Lipschitz continuity Puns

I know how to prove all of these if asked on an exam. But what exactly is different from each of these? visually how do they differ?

In this video we continue on the topic of Lipschitz continuity by presenting a paper which proposes a projection method to enforce it! If you enjoy this video consider watching others which I have on the topic! :) I would love to have discussion here or on the comment section, the goal of this youtube channel is to create knowledge and interesting discussions in this area of ML.

Video: https://www.youtube.com/watch?v=9kxhEdiTwek

Paper: https://arxiv.org/abs/1804.04368

Abstract: We investigate the effect of explicitly enforcing the Lipschitz continuity of neural networks with respect to their inputs. To this end, we provide a simple technique for computing an upper bound to the Lipschitz constant---for multiple p-norms---of a feed forward neural network composed of commonly used layer types. Our technique is then used to formulate training a neural network with a bounded Lipschitz constant as a constrained optimisation problem that can be solved using projected stochastic gradient methods. Our evaluation study shows that the performance of the resulting models exceeds that of models trained with other common regularisers. We also provide evidence that the hyperparameters are intuitive to tune, demonstrate how the choice of norm for computing the Lipschitz constant impacts the resulting model, and show that the performance gains provided by our method are particularly noticeable when only a small amount of training data is available.

Hello

I know there were a lot of posts about No Man's Sky lately. But still i stumbled upon a Tweet from two days ago.
In the Tweet Sean Murray said that the planet terrains now were Lipschitz continuous.

I don't have a lot of Knowledge about mathematical analysis and was wondering if someone could explain what this means in relation to the planet terrain?

Can anyone explain Lipschitz Continuity to me? It came up in a class discussion on global and local truncation error when solving nonlinear ODEs using euler’s method but was never explained clearly

The question just ask to proof that f(x,y)=2xy² is lipschitz function within D(-1,1)×(0,2) range. How can i proof this because in the lecture the teacher just proof function with one variable only so when it comes to xy which is 2 var i kinda lost.Can someone please explain it so i can understand these better.

Still trying to fully understand WGAN here. At this point I've understood that the EM distance is Lipschitz continuous, because you can think about moving infinitesimal masses from one distribution to the other.. and I think I get what Lipschitz buys you. If you have a function that can distinguish two classes, and that function is Lipschitz continuous, then you can be sure that it has a smooth gradient that improves training on it. If I squint, I can even understand why clipping weights or normalizing the local gradients (Improved WGAN) ensures your function is Lipschitz continuous.

So I think that main thing that still escapes me is: I see that D=EM distance -> implies D is Lipschitz -> implies D has a nice gradient, but I don't see how D is Lipschitz -> implies D=EM distance. It seems to me that the papers and blogs I've read just kind of say "EM distance has these great properties, and if we train a discriminator network with this kind of regularization we approximate it," where "this kind of regularization" assures Lipschitz continuity. But I don't see why Lipschitz continuity in the discriminator implies that it is a EM distance approximator. Is there some proof that EM distance is the only Lipschitz continuous divergence measure?

I'm on my second week of university math, which so far has consisted solely of definitions, proofs, and theorems and I don't get a single one of them. For example, limits: I know what a limit is (the value which f(x) approaches as x approaches a certain value) and I can solve problems involving limits, but when I look at the definition I feel like a child.

"For every epsilon greater than zero, there exists a delta such that..." All I see is symbols and all I hear is random words. I feel like I'm learning to speak a sentence in a foreign language without learning the meaning of it.

Same thing with Lipschitz continuity. What the heck is that? Everywhere I ask I get the same answer: the definition. I just solved a problem where I was supposed to find the Lipschitz constant, and I found it, but I have no idea what it represents or what I'm supposed to do with it.

It's only six weeks till the exam and we've got about 14 more definitions to plow through. How can I accomplish this? Am I lacking some sort of fundamental knowledge? How does one go from "calculating" math (using the product rule, integration by parts) to reading, understanding and applying things like ∀Ɛ>0∃δ>0 : 0<|x-x'|<δ ⇒ |f(x)-f '|<Ɛ ?

Let me start off with my motivation: I'm thinking about the clustering assumption of machine learning algorithms (nearby objects have similar function values). Specifically, I'd like to vary the distance function to lower the variance of function values in small neighborhoods around each object.

While trying to think through this problem, it occurred to me that maybe what I'm doing is like trying to lower the Lipschitz constant over small neighborhoods. Which brings me to:

Is there a name for a property roughly similar to local Lipschitz continuity when the domain of a function is discrete?

Let's say we have a countable set X and an edit distance between elements of X.

||f(x) - f(y)|| / distance(x,y) <= K

What would you call this?

Note: This question was cross posted to MathOverflow here.

Let f: R^n -> R be a Lipschitz continuous function with ”strict Lipschitz constant” L > 0 - that is, |f(x) - f(y)| < L|x - y| for all x =/= y.

Question: What is the maximal Hausdorff dimension of the set on which f is differentiable and |Df| = L?

Remarks:

i) A Hausdorff dimension of n-1 is trivially achievable.

ii) By sticking together a maximising sequence, we may achieve the supremal Hausdorff dimension, so we are justified in speaking of the maximum.

For example, style transfer networks have some modules that split the style and the content of input images. How can I come up with an architecture that ensures that the learned function is Lipschitz continuous? Pointing to some papers or repositories is more appreciated!

So if we plot the absolute value of Cos[x] we would get this graph. Now it is clear that every nPi/2 there is a point where f(x) is not differentiatable, yet still continuous. Now since Lipschitz continuity requires a function to be differentiable almost everywhere, does that mean that f(x) fits that definition? Despite having an infinite number of undifferentiable points? I guess it should be ok since those points are not measurable sets.

In the def of Lipschitz continuity what is meant by "differentiable almost everywhere"?

Any help appreciated :)

So, I've stumbled across a few different "theorems" while looking through old scripts. Let A be a Lebesgue-measurable subset of R^(n):

• Let f be a Lipschitz continuous function from R^n to R^n then f(A) is Lebesgue-measurable too.

• Let f be a Lipschitz continuous function from R^n to R^m then f(A) is Lebesgue-measurable too.

• Let f be a continuously differentiable function from R^n to R^n then f(A) is Lebesgue-measurable too.

Now I've seen a proof for the first one and I don't think it's flawed so I guess the theorem is correct. Can someone validate this?

I sadly don't have proofs for the other two and actually I think I've found a counterexample to the second one. Let V be some non-measurable (Lebesgue) subset of R^1 and let f be the projection on the first component from R^2 to R^1 that is f(x,y)=x for all (x,y) in R^2. Then f is Lipschitz continuous (right?) but f(Vx{0}) = V. Vx{0} is a null set in R^2 and therefore Lebesgue-measurable however V is not. So is the second "theorem" wrong or is my counterexample flawed?

And what about the third one? Is it a valid theorem? I don't have a proof for it but I don't really have a counterexample either. I know that a continuously differentiable function is locally Lipschitz but when I look at the proof of the first theorem I don't really see how to generalize it to the locally Lipschitz case. Could someone tell me wether the third "theorem" is correct or not?

Suppose you have a function f(x), but you do not know the analytical expression for f. However, you can still generate an output y=f(x) for any input x in some bounded domain D.

My question is, is there any measure of how continuous this function is on domain D?

Here is my idea: suppose D = [0, 1], for simplicity. We can take a set of inputs X = {0, 0.01, 0.02, ..., 1} and compute the function to get Y = {f(0), f(0.01), ..., f(1)}. Then we can find the maximum value of |f(x_k) - f(x_k+1)| / |x_k - x_k+1| and what we'd get is a lower bound of the Lipschitz constant for f, which we can use as a measure of continuity.

I'd appreciate if you could give me some other ideas or guide me in the right direction.

Image samples from Celeb and LSUN Church

Hi, I present one of the paper I recently participated. We propose a new normalization method (with concrete proof ) for GANs which improves from previous penalty based variants and spectral normalization in terms of FID and Inception score.

Gradient normalization method imposes a hard 1-Lipschitz constraint on the discriminator function, which increases the capacity of the discriminators ( generator have to generate higher "fidelity" images to cheat the discriminator )

If you are interested feel free to checkout out arxiv preprint version:

Arxiv : https://arxiv.org/abs/2109.02235

Readers friendly : https://www.arxiv-vanity.com/papers/2109.02235/

Code is now open source!

Github : https://github.com/basiclab/GNGAN-PyTorch

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

The nurse asked the rabbit, “what is your blood type?”

“I am probably a type O” said the rabbit.

Mentos

(I will see myself out)

The doctor says it terminal.

Alot of great jokes get posted here! However just because you have a joke, doesn't mean it's a dad joke.

THIS IS NOT ABOUT NSFW, THIS IS ABOUT LONG JOKES, BLONDE JOKES, SEXUAL JOKES, KNOCK KNOCK JOKES, POLITICAL JOKES, ETC BEING POSTED IN A DAD JOKE SUB

Try telling these sexual jokes that get posted here, to your kid and see how your spouse likes it.. if that goes well, Try telling one of your friends kid about your sex life being like Coca cola, first it was normal, than light and now zero , and see if the parents are OK with you telling their kid the "dad joke"

I'm not even referencing the NSFW, I'm saying Dad jokes are corny, and sometimes painful, not sexual

So check out r/jokes for all types of jokes

r/unclejokes for dirty jokes

r/3amjokes for real weird and alot of OC

r/cleandadjokes If your really sick of seeing not dad jokes in r/dadjokes

Punchline !

Edit: this is not a post about NSFW , This is about jokes, knock knock jokes, blonde jokes, political jokes etc being posted in a dad joke sub

Edit 2: don't touch the thermostat

Do your worst!

How the hell am I suppose to know when it’s raining in Sweden?

Mathematical puns makes me number

We told her she can lean on us for support. Although, we are going to have to change her driver's license, her height is going down by a foot. I don't want to go too far out on a limb here but it better not be a hack job.

Ants don’t even have the concept fathers, let alone a good dad joke. Keep r/ants out of my r/dadjokes.

But no, seriously. I understand rule 7 is great to have intelligent discussion, but sometimes it feels like 1 in 10 posts here is someone getting upset about the jokes on this sub. Let the mods deal with it, they regulate the sub.

They were cooked in Greece.

I'm surprised it hasn't decade.

He lost May

Now that I listen to albums, I hardly ever leave the house.

Said if she ever hosts a gender reveal party, when it comes time to pop the balloon she'll spray everyone with water.

Gender is fluid.