A cheat sheet for Brascamp-Lieb inequalities
I have ventured into different mathematical principles. Along the way, I've had the chance to read and digest fascinating papers. While some aided me in my work, others were there for a short journey. Still, I want to write my opinions on interesting works that I've taken the time to analyze. Mostly I focus on what snippets I've learned from the writers, whether it's a proof technique or a new concept.
My summary of the convergence of discrete harmonic extensions based on the paper "Random walk on sphere packings and Delaunay triangulations in arbitrary dimension". *first version, before the extensive update* The key idea is the quantitative uniform convergence of the suitably adaptive discrete Dirichlet problem, obtained by leveraging an energy-control estimate.
My thoughts on the coupling of Hebbian learning and preferrential attachment. This coupling suggests the potential for applying free probability to the study of spiking neural networks.
My summary of the paper "A probabilistic Takens theorem". As a fan of Whitney's embedding theorem, I couldn't pass up a chance to learn more about coordinate-delay embedding theorems.
My technical review of the paper "Scaling algorithms for unbalanced optimal transport problems ", written in a simplified manner. I was most interested in understanding how the so-called scaling algorithms came about and the Thompson metric for fixed-point convergence, even though they are only a fraction of the vast amount of information given in the said paper.
My technical review of the paper "Minimax estimation of smooth optimal transport maps ". It is one of the first papers in statistical optimal transport that I've read. Like the previous paper, this one also shares a tremendous amount of information. It's where I learned about min-max theory and statistical wavelet theory.
My technical review of the paper "Optimal sampling rates for approximating analytic functions from pointwise samples ". It is where I learned about the stability problem in sampling theory as well as the Impossibility theorem. The topic is related to one of my works.
My technical review of the paper "Equivalence of approximation by convolutional neural networks and fully-connected networks ". It is one of my first reads on the mathematics of neural networks and undoubtedly one of my favorite reads, just for the sheer rigorousness alone.
My summary/review of the paper "Stable phase retrieval in infinite dimensions ". I learned the many applications of the phase retrieval problem through this paper.
My semi-technical review of the paper "Condensation in preferential attachment models with location-based choice" by Haslegrave et al. This is one of my first exposure to the concept of an evolving, dynamical network.
My cheat sheet for extremizing Brascamp-Lieb inequalities and a discussion of an entropy lemma
Diagram showing how the Loomis-Whitney inequality can be obtained via,
applying a continuous entropy lemma on the hypergraph H=((x,y,z), (yz,xz,xy))
specifying a Brascamp-Lieb datum (B,p)
Amendment 1. An amended version of "Restricted Riemannian geometry for positive semidefinite matrices," which is also available on arXiv. The mathematical content remains unchanged from the original publication in LAA. However, the revised version features updated Figures 1 and 2, which have been arranged for improved clarity.
Amendment 2. An updated version of "Functions of nearly maximal Gowers-Host-Kra norms on Euclidean spaces." This version differs from the published version in the following points:
better presentation (better equation alignments, better referencing)
typesetting errors corrected
references that were pre-prints at the time of publication are now cited with their corresponding appropriate journals.
The mathematical content is unchanged.
Amendment 3. (Probably outdated) Some extra calculations that were cut from "Superiority of GNN over NN in generalizing bandlimited functions." Particularly, I discussed how the approximation scheme using Taylor expansions adapted to a NN structure might require an extra log factor of network weights than what would be required using a GNN.
If you had asked me two months ago whether AI could ever replicate the uniquely human capacity for logic and reasoning, I would have said no. Ask me the same question now, and the answer is a highly probable “yes.” The more appropriate question has shifted from whether to when. When will we invent machine learners that don’t consume the energy budget of a small nation? When will they learn to be as “efficient” as a human brain? (And yes, I’m aware that human brains burn plenty of calories, but at least they don’t require a data center just to generate a short video.) If there is a will, there is a way. And if humanity, or a subset of humanity with enough resources and ambition, decides to make it happen, it probably will. One of my research directions is spiking neural networks, and I’ve been curious whether they might contribute to a new generation of energy-friendly AI models. Or are they even a tiny, respectable imitation of the human brain? (no, but they’re trying their best) As I think more about “intelligence,” my real fear isn’t just that AI might eventually replicate human thought. It’s the possibility that AI might one day read my thoughts, anticipating not just my lunch order, but my desires, attractions, and inner doubts. (Because, what if a human were treated—quite harmlessly, wink—as a reasonably consistent system of thoughts, behaviors, and patterns?)
Anyway, this brings me to an article Google thoughtfully suggested. The piece discusses a new book by a life scientist and author who argues (I paraphrase) that “intelligence is an emergent property of social complexity.” Believable. Many of us have had those moments where we seem to sync brains with someone else. But the article left me unconvinced in a few places. Take this poetic sentence: “The way this explanation differs from others is by offering an incentive rather than simply means to achieve it: yes, free hands, meat diet, and many other factors made our brain possible, but the reason we needed it in the first place was to remember all our friends who helped us fight monsters.” To me, the “meat diet” was driven by the goal toward intelligence. “Free hands” arose through circumstantial evolution and was also driven by the goal toward intelligence. Complex social structures sustain intelligence once the human brain size reaches a certain plateau, after “free hands” and “meat diet.” Otherwise, would horses, when gathering in groups large enough to start their own social clubs, develop human intelligence? Something tells me they won’t. And what about the Neanderthals and Denisovans? They also trended toward intelligence, and there’s no evidence that their group sizes were meaningfully smaller than those of early Homo sapiens. Why did they trend slower, that in the end, they were absorbed by us?
Disagreements aside, it’s a nice read. (The author later drifted into a discussion of human happiness and all that jazz.) What intrigued me, though, is that this short article hints at a key I’ve suspected for a while, for sustainable intelligence and perhaps anti-hallucinations: high-dimensional connections and sizable strongly connected components, so as to essentially invoke a Ramsey-type principle. But then this drops me right back into a familiar loop: reaching that level of complexity requires enormous energy expenditure. I glimpse a light, it flickers, and then I’m lost in my thoughts all over again.