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Vaishnavh Nagarajan

@vaishnavh.bsky.social
3.4K followers 387 following 237 posts

Foundations of AI. I like simple and minimal examples and creative ideas. I also like thinking about the next token 🧮🧸 Google | PhD, CMU | arxiv.org/abs/2504.15266 | arxiv.org/abs/2403.06963 vaishnavh.github.io

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Vaishnavh Nagarajan @vaishnavh.bsky.social · 25/08/2026
Loved this analogy between how we perceive a scientific discovery that takes its full form from nascent glimpses and little elements of that idea and how an orchestra is set up and perceived. From Loren Eiseley's The Firmament of Time.
screenshot of paragraph from book
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 25/06/2026
I learned this from the proof of the "strong lottery ticket hypothesis" in this eye-opening paper: arxiv.org/abs/2002.00585
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 25/06/2026
This means, you don't need exponentially many darts to get close to the target! The full post is here: vaishnavh.github.io/blog/lth/
vaishnavh.github.io
A (dis)analogy for (mis)understanding the lottery ticket hypothesis - Vaishnavh Nagarajan
“For any given task, a sufficiently large neural network will contain within it a good subnetwork that can be isolate...
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 25/06/2026
Our intuition is to imagine each subnetwork in a network as throwing a dart in high-dim space to match a target. But, in fact what's played is *multiple*, *independent*, easier games of dart-throwing in multiple low-dims; the best dart from each game is put together to make the winning subnetwork.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 25/06/2026
In new post, I write about how we misunderstand the famous lottery ticket hypothesis. We tend to think that a network needs to be exponentially large for there to be a subnetwork to win the lottery. This is incorrect---a small network suffices! I use a dart throwing analogy to make sense of this.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/06/2026
I wrote this bec I have always implicitly referred to this analogy when I design talks or give feedback. It helps adapt on the fly rather than memorize rules and practice them. I find it particularly useful in day-to-day collaborations and the beginning stages of an exploration!
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/06/2026
Meta-information also verbalizes the many gestures of spatial guidance like "put this incoming piece of the map over there" or "don't look at that, it's just a distraction!" The essay also discusses how these rules break in story-telling.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/06/2026
Another is that you always provide meta-information (information about information) *before* the information. *Before* saying something you explain why it's needed, where it's headed, so that the audience is psychologically ready to assimilate and compress that information.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/06/2026
These challenges justify the many well-known rules of technical communication. One broad rule is to reveal the pieces of the map in a "contiguous" order and also in a "hierarchical" order. This helps the impatient audience make progress with each step, and with a bird's eye view.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/06/2026
The other challenge is that the marooned audience has limited memory and cognition & they're filled with doubt on a terrifying island: where are we now? why are we here? where are we headed? are were there yet? why are you, my savior, making me climb this nasty wall of equations?
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/06/2026
Link: vaishnavh.github.io/blog/technic... The challenge in such a rescue mission is that you've to verbalize an intricate, loopy, branchy 2-dimensional thing (the map). But you also somehow need to convey gestures like pointing a finger, orienting an object etc., with just words.
vaishnavh.github.io
An analogy for technical communication - Vaishnavh Nagarajan
Rescuing a marooned audienceTechnical/scientific communication—writing, talks, day-to-day collaboration, especially d...
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/06/2026
In my next blogpost, I write about how I view technical communication: it's like trying to communicate an escape route to someone without a map but with a catch: you're not with them. You only have a walkie-talkie. Also, they're in panic.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 14/04/2026
This is just an attempt to weave all my related armchair thoughts into one piece; not a serious deeply-researched cite-able philosophical essay. Happy to be hear more thoughts & refs to expand my thinking!
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 14/04/2026
This exercise made me realize how shocking it is that LLMs got so far with reasoning just by training on text that refers to more text without ever grounding on non-text stimuli!
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 14/04/2026
can develop "vision" from self-referential text, how we fail to visualize higher dimensional & quantum objects & yet manipulate them, and also various fascinating human phenomena (like not having an internal monologue), and some thought expts borrowed from consciousness.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 14/04/2026
I framed this as a discussion between three people who respond "yes/yes" vs. "no/no" vs. "yes/no" which leads to mind-bending questions/analogies, somehow simultaneously philosophical and concrete e.g., the self-referential nature of the dictionary, how eigenvector representations of graphs
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 14/04/2026
Consolidated my armchair thoughts about "how may an LLM (not) differ from a human who thinks in images/text?". I split this as 2 qns: - is text sufficient to be correct about say, a circle? - does correctness imply sharing the same "understanding" as humans? vaishnavh.github.io/blog/what-ll...
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/03/2026
Scientists, just like lawyers, are also bound by codes of conduct that demand integrity in the process. So there's tension underlying their allegiance to their idea vs. making sure they don't cross a line in doing so.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/03/2026
I've a few more arguments in the essay, and also more nuance. e.g., scientists still try to be as neutral as they can, but seemingly at a lower day-to-day level or at the initial stages of an idea.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/03/2026
For a position/idea to stand a *fair chance* against other positions/ideas in this courtroom, it *needs* a dedicated lawyer whose job is to think deeply and creatively about that position & present the best/strongest form of it for time to pass a judgement.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/03/2026
In short, my view is that because science is exploration under uncertainty, at some level, scientists end up gambling and picking a side. These sides fight it out in a courtroom, with time as the judge.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/03/2026
In practice, scientists do not seem to behave like "neutral, rational agents" but rather behave like "zealous advocates" for an idea that has "hired them". I wrote about how I think this "courtroom" view of science works and what I learned from it! vaishnavh.github.io/blog/emotion...
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 18/03/2026
I remember that bad reviews meant you were banned from *reviewing* for a future conference, which sounds like a bad incentive system.... Why are you threatening someone with a good time?
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 18/03/2026
Also, what's the catch with punishing bad reviews by preventing future submissions? Say: if your reviews are egregiously bad as flagged by multiple ACs across at least two conferences, you won't be able to submit papers to the next N conferences. (Possible that I'm missing something here.)
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 18/03/2026
This would not only be more just, it would also disincentivize spamming, paper-count-maxing, chopping up one project into 5 papers etc.,
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 18/03/2026
Curious why conferences don't have a system where the authors of every paper together guarantee N reviews per paper (and they can distribute the load amongst themselves). This way wouldn't we tax authors in proportion to the number of papers they burden the system with?
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 03/03/2026
A recent paper (arxiv.org/abs/2602.18671) made me question something basic: do the logits of a language model model the next-token or the full sequence distribution? It really messed with my brain (in a fun way!). I wrote about the paper to clarify my thinking. vaishnavh.github.io/blog/joint-o...
vaishnavh.github.io
What does a language model model? - Vaishnavh Nagarajan
TL;DR: Does the next-token logit track the conditional or the joint probability of the whole sequence?I had an invisi...
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/02/2026
(the exact observation is even stronger than what I wrote here e.g., the low-rank structure "generalizes" across prompt-response pairs.)
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/02/2026
but it turns out that if you arrange next-token logits from pairs of prompt x response sequences into a matrix (see pic for the exact object), you still get a *linear* *low-rank* structure. neither this linearity, nor the low-rankness follows by design. it somehow emerges from training.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/02/2026
here's my understanding: the low-rank observation is a non-trivial extension of a more straightforward & well-known observation called the softmax bottleneck. If you stack a bunch of next-token logits from various prompts, you'll get a low-rank matrix. this is by *design* (the last layer bottleneck)
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/02/2026
If the low-rank logits really holds across settings, I expect it should have a lot of downstream corollaries & connections waiting to be discovered
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/02/2026
I also like the low-rank logits finding (arxiv.org/abs/2510.24966) because it provides a novel, simple and surprising abstraction to think about what function a trained LLM implements. It took me a *lot* of time to understand, appreciate and buy the exact result here...
arxiv.org
Sequences of Logits Reveal the Low Rank Structure of Language Models
A major problem in the study of large language models is to understand their inherent low-dimensional structure. We introduce an approach to study the low-dimensional structure of language models at a...
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/02/2026
Incredibly, you can select these datapoints through a straightforward method: see whether the given preference is aligned with a model prompted with the target behavior. (i'd have expected that you'd need an exponential search over all possible data subsets to accomplish this)
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/02/2026
This paper discovers another spooky generalization effect: to trigger any target behavior in an LLM, you can carefully subselect from a *completely unrelated* preference dataset such that preference finetuning on that subselected dataset produces that behavior.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 19/02/2026
Really liked this paper which ties up two observations that are equally mindboggling (low-rank logits & subliminal/weird generalization effects) and presents one other such observation arxiv.org/abs/2602.04863
arxiv.org
Subliminal Effects in Your Data: A General Mechanism via Log-Linearity
Training modern large language models (LLMs) has become a veritable smorgasbord of algorithms and datasets designed to elicit particular behaviors, making it critical to develop techniques to understa...
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Declan Campbell @thisisadax.bsky.social · 05/02/2026
The visual world is composed of objects, and those objects are composed of features. But do VLMs exploit this compositional structure when processing multi-object scenes? In our 🆒🆕 #ICLR2026 paper, we find they do – via emergent symbolic mechanisms for visual binding. 🧵👇
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 13/01/2026
He also contrasts the personalities of Hardy and Einstein:
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 13/01/2026
Currently reading "a mathematician's apology" by GH Hardy. This is excerpt the foreword by CP Snow describing Hardy's personality and his work:
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 08/01/2026
in associative memory, the latent space doesn't really encode any interesting distance. imagine you're trying to store which countries share borders. you could simply write down a list of adjacent countries OR you could visualize the world map in your head. this is "associative" vs "geometric".
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 12/01/2026
fascinating!
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 12/01/2026
Would love pointers to related lit! Will DM you about the other question. Thank you for your kind words!
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 09/01/2026
Rare to see such long term efforts these days 🫡
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Andrew Gordon Wilson @andrewgwils.bsky.social · 07/01/2026
We introduce epiplexity, a new measure of information that provides a foundation for how to select, generate, or transform data for learning systems. We have been working on this for almost 2 years, and I cannot contain my excitement! arxiv.org/abs/2601.03220 1/7
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Jeff Dean @jeffdean.bsky.social · 07/01/2026
Please welcome Google's Open Source efforts to Blue Sky at @opensource.google!
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 08/01/2026
for deeper models, they initialize the network in a way that the decomposition of each layer aligns with the previous layer. if you didn't assume this, there'll be "interference" across components which I *suspect* would contribute to associative memorization.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 08/01/2026
thanks for being curious about it :-) I'm basing this off of the assumptions made in this seminal paper arxiv.org/abs/1312.6120 they begin with an analysis of 2-layer (weight-untied) models where the dynamics neatly evolve along each spectral component.
arxiv.org
Exact solutions to the nonlinear dynamics of learning in deep linear neural networks
Despite the widespread practical success of deep learning methods, our theoretical understanding of the dynamics of learning in deep neural networks remains quite sparse. We attempt to bridge the gap ...
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 08/01/2026
now if I ask you "how many countries away is Mongolia from India?", in the lookup table approach, you've to sit and piece together the connections by iterating over a frustratingly long list. in the map approach, you can "see" the answer quickly.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 08/01/2026
in associative memory, the latent space doesn't really encode any interesting distance. imagine you're trying to store which countries share borders. you could simply write down a list of adjacent countries OR you could visualize the world map in your head. this is "associative" vs "geometric".
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 08/01/2026
Thanks for engaging with the work! Could you elaborate? I'm not an expert on graph theory but I'd be interested in any ideas to better understand this.
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Vaishnavh Nagarajan @vaishnavh.bsky.social · 08/01/2026
19/ These findings build on many nascent, fragmented observations in literature not credited here due to low space. There are also caveats in extending all this to natural language (each caveat, an open question ;) ). Please see the full story here: arxiv.org/abs/2510.26745
arxiv.org
Deep sequence models tend to memorize geometrically; it is unclear why
Deep sequence models are said to store atomic facts predominantly in the form of associative memory: a brute-force lookup of co-occurring entities. We identify a dramatically different form of storage...
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