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!

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.

Randomized numerical linear algebra has been common for over two decades. A good resource is Finding Structure with Randomness. The authors on it (Halko, PG Martinsson, Tropp) are very big in the field. Others are Drineas and E. Liberty. The general idea shows up everywhere. My advisor used it for solving classes of PDEs

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!

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

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

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

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

Damn... your videos are getting beyond excellent!

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

its always a pleasure to watch this channel

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

Lovely type, great clarity .

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!

Awesome presentation, thank you!

Amazing video with great presentation. Thank you

thanks a ton! this was eye-opening 😊

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!

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

Excellent video, really well paced.

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

This is a really well made video, nice!

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

This is exactly what i needed. Subscribed

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 feels like a good presentation topic for numerical methods seminar

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.

Amazing video!

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

I enjoyed this video. Thank you.

Great video, thank you!

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

An excellent presentation.

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!

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.

Great video brother! 😍

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

This is a really good video 💯

Great video!

Great stuff

Whoa - very cool stuff !!

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).

Very useful, thanks

Phenomenal video

Excellent ❤

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.

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 !

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

We need more content creators like you ❤

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.

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).

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!

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

Great video! Thank you for producing this!

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.

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

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.

Hi DJ! I love improvements in algorithmic efficiency.

great video!

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.

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

This is very interesting

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

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

Phenomenal! But I have one correction for (25:33): Full rank isn't restricted to square [invertible] matrices, it just means rank = min(m,n) rather than rank = k < min(m,n).

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.

I've been hoping some advances in probabilistic numerics and random matrix theory bring PGM's some love. Computing matmuls/inverses every iteration of MCMC makes me sad :(. As expected, great video!

Danke!

anotha banger by DJ

Nearly 7 years ago when I was still a practicing geneticist, sequenced DNA would usually only be a few nucleotides long, maybe 50, and it would have to get mapped to a genome with billions of possible locations to test. The fastest algorithms ended up being used in the most published papers, so competition was pretty fierce to be the fastest. The gold standard was a deterministic program called BWA/Bowtie, but just before I left the field a new breed of non-deterministic aligners with mapping times orders of magnitude faster were developed, and it really split opinions. Different deterministic programs would give different results (i.e. they had noise/error too, even if they're consistent about it), so in many ways who cared if a program gave different results every time you ran it, particularly if you only intend to run it once... But there were other problems. You couldn't create definitive analyses anymore, you couldn't retrace someone else's steps, you couldn't rely on checksums, total nightmare. The "hidden structures" aspect of the paper was interesting, the structures are in the data, and how the algorithm interacts with the data, which as the programmer you don't have access to by definition - but you also kinda know all you need to know about it too. It feels very similar to making a good meme.

Of course. This isn't a surprise. I've been using these techniques for optimization for a long time. Simulated annealing was proven (decades ago) to scale better than many optimization algorithms. If your big O is bigger than Sim annealing, use sim annealing! Always calculate your big O and THEN measure your implementation to make sure you hit it. Same thing goes for your error... and controlling that can blow out your big O and that's data not algorithm dependent! ALWAYS MEASURE! If you have to pre sort before accumulating to minimize error you are not going to hit your scaling numbers and you're going to murder your cache and memory pipelining. The key with that 1/e term is to recall that floating point math is going to accumulate rounding errors at a precision of about 0.1-1.0 in 1M. This sets your floor and the sensitivity of your eigenvalues ( if they vary by more than about one part in 1M, your answers will be dominated by errors, so you take the hit and use doubles). This kind of stuff used to be explicitly covered in scientific computing classes when resources were limited and the hardware was MUCH less complex. It's interesting that this complexity has managed to hide potential optimizations of order 20-1000 x. But it makes sense, in order to use the HW efficiently you need to be an expert in so many things that the problems you're actually trying to solve becomes something of an afterthought and resources allocation in universities and other organizations focused on numerical methods face the pressures of silos and hyperspecialization. Conaway's law strikes again, as our software matches the organizational structures that create it.

Another beautiful global optima of priceless information to pull me out of my local tunnels :) Thank you as always

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.

Very informative video and I will be waiting for more. I am hooked on randomized linear algebra since Ewin Tang "dequantization" papers. I wonder if randomized algos will have huge impact on ML training performance (not just inference). I also wonder how will it compare in performance and accuracy: low-rank approximations of ML models vs randomized inference on full models.

very nice!

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

I tripped across the Integer relation algorithm at 15, when I wrote a calculator program to calculate how much change you put on the scale just based on the total weight. Thanks to this video (top 10 problems list), I finally know what that's called. Until now I called this the "primeness of unique coin weights" lol

There are also other approaches where you choose for an epsilon and reduce complexity of the problem, like in hierarchical matrices

This has been my thought about deep learning for a while now - we build computers to be deterministic, but deep learning would run best on a different kind of computer that is lossy but as a tradeoff much more energy inefficient. This is a different take though (keep determinism, but instead deliberately code faster but lossy algorithms) that could also do the job,

This is something that I've really been wondering about at a general level. How to add pinches of randomness to improve inference, simulations, etc. I personally wonder how we could use it to improve model accuracy by specifically predicting error then building in a stochastic prediction, Might be a big change in ML and neural nets

Thanks

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

Great video ! I had a compressive sensing class this semester, it sure is a very interesting and promissing topic of reasearch ! But I'm not sure that video games would benefit a lot from it ? If I understood correctly, the gains are mostly at high dimension, while video games linear algebra is basically only 3D, do you have exemples ? Thanks again !

Simply phenomenal if it get implemented in deep learning framework

Let me try to phrase it for people who have no math degree education, but rather engineering level: You effectively select the best fitting equations of the linear problem which is originally highly overdefined for your x vector to sufficiently represent the complete system with a small subset of the original equations. - correct? That's not directly "inducing random noise" but rather a simplification by omission of probably irrelevant equations. This reminds me of how we did such a scheme for a "bundle block adjustment" application: We used the drastic performance boost from simplification to do multiple simple bba within each reaction step of the system with different drastically simplified subsets from the data, to then compare the results with the expected outcome (low rest error, good alignment with the continuation of the coordinates of our x vector from the previous step), then performing a final selection based on those outcomes and then performing a final error minimizing solving with those perfectly selected equations. That gives the best from both worlds: Speed up but without sacrificing correctness. And there is no magic at all (and no "introduced random noise"). Just a "try simple" first iteration, then based on that a selected final iteration. Basically engineering optimization based on working standard linear algebra systems.

"They're quite good" - Understatement of the decade 😄

Using that engineering mindset, close enough is good enough! Worked for fast inverse square, and like you said floating is pretty inaccurate anyway but still works, might as well try to guess and see how much you can get away with. AI basically does that with quantization (and in general). I should really finish watching vids before commenting since he mentions ML literally 30 seconds after I made this.

I like how you think around the problem at 27:48

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???

We can think of this as the "Battleship" problem, least number of guesses to take out a fleet, I was shown this in primary school when I was taught the value of estimation.