"Cumulative Dissipation Gramian" Ws = Observability Gramian (from Control Theory). For example the spectral cutoff is exactly the Hankel singular value truncation from model reduction.

"Signal Channel" / "Reservoir" is Controllable/Observable vs. Uncontrollable/Unobservable Subspaces. Using Adamjan-Arov-Krein (AAK) theory gives the optimal nonlinear reduced model answering the optimal compression question.

"Drift–Diffusion Separation" is Freidlin-Wentzell Large Deviation Theory. They can predict "grokking" time from the FW action.

"Population-Risk Gate" is Quantum Weak Value / Postselection (Aharonov)

So for the follow-up problems

Control theory gives the truncation error bounds for model compression. Large deviation theory gives the grokking time predictions. Quantum measurement theory gives the imaginary preconditioners. Information geometry gives the optimal continuous relaxation of the gate.

Some nice implications of new ways of doing stuff which are nice to see formalized here:

Old: Pick architecture, hope it generalizes New: Design architecture to maximize observability Gramian rank (Honestly we pull a lot from control theory here)

Old: Use validation set to detect overfitting New: Monitor λ(Ws) spectrum during training; no validation needed

Old: Prune post-hoc based on magnitude New: Prune during training based on ker(Ws) membership

Old: Fixed learning rate New: Spectral learning rate

Note that I said "predict" not "describe". It feels like we're still in the era of Kepler, not Newton.

[1] https://arxiv.org/pdf/2604.21691

[2] https://news.ycombinator.com/item?id=47893779

1. Older ML models encoded in their architecture and lack of expressivity a bias to simplicity; which aided interpolation.

2. Overparameterized models instead use regularization to nudge parameters to simpler and more robust representations, while still memorizing the noise. In this manner, we still achieve generalization performance OOD. Moreover, the softer nudging and fundamental architectural expressivity allows for "data-specific" generalizations and representations that may be impossible to represent in small models. 3. At the critical point between the two regimes, the model is expressive enough to memorize; but not expressive enough to simultaneously both do that and encode general patterns.

I wonder how this understanding translates to these researchers' models of deep learning.

But at what computational cost?

Does anyone understand the formula they expressed above this sentence? is this just the classic "skip updating parameters with high gradient/loss variance in multiple batches/samples" ?

As a fellow tufte css enjoyer, Why is user select turned off on the sidenotes? I would like to be able to copy paste them quite badly.

We're given a signal channel and a reservoir. Signal lives in the channel, noise lives in the reservoir, and the reservoir supposedly doesn’t show up at test time.

Okay, but then we have: why would SGD put the right things in the right bucket?

If the answer is “because the reservoir is defined as the stuff that doesn’t transfer to test,” then this is close to circular.

The Borges/Lavoisier stuff is a tell. "We have unified the field” rhetoric should come after nontrivial predictions and results. Claiming to solve benign overfitting, double descent, grokking, implicit bias, risk of training on population, how to avoid a validation set, and last but not least, skipping training by analytically jumping to the end is 6 theory papers, 3 NeurIPS winners, and a $10B startup. Let's get some results before we tell everyone we unified the field. :) I hope you're right.

https://arxiv.org/pdf/2605.01172 is the current version. The money graphs are page 8 and on where they show (some weirdly thick) line charts with loss results reached in roughly 1/5 the number of steps that Adam takes, just what the blog post mentions.

They also claim holding back test data is not needed, also with more graphs.

I'm not an ML scientist, and I did not attempt to seriously parse the math. It reads to me as something precisely in that liminal space some math papers do where there's enough new terminology that actually parsing through it all is going to take real, concerted effort, possibly with mild brain damage as a risk.

Their 3d graphs of "kernel eigenstructure" also do double duty for me as totally impenetrable and possibly part of an April fool's ML paper that's hilarious to insiders. Or maybe they show something really amazing; they definitely seem to converge into a shape...What does that shape mean??? Why??? What is an eigenstructure? Is it just 3D eigenvectors of some matrices? Is it natural to have a 3D shape representing these large matrices? If not, how and why were these projected down? And why are they different colors in the paper?? You get the feel for my level of understanding.

I think it would frankly just be easier to validate this claim than parse the whole paper. If only I could understand

  > Each one-step kernel increment ηKMtSS integrates into WMS , so a sequence of one-step rate-maximizers is the greedy policy whose integral is the signal-channel content of the trajectory through G, exactly as plain SGD is the greedy step whose integral is empirical-risk descent through D. The diagonal cutoff µ2 k >σ2 k/(b−1) is the optimal first-order preconditioner for population risk on any diagonal base, and a streaming variance EMAˆst of squared gradient deviations realizes it as a one-line change to AdamW: one extra parameter-sized state vector and a per parameter gate that multiplies the standard moment update

Also of note to me, they talk about grokking which I found SUUUPER fascinating when it was first reported, and have never heard about since. So I was really glad to read about it and read that there has been a little academic work on the phenomenon.

Finally, of the three models they repot results on, two are extremely tiny, the last is a DPO round on Qwen 0.5B -- if the code for that is published, I imagine it would be easy to adapt and evaluate in other regimes.