Nothing is “mere.” I, too, can see the stars on a desert night, and feel them. But do I see less or more? The vastness of the heavens stretches my imagination—stuck on this carousel, my little eye can catch one-million-year-old light. A vast pattern—of which I am a part—perhaps my stuff was belched from some forgotten star, as one is belching there. Or see them with the greater eye of Palomar, rushing all apart from some common starting point when they were perhaps all together. What is the pattern, or the meaning, or the why? It does not do harm to the mystery to know a little about it.
- Richard P. Feynman on The Relation of Physics to Other Sciences
Dalcy
Karma: 928
Becoming Stronger™ (Sep 28 - Oct 12)
Notes and reflections on the things I’ve learned while Doing Scholarship the last two week (i.e. studying math).
Mostly the past two weeks were on differential geometry (Lee):
Ch 4 (Submersion, Immersion, Embedding) comments:
Conceptually, by the Constant rank theorem, constant rank maps (smooth maps whose differential is constant rank at all ) are precisely the maps with a linear local coordinate representation (thus are maps well-modeled locally by its differentials).
Basically a nonlinear version of the linear algebra theorem that any square matrix can be expressed as . The proof is much more complicated however: basically a clever choice of coordinate transformation via the inverse function theorem.
The point of the chapter is to come up with various characterizations of submersion, immersion, embedding. For example, 1) smooth immersion iff locally smooth embedding, 2) smooth submersion iff every point is an image of a local section, 3) surjective maps ⇒ submersion & injective ⇒ immersion …
The proof of 3) is a very cool application of the Baire category theorem. Baire category theorem says the countable union of nowhere dense sets has empty interior; this is not very motivating, but reading Bredon[1] helped clarify its conceptual significance.
Namely, consider the more illuminating contrapositive statement: countable intersection of dense open sets is dense. Conceptually, the space is some configuration space, and dense open sets represent configurations that satisfy certain generically satisfied constraints (polynomial being nonzero is a prototypical example, which is a dense & open set). Then, the question is whether the property of a countable number of these constraints being satisfied at the same time is still generic, i.e. dense. The Baire category theorem says this is indeed the case (for locally compact Hausdorff spaces).
Sections are just right inverses, and their intuitive geometric content was a bit confusing until I read the wikipedia page: a section of f is an abstraction of a graph by viewing f as a sort of “projection map.” That makes sense! I’m sure this will come up later in the fiber bundle context.
The “figure-eight curve” and “dense torus map” as prototypical examples of smooth immersions that isn’t a smooth embedding, due to topological considerations.
Ch 5 (Submanifold) comments:
Similar to Ch 4, many useful characterizations of submanifolds and how to generate them. eg embedded submanifold iff locally a “slice” of the ambient manifold’s coordinate chart. embedded submanifold iff image of smooth embedding, immersed submanifold iff image of smooth immersion. Level sets of a smooth map at a “regular value” are embedded submanifolds …
Ch 6 (Sard’s theorem) comments:
Finally, one of the more fun chapters! Finally learned the proof of the Whitney embedding / immersion theorem that I’ve heard a lot about.
The compact case of the Whitney embedding theorem is much more conceptually straightforward:
Given a (finite, possible since compact) chart of the -dim manifold, literally just adjoin them while multiplying them with appropriate partitions of unity to get a map, and adjoin the m partitions of unity (a “chart indicator variable”) to get a map. This turns out to be an immersion, and thus an embedding since M is compact.
Apply the projection map with a 1-dim kernel . By Sard’s theorem, this turns out to be an immersion (when restricted to ) for almost any choice of , as long as . Repeatedly apply this to the massive codomain to get an immersion to .
This projection map can in fact be promoted to an embedding, given that the original immersion of M to R^n is an embedding.
High-level takeaways:
The most dumb and obvious way of interpolating coordinate charts into a global map via partitions of unity, with slight modifications, gives a bona fide immersion of a manifold into !!
It was interesting to learn that there was a 1-2 decade period of foundational uncertainty (between the first proposal of the abstract manifold definition and Whitney’s above proof) where people didn’t know whether the abstract manifold definition was actually more general than or not.[2]
Partitions of unity really is used everywhere. I wonder how the theory of complex analytic manifolds ever do anything when analytic partitions of unity don’t exist.
Proof strategy of promoting a smooth map to a proper map (at the cost of increased dimensionality of the codomain) by literally adjoining a proper map next to it. Clever!
I presume this is the main motivation behind exhaustion functions ( s.t. is compact ). It’s a proper map, it exists for any manifolds (again, shown by partitions of unity), and has codomain of dimension 1 so it minimally increases the function codomain dimension.
More applications on Whitney approximation theorems and transversality arguments.
The latter, including the transversality homotopy theorem (actually learned this a year ago in my difftop class, though that class used Guillemin’s book where manifolds are always embedded in - so it’s good to learn them from a more intrinsic perspective) is very interesting.
It also ties to one of my motivation for all this math learning, backchaining from trying to do good alignment theory work, which is learning the math of structural stability and its role in the theory of forms (morphogenesis) cf Thom, Structural Stability and Morphogenesis (thank you Dan Murfet for explaining this perspective).
Rabbit holes that I could not afford to pursue:
The category of smooth manifolds is an idempotent-splitting completion of the category of open subspaces of findim cartesian spaces?!?!?! My mind is blown.
So much more elegant than the standard definition via charts and maximal smooth structures and such. Unsure of the utility of this characterization though, lol (read Lawvere’s paper).
There is a duality between the category of smooth manifolds and the category of R-algebras. Fascinating how such dualities between algebra and geometry seem to be a very common motif throughout different fields, I’m sure this will come up in Vakil’s book later. Also curious about Gelfand’s duality on this for topological spaces.
“It is better to have a good category with bad objects than a bad category with good objects.”—Grothendieck (probably not). For example, the category of smooth manifolds is not nice, motivating smooth sets, diffeological spaces, and so on.
Dichotomy between nice objects and nice categories: in the context of alignment theory, maybe I can view Programs as Singularities as enlarging an instantiation of this idea by enlarging the class of Turing machines.
I found this intuition for adjoint functors illuminating. Specifically, note set maps and being inverses are equivalent to the condition that their graphs are mirrored along the diagonal, i.e. . Rephrase this using Kronecker delta, . Now can be seen as expressing a “relation” that could be exhibited by two elements of a set, i.e. equality (1) or inequality (0). But in general categories, objects can exhibit more relations—so replace by - you get adjoint functors!
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Example of how reading books in parallel improves learning efficiency.
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Why that long? The dimensionality reduction by projection is perhaps more nontrivial because of Sard, but the obvious gluing should have been sufficient to construct an immersion at least, albeit at the cost of inefficient codomain dimension. Maybe the historically difficult part was the concept of partition of unity and that it always exist in manifolds?
Thanks for the recommendation! Woit’s book does look fantastic (also as an introduction to quantum mechanics). I also known Sternberg’s Group Theory and Physics to be a good representation theory & physics book.
I did encounter Brown’s book during my search for algebraic topology books but I had to pass it over Bredon’s because it didn’t develop the homology / cohomology to the extent I was interested in. Though the groupoid perspective does seem very interesting and useful, so I might read it after completing my current set of textbooks.
Becoming Stronger™ (Sep 13 - Sep 27)
Notes and reflections on the things I’ve learned while Doing Scholarship this week (i.e. studying math)[1].
I am starting to see the value of categorical thinking.
For example from [FOAG], it was quite mindblowing to learn that stalk (the set of germs at a point) can be equivalently defined as a simple colimit of sections of presheaf over open sets of containing a point, and this definition made proving certain constructions (eg inducing a map of stalks from a map ) very easy.
Also, I was first introduced the concept of presheaf as an abstraction of a map that takes open sets and returns functions over it, abstracting properties like there existing a restriction map that composes naturally. Turns out (punchline, presumably) this is just a functor !
Yoneda lemma is very cool. I recall seeing some of the ideas from Programs as Singularities (paper), where there are ideas of embedding programs (for the analogue in Yoneda lemma, the Yoneda embedding being a fully faithful functor from some category … ) into a different space (… to the category …) that contains “almost programs” ( … because the Yoneda embedding is not surjective, let alone essentially surjective), and that studying this enlarged space lends insight into the original space.
Rabbit hole: Yoneda lemma as expressing consistency conditions on Lawvere’s idea of Space and Quantity being presheaf and copresheaf??
I am also starting to more appreciate the notion of sheaf or ringed spaces from [FOAG] - or more generally, the notion that a “space” can be productively studied by studying functions defined on it. For example I learned from [Bredon] that a manifold, whose usual definition is a topological space locally homeomorphic to a Euclidean space, can equivalently be defined as a ringed space whose structure sheaf is valued in some subalgebra of continuous maps over a given open set. Very cool!
Started reading [Procesi] to learn invariant theory and representation theory because it came up quite often as my bottleneck in my recent work (eg). Also interpretability, apparently. So far I just read pg 1-9, reviewing the very basics of group action (e.g., orbit stabilizer theorem). Lie groups aren’t coming up until pg ~50 so until then I should catch up on the relevant Lie group prerequisites through [Lee] or [Bredon].
Also reviewed some basic topology by skimming pg 1-50 of [Bredon]. So many rabbit holes just in point-set topology that I can’t afford (time-wise) to follow, e.g.,
(1) nets generalize sequences and successfully characterize topological properties—I once learned of filters, and I do not yet know how they relate (and don’t relate constructively + why constructively filters are more natural) and especially universal net vs ultrafilter
(2) I didn’t know that manifolds are metrizable, but yes they are by an easy consequence of the Urysohn metrization theorem (second-countable & completely regular ⇒ metrizable). But I would like to study this proof in more detail. Also, how to intuitively think about the metric of a manifold?
I didn’t know that the proof to Urysohn metrization was this nice! It’s a consequence of the following lemma: recall, “completely regular” means given a point and a closed set , there exists a continuous s.t. and . BUT adding second-countability to the hypothesis then lets you choose this from a fixed, countable family .
Then, mapping under this countable family of functions (thus taking value in ) turns out to be an embedding—and can be metrized, so can be metrized as well.
(3) I learned about various natural variants / generalizations of compactness (-compactness, local compactness, paracompactness). My understanding of their importance is because:
(a) paracompactness implies the existence of partition of unity subordinate to any open cover (a consequence of paracompactness ⇒ normal, and Urysohn’s lemma, also paracompactness by definition allowing you to find the open refinement of the given open cover as required by the definition of partition of unity subordinate to an open cover.)
(b) for locally compact Hausdorff X, we can characterize paracompactness by “disjoint union of open -compact subsets,” which is much easier to check than the definition of paracompactness as locally finite open refinement of open covers.
e.g., from this, it is immediate that manifolds are paracompact: (1) locally Euclidean ⇒ locally compact. (2) Second-countable ⇒ Lindelof. (3) Lindelof & locally compact ⇒ -compact. (1) & (2) & (3) + above ⇒ manifolds are paracompact. From which other properties of manifolds immediately follow from that of paracompactness, eg manifolds always admit a partition of unity subordinate to any open cover.
But rabbit hole: recall, open sets axiomatize semidecidable properties. What is, then, the logical interpretation of compactness, -compactness, local compactness, paracompactness?
This week, I’ll start tracking the exercises I solve and pages I cover and post them in next week’s shortform (EDIT: biweekly), so that I can keep track of my progress + additional accountability.
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I am self-studying math. The purpose of this shortform is to publicly write down:
things I’ve learned each week,
my progress through the textbooks I’ve committed to read[2], and
other learning-related reflections,
with the aim of:
allocating some time each week reflecting on what I’ve learned to self-improve,
be kept socially accountable by publicly committing on making progress and actually sticking through the textbooks,
and write / think in public which I enjoy doing so.
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I am currently reading the following textbooks:
[Lee]: Lee, Smooth Manifolds
[Bredon]: Bredon, Topology and Geometry
[FOAG]: Vakil, Foundations of Algebraic Geometry
[Procesi]: Procesi, Lie Groups: An Approach through Invariants and Representations
and I plan to do most of the exercises for each of the textbooks unless I find some of them too redundant. For this week’s shortform I haven’t written down my progress this week on each of these books nor the problems I’ve solved because I haven’t started tracking them, so I’ll do them starting next week.
A simple sketch of the role data structure plays in loss landscape degeneracy.
The RLCT[1] is a function of both and . The role of is clear enough, with very intuitive examples[2] of local degeneracy arising from the structure of the parameter function map. However until recently the intuitive role of really eluded me.
I think I now have some intuitive picture of how structure in influences RLCT (at least particular instances of it). Consider the following example.
Toy Example: G-invariant distribution, G-equivariant submodule
Suppose the true distribution is (1) realizable ( for some ), (2) invariant under some group action, . Now, suppose that the model class is that of exponential models, i.e. . In particular, suppose that , the fixed feature map, is -equivariant, i.e. such that .
Claim: There is a degeneracy of the form , and in particular if is a Lie group, the rank upper bound of RLCT decreases by .
This is nothing nontrivial. The first claim is an immediate consequence of the definitions:
and implies
Then, we have the following:
… and the latter claim on RLCT is a consequence of reducing the rank of at by together with the rank upper bound result here.
High-level idea: Emulability of input symmetry
While this model is very toy, I think the high-level idea for which this a concrete model of is interesting: Abstracting out, the proof of how data structure influence degeneracy routes through two steps:
The true distribution has some structure / symmetry, say, (with as a function of , indicating some infinitesimal change; all of this is meant to be taken heuristically), which gets imparted onto by realizability, i.e. .
Emulatability: At , the model can “emulate” certain classes of perturbations to certain classes of input by instead perturbing the parameters, i.e. .[3]
Basically, (1) realizablity imparts input-symmetry to , and (2) emulatability essentially “push-forwards” this to a symmetry in the parameters[4]. I think this is very interesting!
Story: Suppose I am tasked with image segmentation, but my visual cortex is perturbed by , causing me to perceive colors with a slightly different hue. Then, if my visual cortex wasn’t perturbed but rather the world’s color shifted to that hue i.e. , then I would virtually not notice anything and be making the same predictions .
Going back to the exponential model, the most unrealistic part of it (even after taking into account that it is a toy instantiation of this high-level idea) is the fact that its symmetry is generic: holds for ALL , since the -equivariant is independent of . A more realistic model would look something like where also depends on and importantly, whether satisfies -equivariance depends on the value of .
Then, if but makes -equivariant while doesn’t, then the rank upper bound of the RLCT for the former is lower than that of the latter (thus would be represented much more greatly in the Bayesian posterior).
This is more realistic, and I think sheds some light on why training imparts models with circuits / algorithms / internal symmetries that reflect structure in the data.
(Thanks to Dan Murfet for various related discussions.)
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Very brief SLT context: In SLT, the main quantity of interest is RLCT, which broadly speaking is a measure of degeneracy of the most degenerate point among the optimal parameters. We care about this because it directly controls the asymptotics of the Bayesian posterior. Also, we often care about its localized version where we restrict the parameter space to an infinitesimal neighborhood (germ) of a particular optimal parameter we’re interested in measuring the degeneracy of.
RLCT is a particular invariant of the average log likelihood function , meaning it is a function of the true distribution and the parametric model (the choice of the prior doesn’t matter under reasonable regularity conditions).
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Given a two layer feedforward network with ReLU, multiply the first layer by and dividing the next by implements the same function. Many other examples, including non-generic degeneracies which occur at particular weight values unlike the constant multiplication degeneracy which occurs at every ; more examples in Liam Carroll’s thesis.
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This reminds me of the notion of data-program equivalence (programs-as-data, Gödel numbering, UTM). Perhaps some infinitesimal version of it?
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Let the input-side symmetry to be trivial (i.e. ), and we recover degeneracies originating from the structure of the parameter-function map alone as a special case.
Found a proof sketch here (App. D.3), couldn’t it find elsewhere in canonical SLT references eg gray book. Idea seems simple:
(We’ll prove it for the local RLCT because the splitting lemma most naturally applies when dealing with local diffeomorphisms—but if you’re interested in the statement for the global RLCT, then since RLCT is min over local RLCT (3.2.2), just work with the local RLCT at the minimizing point and the rest of the argument follows)
Local RLCT is invariant under local diffeomorphism, easy to prove by the volume scaling formula (check exercise 17).
Apply the splitting lemma (elementary proof in Gilmore 3.4) to locally express the function as a quadratic in the nondegenerate plus the rest.
Use Remark 7.2.3 in the gray book which says that if and (which the change of variable in the splitting lemma satisfies), then (also easy to prove with the volume scaling formula).
So is lower-bounded by the RLCT in the quadratic part, which is (again easy to prove with the volume scaling formula, check pg 17 here)
There shouldn’t be a negative sign here (14a).
(will edit this comment over time to collect typos as I find them)
The fourth one is great.
Conventionally is a random variable, just like how is a random variable. To be fair the conventions are somewhat inconsistent, given that (as you said) is a number.
Previous discussion, comment by johnswentworth:
Relevant slogan: Goodheart is about generalization, not approximation.
[...]
In all the standard real-world examples of Goodheart, the real problem is that the proxy is not even approximately correct once we move out of a certain regime.
Speaking from the perspective of someone still developing basic mathematical maturity and often lacking prerequisites, it’s very useful as a learning aid. For example, it significantly expanded the range of papers or technical results accessible to me. If I’m reading a paper containing unfamiliar math, I no longer have to go down the rabbit hole of tracing prerequisite dependencies, which often expand exponentially (partly because I don’t know which results or sections in the prerequisite texts are essential, making it difficult to scope my focus). Now I can simply ask the LLM for a self-contained exposition. Using traditional means of self-studying like [search engines / Wikipedia / StackExchange] is very often no match for this task, mostly in terms of time spent or wasted effort; simply having someone I can directly ask my highly specific (and often dumb) questions or confusions and receive equally specific responses is just really useful.
Non-Shannon-type Inequalities
The first new qualitative thing in Information Theory when you move from two variables to three variables is the presence of negative values: information measures (entropy, conditional entropy, mutual information) are always nonnegative for two variables, but there can be negative triple mutual information .
This so far is a relatively well-known fact. But what is the first new qualitative thing when moving from three to four variables? Non-Shannon-type Inequalities.
A fundamental result in Information Theory is that always holds.
Given random variables and , from now on we write with the obvious interpretation of the variables standing for the joint variables they correspond to as indices.
Since always holds, a nonnegative linear combination of a bunch of these is always a valid inequality, which we call a Shannon-type Inequality.
Then the question is, whether Shannon-type Inequalities capture all valid information inequalities of variable. It turns out, yes for , (approximately) yes for , and no for .
Behold, the glorious Zhang-Yeung inequality, a Non-Shannon-type Inequality for :
Explanation of the math, for anyone curious.
Given random variables and , it turns out that is equivalent to (submodularity), if , and .
This lets us write the inequality involving conditional mutual information in terms of joint entropy instead.
Let then be a subset of , each element corresponding to the values of the joint entropy assigned to each subset of some random variables . For example, an element of would be for some random variables and , with a different element being a different tuple induced by a different random variable .
Now let represent elements of satisfying the three aforementioned conditions on joint entropy. For example, ’s element would be satisfying e.g., (monotonicity). This is also a convex cone, so its elements really do correspond to “nonnegative linear combinations” of Shannon-type inequalities.
Then, the claim that “nonnegative linear combinations of Shannon-type inequalities span all inequalities on the possible Shannon measures” would correspond to the claim that for all .
The content of the papers linked above is to show that:
but (closure[1])
and , and also for all .
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This implies that, while there exists a -tuple satisfying Shannon-type inequalities that can’t be constructed or realized by any random variables , there does exist a sequence of random variables whose induced -tuple of joint entropies converge to that tuple in the limit.
Relevant: Alignment as a Bottleneck to Usefulness of GPT-3
between alignment and capabilities, which is the main bottleneck to getting value out of GPT-like models, both in the short term and the long(er) term?
By the way, Gemini 2.5 Pro and o3-mini-high is good at tic-tac-toe. I was surprised because the last time I tested this on o1-preview, it did quite terribly.
Where in the literature can I find the proof of the lower bound?
Previous discussion, comment by A.H. :
Sorry to be a party pooper, but I find the story of Jason Padgett (the guy who ‘banged his head and become a math genius’) completely unconvincing. From the video that you cite, here is the ‘evidence’ that he is ‘math genius’:
He tells us, with no context, ‘the inner boundary of pi is f(x)=x sin(pi/x)’. Ok!
He makes ‘math inspired’ drawings (some of which admittedly are pretty cool but they’re not exactly original) and sells them on his website
He claims that a physicist (who is not named or interviewed) saw him drawing in the mall, and, on the basis of this, suggested that he study physics.
He went to ‘school’ and studied math and physics. He says started with basic algebra and calculus and apparently ‘aced all the classes’, but doesn’t tell us what level he reached. Graduate? Post-graduate?
He was ‘doing integrals with triangles instead of integrals with rectangles’
He tells us ‘every shape in the universe is a fractal’
Some fMRI scans were done on his brain which found ‘he had conscious access to parts of the brain we don’t normally have access to’.
I wrote “your brain can wind up settling on either of [the two generative models]”, not both at once.
Ah that makes sense. So the picture I should have is: whatever local algorithm oscillates between multiple local MAP solutions over time that correspond to qualitatively different high-level information (e.g., clockwise vs counterclockwise). Concretely, something like the metastable states of a Hopfield network, or the update steps of predictive coding (literally gradient update to find MAP solution for perception!!) oscillating between multiple local minima?
Curious about the claim regarding bistable perception as the brain “settling” differently on two distinct but roughly equally plausible generative model parameters behind an observation. In standard statistical terms, should I think of it as: two parameters having similarly high Bayesian posterior probability, but the brain not explicitly representing this posterior, instead using something like local hill climbing to find a local MAP solution—bistable perception corresponding to the two different solutions this process converges to?
If correct, to what extent should I interpret the brain as finding a single solution (MLE/MAP) versus representing a superposition or distribution over multiple solutions (fully Bayesian)? Specifically, in which context should I interpret the phrase “the brain settling on two different generative models”?
I just read your koan and wow it’s a great post, thank you for writing it. It also gave me some new insights as to how to think about my confusions and some answers. Here’s my chain of thought:
if I want my information theoretic quantities to not degenerate, then I need some distribution over the weights. What is the natural distribution to consider?
Well, there’s the Bayesian posterior.
But I feel like there is a sense in which an individual neural network with its weight should be considered as a deterministic information processing system on its own, without reference to an ensemble.
Using the Bayesian posterior won’t let me do this:
If I have a fixed neural network that contains a circuit that takes activation (at a particular location in the network) to produce activation (at a different location), it would make sense to ask questions about the nature of information processing that does, like .
But intuitively, taking the weight as an unknown averages everything out—even if my original fixed network had a relatively high probability density in the Bayesian posterior, it is unlikely that and would be related by similar circuit mechanisms given another random sample weight from the posterior.
Same with sampling from the post-SGD distribution.
So it would be nice to find a way to interpolate the two. And I think the idea of a tempered local Bayesian posterior from your koan post basically is the right way to do this! (and all of this makes me think papers that measure mutual information between activations in different layers via introducing a noise distribution over the parameters of are a lot more reasonable than I originally thought)
I like the definition, it’s the minimum expected code length for a distribution under constraints on the code (namely, constraints on the kind of beliefs you’re allowed to have—after having that belief, the optimal code is as always the negative log prob).
Also the examples in Proposition 1 were pretty cool in that it gave new characterizations of some well-known quantities—log determinant of the covariance matrix does indeed intuitively measure the uncertainty of a random variable, but it is very cool to see that it in fact has entropy interpretations!
It’s kinda sad because after a brief search it seems like none of the original authors are interested in extending this framework.
This seems like a misleading example of doomers being wrong (agree denotationally, disagree connotationally), since I think it’s plausible that Y2K was not a big deal (to such an extent that “most people think it was a myth, hoax, or urban legend”) precisely because of the mitigation efforts stemmed by the doomsayers’ predictions.