Short introductory post for my research direction: Gaussian Natural Latents. I explain the motivation and give a preview of the forthcoming results.

The Natural Abstractions agenda, in my view, is a promising research program that asks important theoretical questions about the nature of agency and optimization.

Here's an excerpt from Nate Soares' excellent post:

Imaginary John: I suspect there's a common format to concepts, that is a fairly objective fact about the math of the territory, and that—if mastered—could be used to understand an AGI's concepts. And perhaps select the ones we wish it would optimize for. Which isn't the whole problem, but sure is a big chunk of the problem. (And other chunks might well be easier to address given mastery of the fairly-objective concepts of "agent" and "optimizer" and so on.)

Nate: This does seem to me like it's trying to attack the actual problem! I have my doubts about this particular line of research (and those doubts are on my list of things to write up), but hooray for a proposal that, if it succeeded by its own lights, would address this hard problem!

I think that Natural Abstractions ideally wants to grow into a mature, interdisciplinary research program bridging information theory, statistical learning theory, math, and physics. A rigorous theory explaining what concepts are and how they form would have key theorems, on top of which you could build more theorems, based on which you could make predictions, based on which you could run experiments, etc.

That is to say, it would very cool if this did happen, but it mostly hasn't happened yet.

The main problem as I see it is that theorems about "abstraction in general" are hard to even state, let alone prove. What we want to do is to build up the program as we would in a subfield of mathematics, but many of the core objects in the framework^[1] don't have a generic closed-form representation. It's hard to algebraically manipulate objects and prove theorems about them when you can't write them down.

So I argue we should try thinking like physicists.

Why Gaussians?

Gaussian distributions are the spherical cows of probability theory.

I.e. they are simple, well-behaved objects: any Gaussian distribution is fully described by its mean and covariance. Conditional independence becomes a linear algebra condition on the covariance matrix. Mediation error, redundancy error, mutual information, everything in the framework turns into a function of eigenvalues of matrices you can write down. The key technique is canonical correlation analysis: any pair of Gaussian vectors can be rotated into a stack of independent scalar pairs with correlations , and every question about what the two vectors share reduces to a question about that spectrum.

Also, it turns out that the natural latent conditions have been exactly characterized by information theory, 50 years ago. The Gaussian case of that literature was recently solved. So most of the work was recognizing that someone had already done the hard part and translating between framings.

The result: in the restricted case where all variables are jointly Gaussian, every object in the framework has well-behaved closed-form representations. We get theorems! This isn't immediately obvious in advance (at least, it wasn't to me).

I claim this gives us an Exact Theory of Gaussian Natural Latents.

And there are some unexpectedly clean results which have suggestive interpretations, in terms of how we might approach the general case.

Preview of Results

This is a preview, proofs will be in the forthcoming posts.

Exact natural latents don't exist

In any nondegenerate jointly Gaussian system, a latent satisfying the exact redundancy conditions is independent of the entire system.

Approximate natural latents have exact prices

For two Gaussian views with correlation , there is an exact (closed-form) tradeoff curve between mediation error and redundancy error, and it has a floor. As , even the "cheapest" natural latent costs bits of error.

Characterizing the behavior of optimal latents

For vector-valued views with many correlated modes, the best approximate natural latent is shown to pull every strong mode down to a common residual correlation—a "waterline"—abandoning weak modes entirely. The error budget controls the waterline, and the waterline controls the dimensionality of concept being represented.

Combined with the observation that correlations between separated observers decay with distance, this predicts that shared concepts won't blur, but instead shed components one at a time in discrete phase transitions.

The Gaussian case of an open conjecture

Wentworth and Lorell's $500 bounty asks whether stochastic natural latents can always be replaced by deterministic ones. I claim the aforementioned tradeoff curve settles this for Gaussian views^[2] with linear error transfer.

What this Means

Of course, the underlying structure of our universe is not even approximately jointly Gaussian. The jointly Gaussian setting is the easy toy case where we've set all the complications aside.

Despite the simplification, I believe this direction is potentially useful for expanding our conceptual understanding. For instance, the structural insight relating the error tradeoff curve to a question about Wyner common information applies in general. AFAICT, nobody has made this connection before.

In the next few posts, I hope to demonstrate that natural latents have a particular geometry in this jointly Gaussian setting. I think this structure is interesting for its own sake, but it's unclear how much it tells us about the general case. At the end of this sequence, I'll revisit this point to discuss the implications and open questions in more detail.

1. ^{^}

   E.g. mediation error and redundancy error.

2. ^{^}

   I plan to share code which verifies this claim computationally, along with the post. I have not proved the generic (non-Gaussian) version of the conjecture: I don't claim the $500 bounty at this time.