Joseph Van Name answers Has anyone actually tried to convince Terry Tao or other top mathematicians to work on alignment?

[Joseph Van Name]

Um. If you want to convince a mathematician like Terry Tao to be interested in AI alignment, you will need to present yourself as a reasonably competent mathematician or related expert and actually formulate an AI problem in such a way so that someone like Terry Tao would be interested in it. If you yourself are not interested in the problem, then Terry Tao will not be interested in it either.

Terry Tao is interested in random matrix theory (he wrote the book on it), and random matrix theory is somewhat related to my approach to AI interpretability and alignment. If you are going to send these problems to a mathematician, please inform me about this before you do so.

My approach to alignment: Given matrices $A_1,\dots ,A_r;B_1,\dots ,B_r$, define a superoperator $\Gamma (A_1,\dots ,A_r;B_1,\dots ,B_r)$ by setting

$\Gamma (A_1,\dots ,A_r;B_1,\dots ,B_r)\left(X\right)={\sum }_{k=1}^r{A}_{k}X{B}_k^{\ast }$, and define $\Phi (A_1,\dots ,A_r)=\Gamma (A_1,\dots ,A_r;A_1,\dots ,A_r)$. Define the $L_2$-spectral radius of $A_1,\dots ,A_r$ as ${\rho }_2(A_1,\dots ,A_r)=\rho \left(\Phi \right(A_1,\dots ,A_r){)}^{1/2}$. Here, $\rho \left(A\right)={lim}_{n\to \infty }\parallel A^n{\parallel }^{1/n}$ is the usual spectral radius.

Define ${\rho }_{2,d}^K(A_1,\dots ,A_r)=max\{\frac{\rho \left(\Gamma \right(A_1,\dots ,A_r;X_1,\dots ,X_r\left)\right)}{{\rho }_2(X_1,\dots ,X_r)}:X_1,\dots ,X_r\in M_d(K\left)\right\}$. Here, $K$ is either the field of reals, field of complex numbers, or division ring of quaternions.

Given matrices $A_1,\dots ,A_r;B_1,\dots ,B_r$, define

$\parallel (A_1,\dots ,A_r)\simeq (B_1,\dots ,B_r)\parallel =\frac{\Gamma (A_1,\dots ,A_r;B_1,\dots ,B_r)}{{\rho }_2(A_1,\dots ,A_r){\rho }_2(B_1,\dots ,B_r)}$. The value $\parallel (A_1,\dots ,A_r)\simeq (B_1,\dots ,B_r)\parallel$ is always a real number in the interval $[0,1]$ that is a measure of how jointly similar the tuples $(A_1,\dots ,A_r),(B_1,\dots ,B_r)$ are. The motivation behind ${\rho }_{2,d}^K(A_1,\dots ,A_r)$ is that $\frac{{\rho }_{2,d}^K(A_1,\dots ,A_r)}{{\rho }_2(A_1,\dots ,A_r)}$ is always a real number in $[0,1]$ (well except when the denominator is zero) that measures how well $A_1,\dots ,A_r$ can be approximated by $d\times d$-matrices. Informally, $\frac{{\rho }_{2,d}^K(A_1,\dots ,A_r)}{{\rho }_2(A_1,\dots ,A_r)}$ measures how random $A_1,\dots ,A_r$ are where a lower value of $\frac{{\rho }_{2,d}^K(A_1,\dots ,A_r)}{{\rho }_2(A_1,\dots ,A_r)}$ indicates a lower degree of randomness.

A better theoretical understanding of ${\rho }_{2,d}^K(A_1,\dots ,A_r)$ would be great. If $X_1,\dots ,X_r\in M_d\left(K\right)$ and $\frac{\rho \left(\Gamma \right(A_1,\dots ,A_r;X_1,\dots ,X_r\left)\right)}{{\varphi }_2(X_1,\dots ,X_r)}$ is locally maximized, then we say that $(X_1,\dots ,X_r)$ is an LSRDR of $(A_1,\dots ,A_r)$. Said differently, $(X_1,\dots ,X_r)\in M_d\left(K\right)$ is an LSRDR of $(A_1,\dots ,A_r)$ if the similarity $\parallel (A_1,\dots ,A_r)\simeq (X_1,\dots ,X_r)\parallel$ is maximized.

Here, the notion of an LSRDR is a machine learning notion that seems to be much more interpretable and much less subject to noise than many other machine learning notions. But a solid mathematical theory behind LSRDRs would help us understand not just what LSRDRs do, but the mathematical theory would help us understand why they do it.

Problems in random matrix theory concerning LSRDRs:

1. Suppose that $U_1,\dots ,U_r$ are random matrices (according to some distribution). Then what are some bounds for ${\rho }_{2,d}^K(U_1,\dots ,U_r)$.

2. Suppose that $U_1,\dots ,U_r$ are random matrices and $A_1,\dots ,A_r$ are non-random matrices. What can we say about the spectrum of $\Gamma (A_1,\dots ,A_r;U_1,\dots ,U_r)$? My computer experiments indicate that this spectrum satisfies the circular law, and the radius of the disc for this circular law is proportional to ${\rho }_2(A_1,\dots ,A_r)$, but a proof of this circular law would be nice.

3. Tensors can be naturally associated with collections of matrices. Suppose now that $U_1,\dots ,U_r$ are the matrices associated with a random tensor. Then what are some bounds for ${\rho }_{2,d}^K(U_1,\dots ,U_r)$.

P.S. By massively downvoting my posts where I talk about mathematics that is clearly applicable to AI interpretability and alignment, the people on this site are simply demonstrating that they need to do a lot of soul searching before they annoy people like Terry Tao with their lack of mathematical expertise.

P.P.S. Instead of trying to get a high profile mathematician like Terry Tao to be interested in problems, it may be better to search for a specific mathematician who is an expert in a specific area related to AI alignment since it may be easier to contact a lower profile mathematician, and a lower profile mathematician may have more specific things to say and contribute. You are lucky that Terry Tao is interested in random matrix theory, but this does not mean that Terry Tao is interested in anything in the intersection between alignment and random matrix theory. It may be better to search harder for mathematicians who are interested in your specific problems.

P.P.P.S. To get more mathematicians interested in alignment, it may be a good idea to develop AI systems that behave much more mathematically. Neural networks currently do not behave very mathematically since they look like the things that engineers would come up with instead of mathematicians.

P.P.P.P.S. I have developed the notion of an LSRDR for cryptocurrency research because I am using this to evaluate the cryptographic security of cryptographic functions.