and everything in experienced life is continuous) the question is not "is it linear," but "on what scale is it functionally linear," which makes calculations of highly complex situations much simpler.

You discussed all the priors incredibly well. I didn’t even understand the premise of random in this context and now I leave with a lot more. Keep it up man ur videos are the bomb

I started reading this paper when you mentioned it on Twitter, forgot it was you who I got it from and was now so happy to see a video about it!

Damn... your videos are getting beyond excellent!

Outstanding content, instant sub. Keep up the good work!

its always a pleasure to watch this channel

My first thought was "this is like journal club with DJ"! Great stuff - well researched and crisply delivered. More of this, if you please.

It feels like this video was made to match my exact interests LOL I've been interested in NLA for a while now, and I've recently studied more "traditional" randomized algorithms in uni for combinatorial tasks (e.g. Karger's Min-cut). It's interesting to see how they've recently made ways to combine the 2 paradigms. I'm excited to see where this field goes. Thanks for the video and for introducing me to the topic!

This is a world class video. Thanks for posting this and keep it up!

Excellent video, really well paced.

Wow, so much research went into this! It makes me even more motivated to read papers and produce videos 😀

Lovely type, great clarity .

Incredible video with very good animations and script. Thank you !

Amazing video!

Amazing video. Had me hooked from start to finish. You gained a new subscriber

What a terrific video and channel. Great work! Subbed.

This is a really well made video, nice!

Great video brother! 😍

You are an amazing teacher! Thank you for explaining the topic in this manner. It really motivates me to continue learning about all things linear algebra!

Awesome presentation, thank you!

Amazing explanation of a complex topic! You've got yourself a new subscriber.

Loving it loving it loving it!! Amazing video, amazing topic 👏

This is exactly what i needed. Subscribed

thanks a ton! this was eye-opening 😊

Amazing video with great presentation. Thank you

I used one time random matrices for eigenvalue counts on intervals and it was amazing! Di Napoli, E., Polizzi, E., & Saad, Y. (2016). Efficient estimation of eigenvalue counts in an interval. Numerical Linear Algebra with Applications, 23(4), 674-692.

you've GOT to post more, this stuff is amazing, im still in high school but learning about so-called 'mature' processes which become completely revolutionised really inspires me, thanks for this :)

Great video!

Very good video on a very interesting topic. Who would have thought that there is this much to gain in such a commonly used piece of mathematics.

this feels like a good presentation topic for numerical methods seminar

Adequate density of new information, and sublime narrative. Also, on point visuals

We need more content creators like you ❤

Great video, thank you!

Been a while since ur last post thanks Please make more often I like what u make

Oh my god, this forms a small part of my PhD thesis where I've been trying to understand LAPACK's advantage/disadvantage when it comes to inverting matrices. Having this video really helps me put things into contex! Thank you very much for making this!

I enjoyed this video. Thank you.

Whoa - very cool stuff !!

This is a really good video 💯

An excellent presentation.

Fascinating. Thanks very much for filling us then on this.

This was a nice walk down memory lane for me, and a good introduction to the beginner. It's nice to see SWE getting interested in these techniques, which have a very long history (like solving finite elements with diffusion decades ago, and compressed sensing). I enjoyed your video. A few notes: It's useful to note that "random" projections started out as Gaussian, but it turns out very simple, in-memory, transformations let you use binary random numbers at high speed with little to no loss of accuracy. I think you had this in mind when talking about the random matrix S in sketch-and-solve. BLAS sounds like blast, but without the t. I'm sure there's people who pronounce it like blahs. Software engineers mangle the pronunciation of everything, including other SWE packages, looking at you, Ubuntu users. However the first pronunciation is the pronunciation I have always heard in the applied linear algebra field. FORTRAN doesn't end like "fortune," but rather ends with "tran," but maybe people pronounce "fortran" (uncapitalized) that way these days - IDK (see note above re: mangling; FORTRAN has been decapitalized since I started working with it). Cholesky starts with a hard "K" sound, which is the only pronunciation you'll ever hear in NLA and linear algebra. It certainly is the way Cholesky pronounced it. Me, I always pronounce Numpy to sound like lumpy just to tweak people, even though I know better ☺. I've always pronounced CQRRPT as "corrupt," too, but because that's what the acronym looks like (my eyes are bad). One way to explain how these work intuitively is to look at a PCA, similar to what you touched on with the illustration of covariance. If you know the rank is low, then there will be, say, k large PCA directions, and the rest will be small. If you perform random projection on the data, those large directions will almost certainly show up in your projections, with the remaining PCA directions certainly being no bigger than they were originally (projection is always non-expanding). This means the random projections will still contain large components of the strong PCA directions, and you only need to make sure you took enough random projections to avoid being unlucky enough to accidentally be very nearly normal with the strong PCA directions every time. The odds of you being unlucky go down with every random projection you add. You'd have to be very unlucky to take a photo of a stick from random directions, and have every photo of the stick be taken end-on. In most photos, it will look like a stick, not a point. Similarly, taking a photo of a piece of paper from random directions will look like a distorted rectangle, not a line segment It's one case where the curse of dimensionality is actually working in your favor - several random projections almost guarantees they won't all be projections to an object that's the thickness of the paper. I've been writing randomized algos for a long time (I have had arguments w engineers about how random SVD couldn't possibly work!), and love seeing random linear algebra libraries that are open and unit tested. I agree with your summary - a good algorithm is worth far more than good hardware. Looking forward to you tracking new developments in the future.

the quality on this editing is top notch, congratulations!!!

Great stuff

I think you made the best numerical linear algebra in the world, we really need more content like this. Keep up the good work.

"And if you aren't, you're probably doing something wrong." So very very true. Don't roll your own NLA code. You won't get it right and it certainly won't be faster. The corollary is "If you're inverting a matrix, you're probably doing something wrong." But that's a different problem I have to solve with newbies.

I realize that YT comments are not the best place to explain "complex" ideas, but here it goes anyway: The head bending relative difference piece reply is "just" a coordinate transformation. At 29:45, you lay ellipses atop each other and show the absolute approximation difference between the full sample and the sketch. The "trick" is to realize that this happens in the common (base) coordinate system and that nothing stops you from changing this coordinate system. For example, you can (a) move the origin to the centroid of the sketch, (b) rotate so that X and Y align with the semi-axis of the sketch, and (c) scale (asymmetrically) so that the sketches semi-axis have length 1. What happens to the ellipsoid of the full sample in this "sketch space"? Two things happen when plotting in the new coordinate system: (1) the ellipsoid of the sketch becomes a circle around the origin (semi-axes are all 1) by construction. (2) the ellipsoid of the full sample becomes an "almost" circle depending on the quality of the approximation of the full sample by the sketch. As sample size increases, centroids converges, semi-axes start aligning, and (importantly) semi-axes get stretched/squashed until they reach length 1. Again, this is for the full sample - the sketch is already "a perfect circle by construction". In other words, as we increase the sample size of the sketch the full sample looks more and more like a unit circle in "sketch space". We can now quantify the quality of the approximation using the ratio of the full sample's semi-axis in "sketch space". If there are no relative errors (perfect approximation), these become the ratio of radii of a circle which is always 1. Any other number is due to (relative) approximation error, lower is better, and it can't be less than 1. The claim now is that, even for small samples, this ratio is already low enough for practical use, i.e., sketches with just 10 points already yield good results.

Really superb presentation!

Awesome video

Great and clear video! Makes me wanna study more numerical LA...combined with probability theory because it shows how likely inefficient many algorithms use currently are, and that randomized algorithms are usually insanely much faster, while being approximately correct. So those randomized algorithms basically can be used anywhere when we don't need to be 100% sure about the result (which is basically always, because our mathematical models are only approximations of what's going on in the world and thus are inaccurate anyways and as you mentioned, if data is used, it's noisy).

Great video! Thank you for producing this!

Very useful, thanks

awesome video! also the soundtrack at the start is beautiful, which piece is it?

Another neat explanation for why the randomized least-squares problem works is the Johnson-Lindenstrauss lemma. That lemma states that most vectors don't change length a lot when you multiply them by a random gaussian matrix, so the norm of S(Ax - b) is within (1-eps) to (1+eps) of the norm of Ax-b with high probability.

great video!

Hi DJ! I love improvements in algorithmic efficiency.

The trick of multiplying by random S in argmin (SAx-Sb)^2 reminds me of the similar trick in the Freivalds' algorithm: instead of verifying matrix multiplication A*B==C we check A*B*x==C*x for a random vector x. Random projections FTW???

Very interesting content! I did find that the video felt longer than expected. I was intrigued by the thumbnail and the promise of at least 10x speed improvement. However, it took quite a while to get to the papers and even longer to get to the explanation. The history definitely deserves its own video and most chapters could be much shorter.

linearity @ 4:43 is diffirent linearity. linear functions in the sense of linear algebra must always pass through (0,0)

I understood this. Thank you, great education!

Phenomenal video

man j really wish these kinda of videos existed when I was in school. I would have reached my math potential instead of getting bored and losing interest because my teachers didn’t know how to teach.

Excellent ❤

As a grad student in theoretical machine learning, I have to say i'm blown away by the quality of your content, please keep videos like these coming !

Really great thank you.

I love how Volker Strassen did things so different from each other.

The first program I fed a computer was one I wrote in FORTRAN IV. It almost exhausted the memory capacity of the IBM machine, which was about 30 KBytes for the user (it used memory overloads, which we'd call banked memory today, in order to not exceed the available memory for programs).

Great video! One thing: “processor registers” not “registries”

Of course i get this in my recommended a few days after my first numerical analysis lecture

As always great (very educative) content. I very much appreciate all the work you put into those videos!

Whenever randomness is involved you got me wanting to use Analogue processors for fast and low-power processing

it's really mind-blowing how random numbers can achieve something such fast

Excellent video! This topic is not easy for the layperson (admittedly, the layperson that likes Linear Algebra), but it was clearly and very well structured.

Amazing!!

The math presentation and explanation alone was worth a sub, let alone the interesting topic.

very nice!

Reminds me of what my high school maths teacher said about being able to assess product quality on a production line with high accuracy by only sampling a few percent of the product items.

I've been through some basic linear algebra courses, but really the covariance problem struck me as one obviousness to a statician. A statician would never go and sample everybody, they would first determine how accurate they needed to be in their certainty, and then go about sampling exactly the number of people that satisfies that equation. I actually had to do this in my job! I can totally see how this will be a great tool used with data prediction and maybe hardware accelerators to make MASSIVE gains. We are in for a huge wild ride! Thanks for the video!

If you take white noise. And put a filter on it. You can produce every note, because every tone and semi tone is in the noise.

Some small correction: At 4:50, assuming the plotted values follow y=f(x), f is actually not linear, since in the graph we see that f(0)/=0. At 8:22, you incorrectly refer to the computer's registers as "registries", but more importantly, data access speed depends much more on cache size than register size, as the latter can generally only hold 1-4 values (32-bit float in 128-bit register), which, while allowing the use of SIMD, is very restrictive in its use. A computer's cache is some intermediate between CPU and disk, which, if used efficiently, can indeed greatly reduce runtime.

wow this is an amazingly well produced and scripted video and delivered perfectly, how long did it take you to plan and execute it?

Worth pointing out that there is a modern sparse linear algebra package called GraphBLAS, that can be used not just for graphs (which generalize to sparse matrices) but also to any sparse matrix multiplication operation.

Danke!

No better way to start the day than with an MI upload 🥳

great stuff

Nice video! Nice channel! The complicated part isn't multiplying ... it's inverting!

This is very interesting