dbranes

Nice to see Jared Kaplan branching out into ML. He did fundamental work on CFTs/bootstrap in physics.

No part of this comment answers my question.

Can you state what's the mechanism that makes it fast? Is this a breakthrough in rendering optimizations or does it introduce smarter data structures? Or is it an accumulation of small optimizations everywhere?

Cool, but this seems to contain very few illustrations.

It looks like she wants to talk about symmetric monoidal categories, not just monoidal categories, for that’s where the calculus of string diagrams is actually useful.

Tensor categories is another word for symmetric monoidal categories, indeed it’s less of a mouthful, and perhaps more “intuitive” - it’s conventional to write the symmetric monoidal structure as a tensor product.

Genuinely tried reading the "examples" you wrote in this thread, can't make any sense out of it. Happy to discuss it if you clarify what you mean.

Just to address your original comment in this thread, perhaps it's relevant to note the following. Consider the homotopy theory of the category of nice topological spaces. The full subcategory of topological spaces supported on discrete topological spaces inherits a homotopy theory. This inherited homotopy theory is equivalent to the trivial homotopy theory on the category of sets: where weak equivalences are isomorphisms. This is the sense in which discrete spaces don't have an interesting homotopy theory, at least naively.

(This statement you can precise in your favorite model for the homotopy theory of spaces, via infinity categories, model categories etc.)

(the embedding is full & faithful)

The category of sets embeds into the category of topological spaces as a full subcategory, the essential image of which are the discrete spaces. Hence the equivalence I claimed is a precise statement.

Look up ring localization.

As you observe one can't really recover any non-trivial number theoretic things by looking at integers with the discrete topology. The theory of discrete topological spaces is just the theory of sets.

Instead you can look at things like prime ideals of integers localized at some prime, and consider algebro-geometric topologies on that

What a confusing name in an era where homotopy-theoretic methods in arithmetic geometry are flourishing. Looks Hatcher covers non of that and use "topology" to mean "geometrically motivated".

Some context would be nice as this (very nice library) has been around for a while: was there a new release? What are we supposed to be looking for Here?

Ugh, I'm cringing a little that they consider adjunctions 'advanced mathematics'. Adjunctions are ubiquitous even in elementary math. For example, in linear algebra whenever one writes down a matrix to represent a operator in some basis, this is using the tensor-hom adjunctions for modules.

Tangent: I like that the logo for the organization 'adjoint' resembles the notation for adjoint functors.

Yes absolutely. Given a matrix as an array of numbers there's a number of natural ways to interpret it as a tensor. What are you uncomfortable about?

Sorry this is just wrong. Maybe you’re trying to get at the “co/contravariant” properties of tensors, in which case your statement can be more clearly stated as, e.g., the space rank (2, 0), and rank (1,1) vectors, admit different interpretations as internal hom spaces of vector spaces. But in any interpretation of your statement the distinction is never important because all spaces distinguished by this distinction are isomorphic via cononical isomorphisms.

Had to scroll down the page for a while before understanding the point of this: drag-and-drop data transformation pipelines that comes with app integrations at both ends. It’s a great idea!

Not an answer but a general comment on learning languages: instead of setting your goal as “to learn as much of a languageas possible”, instead you can go for “I want to be able to do this class of things that will be made very easy once i know thia language”.

If you do the former it’s very easy to learn something then forget it. Furthermore, echoing other comments, you would lack clear measures of success because it almost makes no sense to ask whether you truly “know” a language.

As someone with a high energy theory background but has only read titles/headlines about this time crystal business, can you give a precise TL;DR on what a time crystal is?

Love the idea - looks like a well-thought-out and much needed abstraction. The way I’d pitch it is that it allows one to create UI in a completely declarative and stateless manner.

pretty sure that’s a windows thing, compare with the screenshot of current cheome in the article.

To each their own, but I’d hate the firefox-style scrolling tabs UI suggested by the article.

1. I don’t want to be surprised to find some old tab hidden by the scroll.

2. Id like to be able to close tabs without first scrolling to it.

3. If I control tab past the last tab does it scroll to the first tab? Id then have to scroll back to find my latest tabs?

4. Again, personal preference, but if I had 50 tabs open and wanted to go back to the 24th one, id rather navigate to it through a search engine or bookmarks, rather than trying to locate that particular tab.

This is excellent. Amazing (equation + picture)/text ratio.

Complaint: you don't define your notion of "space". In chapter 1 it's some informal notion that you use to motivate the definition of a set (??), in 1.3 and 1.4 it becomes clear by space you mean "set". Then later you start talking about dimension of spaces, implying not only do they come with a topology now they have a well defined dimension, so a locally Euclidean Hausdorff space or something - but maybe you just mean R^n.

Comment for other commentators in this thread: not all expositions is tailored for the masses. A piece of pedagogical literature that does not appeal to your background doesn't mean it's not good. There's a very clear need for exposition on basic structures in probability theory and this fits there.

Very cool. A question and a comment:

- What about higher(>3)-rank tensors?

- It's not clear to me what distinguishes your picture for a convolution layer from any other tensor of the same shape. Would be nice to visualize the filters/kernels somehow.

For people wondering about the type of thing that the generator supports, there's this config file https://github.com/wonderunit/storyboarder/blob/f0a7cc75beb1...

It's a neural network trained in an adversarial manner which generates things, so yes.

Observation: an alternative way to accomplish the same thing (maybe with a slightly different workflow) is to just call tf-serving from go. See e.g. this gist https://gist.github.com/mauri870/1f953a183ee6c186e70a0a72e78...

You're right I miscounted.

Here's a generalization that yields an infinite family of similar observations.

Step 1. Take any "torus knot with n marked points": which for our purposes will mean homeomorphism classes of embeddings from a circle with n marked points, to the torus.

Step 2. Draw it on paper.

In the case of the Coltrane drawing, n = (5 octaves * 13 notes per octive) = 65. The author exhibited 3 non-homeomrphic embeddings of this circle into the torus in the three images below the protractor picture. In particular these are embeddings generated by iterated Dehn twists on the unknot. You can classify them by winding number.

The image right below the aforementioned ones also shows an embedding of a circle with marked points into the torus if we choose to identify the 2 'c's. This time there's only 13 marked points. This suggests the following. To a circle with marked points, one can associate a canonical family, labelled by integers, of circles with marked points as follows: take any covering map of the circle, and declare the union of the fibers over the marked points to be the new marked points. In the case of the music scale analogy, we take the circle with 13 marked points, and this in particular gives a family of circles with 13 * k marked points for any positive integer k, and the musical interpretation is that k is how many octave one chooses to have. In the images mentioned above k = 3 and k = 1 are exhibited.

There's a simple further generalization of all this: we can replace the torus with any topological space. For example, doing this on a higher genus surface or a non-orientable surface would both yield probably interesting-looking diagrams.