Joseph Van Name’s Shortform

[Joseph Van Name]

Every entry in a matrix counts for the $L_2$-spectral radius similarity. Suppose that $A_1,\dots ,A_r,B_1,\dots ,B_r$ are real $n\times n$-matrices. Set $A^{\otimes 2}=A\otimes A$. Define the $L_2$-spectral radius similarity between $(A_1,\dots ,A_r)$ and $(B_1,\dots ,B_r)$ to be the number

$\frac{\rho (A_1\otimes B_1+\cdots +A_r\otimes B_r)}{\rho (A_1^{\otimes 2}+\cdots +A_r^{\otimes 2}{)}^{1/2}\rho (B_1^{\otimes 2}+\cdots +B_r^{\otimes 2}{)}^{1/2}}$. Then the $L_2$-spectral radius similarity is always a real number in the interval $[0,1]$, so one can think of the $L_2$-spectral radius similarity as a generalization of the value $\frac{|⟨u,v⟩|}{\parallel u\parallel \cdot \parallel v\parallel }$ where $u,v$ are real or complex vectors. It turns out experimentally that if $A_1,\dots ,A_r$ are random real matrices, and each $B_j$ is obtained from $A_j$ by replacing each entry in $B_j$ with $0$ with probability $1-\alpha$, then the $L_2$-spectral radius similarity between $(A_1,\dots ,A_r)$ and $(B_1,\dots ,B_r)$ will be about $\sqrt{\alpha }$. If $u=(A_1,\dots ,A_r),v=(B_1,\dots ,B_r)$, then observe that $\frac{|⟨u,v⟩|}{\parallel u\parallel \cdot \parallel v\parallel }\approx \sqrt{\alpha }$ as well.

Suppose now that $A_1,\dots ,A_r$ are random real $n\times n$ matrices and $C_1,\dots ,C_r$ are the $m\times m$ submatrices of $A_1,\dots ,A_r$ respectively obtained by only looking at the first $m$ rows and columns of $A_1,\dots ,A_r$. Then the $L_2$-spectral radius similarity between $A_1,\dots ,A_r$ and $C_1,\dots ,C_r$ will be about $\sqrt{m/n}$. We can therefore conclude that in some sense $C_1,\dots ,C_r$ is a simplified version of $A_1,\dots ,A_r$ that more efficiently captures the behavior of $A_1,\dots ,A_r$ than $B_1,\dots ,B_r$ does.

If $A_1,\dots ,A_r,B_1,\dots ,B_r$ are independent random matrices with standard Gaussian entries, then the $L_2$-spectral radius similarity between $(A_1,\dots ,A_r)$ and $(B_1,\dots ,B_r)$ will be about $1/\sqrt{r}$ with small variance. If $u,v$ are random Gaussian vectors of length $r$, then $\frac{|⟨u,v⟩|}{\parallel u\parallel \cdot \parallel v\parallel }$ will on average be about $c/\sqrt{r}$ for some constant $c$, but $\frac{|⟨u,v⟩|}{\parallel u\parallel \cdot \parallel v\parallel }$ will have a high variance.

These are some simple observations that I have made about the spectral radius during my research for evaluating cryptographic functions for cryptocurrency technologies.

[Algon]

Your notation is confusing me. If r is the size of the list of matrices, then how can you have a probability of 1-r for r>=2? Maybe you mean 1-1/​r and sqrt{1/​r} instead of 1-r and sqrt{r} respectively?

[Joseph Van Name]

Thanks for pointing that out. I have corrected the typo. I simply used the symbol $r$ for two different quantities, but now the probability is denoted by the symbol $\alpha$.

[Joseph Van Name]

In this post, I will post some observations that I have made about the octonions that demonstrate that the machine learning algorithms that I have been looking at recently behave mathematically and such machine learning algorithms seem to be highly interpretable. The good behavior of these machine learning algorithms is in part due to the mathematical nature of the octonions and also the compatibility with the octonions and the machine learning algorithm. To be specific, one should think of the octonions as encoding a mixed unitary quantum channel that looks very close to the completely depolarizing channel, but my machine learning algorithms work well with those sorts of quantum channels and similar objects.

Suppose that $K$ is either the field of real numbers, complex numbers, or quaternions.

If $A_1,\dots ,A_r\in M_m\left(K\right),B_1,\dots ,B_r\in M_n\left(K\right)$ are matrices, then define an superoperator $\Gamma (A_1,\dots ,A_r;B_1,\dots ,B_r):M_{m,n}\left(K\right)\to M_{m,n}\left(K\right)$

by setting$\Gamma (A_1,\dots ,A_r;B_1,\dots ,B_r)\left(X\right)=A_{1}X{B}_1^{\ast }+\cdots +A_{r}X{B}_r^{\ast }$

(the domain and range of )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 similarity $\parallel (A_1,\dots ,A_r)\simeq (B_1,\dots ,B_r){\parallel }_2$ by setting

$\parallel (A_1,\dots ,A_r)\simeq (B_1,\dots ,B_r){\parallel }_2$

$=\frac{\rho \left(\Gamma \right(A_1,\dots ,A_r;B_1,\dots ,B_r\left)\right)}{\rho \left(\Phi \right(A_1,\dots ,A_r\left){)}^{1/2}\rho \right(\Phi (B_1,\dots ,B_r){)}^{1/2}}$ where $\rho$ denotes the spectral radius.

Recall that the octonions are the unique (up-to-isomorphism) 8 dimensional real inner product space $V$ together with a bilinear binary operation $\ast$ such that$\parallel x\ast y\parallel =\parallel x\parallel \cdot \parallel y\parallel$ and $1\ast x=x\ast 1=x$ for all $x,y\in V$.

Suppose that $e_1,\dots ,e_8$ is an orthonormal basis for $V$. Define operators $(A_1,\dots ,A_8)$ by setting $A_{i}v=e_j\ast v$. Now, define operators $(B_1,\dots ,B_{64})$ up to reordering by setting $\{B_1,\dots ,B_{64}\}=\{A_i\otimes A_j:i,j\in \{1,\dots ,8\left\}\right\}$.

Let $d$ be a positive integer. Then the goal is to find complex symmetric $d\times d$-matrices $(X_1,\dots ,X_{64})$ where $\parallel (A_1,\dots ,A_{64})\simeq (X_1,\dots ,X_{64}){\parallel }_2$ is locally maximized. We achieve this goal through gradient ascent optimization. Since we are using gradient ascent, I consider this to be a machine learning algorithm, but the function mapping $A_j$ to $X_j$ is a linear transformation, so we are training linear models here (we can generalize this fitness function to one where we train non-linear models though, but that takes a lot of work if we want the generalized fitness functions to still behave mathematically).

Experimental Observation: If $1\le d\le 8$, then we can easily find complex symmetric matrices $(X_1,\dots ,X_{64})$ where $\parallel (A_1,\dots ,A_{64})\simeq (X_1,\dots ,X_{64}){\parallel }_2$ is locally maximized and where $\parallel (A_1,\dots ,A_{64})\simeq (X_1,\dots ,X_{64}){\parallel }_2^2=(2d+6)/64=(d+3)/32.$

If $7\le d\le 16$, then we can easily find complex symmetric matrices $(X_1,\dots ,X_{64})$ where $\parallel (A_1,\dots ,A_{64})\simeq (X_1,\dots ,X_{64}){\parallel }_2$ is locally maximized and where$\parallel (A_1,\dots ,A_{64})\simeq (X_1,\dots ,X_{64}){\parallel }_2^2=(2d+4)/64=(d+2)/32.$

.

[Joseph Van Name]

It is time for us to interpret some linear machine learning models that I have been working on. These models are linear, but I can generalize these algorithms to produce multilinear models which have stronger capabilities while still behaving mathematically. Since one can stack the layers to make non-linear models, these types of machine learning algorithms seem to have enough performance to be more relevant for AI safety.

Our goal is to transform a list of $n\times n$-matrices $(A_1,...,A_r)$ into a new and simplified list of $d\times d$-matrices $(X_1,\dots ,X_r)$. There are several ways in which we would like to simplify the matrices. For example, we would sometimes simply like for $d<n$, but in other cases, we would like the matrices $X_j$ to all be real symmetric, complex symmetric, real Hermitian, complex Hermitian, complex anti-symmetric, etc.

We measure similarity between tuples of matrices using spectral radii. Suppose that $(A_1,\dots ,A_r)$ are $n\times n$-matrices and $(X_1,\dots ,X_r)$ are $d\times d$-matrices. Then define an operator $\Gamma (A_1,\dots ,A_r:X_1,\dots ,X_r)$ mapping $n\times d$ matrices to $n\times d$

-matrices by setting $\Gamma (A_1,\dots ,A_r:X_1,\dots ,X_r)\left(X\right)=A_{1}X{X}_1^{\ast }+\dots A_{r}X{X}_r^{\ast }$. Then define $\Phi (X_1,\dots ,X_r)=\Gamma (X_1,\dots ,X_r;X_1,\dots ,X_r)$. Define the similarity between $(A_1,\dots ,A_r)$ and $(X_1,\dots ,X_r)$ by setting

$\parallel (A_1,\dots ,A_r)\simeq (X_1,\dots ,X_r){\parallel }_2$

$=\frac{\rho \left(\Gamma \right(A_1,\dots ,A_r;X_1,\dots ,X_r\left)\right)}{\rho \left(\Phi \right(A_1,\dots ,A_r\left){)}^{1/2}\rho \right(\Phi (X_1,\dots ,X_r){)}^{1/2}}$

where $\rho$ denotes the spectral radius. Here, $\parallel (A_1,\dots ,A_r)\simeq (X_1,\dots ,X_r){\parallel }_2$ should be thought of as a generalization of the cosine similarity to tuples of matrices. And $\parallel (A_1,\dots ,A_r)\simeq (X_1,\dots ,X_r){\parallel }_2$ is always a real number in $[0,1]$, so this is a sensible notion of similarity.

Suppose that $K$ is either the field of real or complex numbers. Let $M_n\left(K\right)$ denote the set of $n$ by $n$ matrices over $K$.

Let $n,d$ be positive integers. Let $T:M_d\left(K\right)\to M_d\left(K\right)$ denote a projection operator. Here, $T$ is a real-linear operator, but if $K$ is not complex, then $T$ is not necessarily complex linear. Here are a few examples of such linear operators $T$ that work:

$K=C:T_1\left(X\right)=(X+X^T)/2$ (Complex symmetric)

$K=C:T_2\left(X\right)=(X-X^T)/2$ (Complex anti-symmetric)

$K=C:T_3\left(X\right)=(X+X^{\ast })/2$ (Complex Hermitian)

$K=C:T_4\left(X\right)=\text{Re}\left(X\right)$ (real, the real part taken elementwise).

$K=R:T_5\left(X\right)=(X+X^T)/2$ (Real symmetric)

$K=R:T_6\left(X\right)=(X-X^T)/2$ (Real anti-symmetric)

$K=C:T_7\left(X\right)=\text{Re}\left(X\right)+\text{Re}(X{)}^T$ (real symmetric)

$K=C:T_8\left(X\right)=\text{Re}\left(X\right)-\text{Re}(X{)}^T$ (real anti-symmetric)

Caution: These are special projection operators on spaces of matrices. The following algorithms do not behave well for general projection operators; they mainly behave well for $T_1,\dots ,T_8$ along with operators that I have forgotten about.

We are now ready to describe our machine learning algorithm’s input and objective.

Input: $r$-matrices $A_1,\dots ,A_r\in M_n\left(K\right)$

Objective: Our goal is to obtain matrices $(X_1,\dots ,X_r)\in M_d\left(K\right)$ where $T\left(X_j\right)=X_j$ for all $j$ but which locally maximizes the similarity$\parallel (A_1,\dots ,A_r)\simeq (X_1,\dots ,X_r){\parallel }_2$.

In this case, we shall call $(X_1,\dots ,X_r)$ an $L_{2,d}$-spectral radius dimensionality reduction (LSRDR) along the subspace $\text{im}\left(T\right).$

LSRDRs along subspaces often perform tricks and are very well-behaved.

If $(X_1,\dots ,X_r),(Y_1,\dots ,Y_r)$ are LSRDRs along subspaces, then there are typically some $\lambda ,C$ where $Y_j=\lambda C{X}_j{C}^{-1}$ for all $j$. Furthermore, if $(X_1,\dots ,X_r)$ is an LSRDR along a subspace, then we can typically find some matrices $R,S$ where$X_j=T\left(R{A}_{j}S\right)$ for all $j$.

The model $(X_1,\dots ,X_r)$ simplifies since it is encoded into the matrices $R,S$, but this also means that the model $(X_1,\dots ,X_r)$ is a linear model. I have just made these observations about the LSRDRs along subspaces, but they seem to behave mathematically enough for me especially since the matrices $R,S$ tend to have mathematical properties that I can’t explain and am still exploring.

[Joseph Van Name]

I am going to share an algorithm that I came up with that tends to produce the same result when we run it multiple times with a different initialization. The iteration is not even guaranteed convergence since we are not using gradient ascent, but it typically converges as long as the algorithm is given a reasonable input. This suggests that the algorithm behaves mathematically and may be useful for things such as quantum error correction. After analyzing the algorithm, I shall use the algorithm to solve a computational problem.

We say that an algorithm is pseudodeterministic if it tends to return the same output even if the computation leading to that output is non-deterministic (due to a random initialization). I believe that we should focus a lot more on pseudodetermistic machine learning algorithms for AI safety and interpretability since pseudodeterministic algorithms are inherently interpretable.

Define $f\left(z\right)=3z^2-2z^3$ for all complex numbers $z$. Then $f\left(0\right)=0,f\left(1\right)=1,f^{\prime }\left(0\right)=f^{\prime }\left(1\right)=0$, and there are neighborhoods $U,V$ of $0,1$ respectively where if $x\in U$, then $f^N\left(x\right)\to 0$ quickly and if $y\in V$, then $f^N\left(y\right)\to 1$ quickly. Set $f^{\infty }={lim}_{N\to \infty }{f}^N$. The function $f^{\infty }$ serves as error correction for projection matrices since if $Q$ is nearly a projection matrix, then $f^{\infty }\left(Q\right)$ will be a projection matrix.

Suppose that $K$ is either the field of real numbers, complex numbers or quaternions. Let $Z\left(K\right)$ denote the center of $K$. In particular, $Z\left(R\right)=R,Z\left(C\right)=C,Z\left(H\right)=R$.

If $A_1,\dots ,A_r$ are $m\times n$-matrices, then define $\Phi (A_1,\dots ,A_r):M_n\left(K\right)\to M_m\left(K\right)$ by setting $\Phi (A_1,\dots ,A_r)={\sum }_{k=1}^r{A}_{k}X{A}_k^{\ast }$. Then we say that an operator of the form $\Phi (A_1,\dots ,A_r)$ is completely positive. We say that a $Z\left(K\right)$-linear operator $E:M_n\left(K\right)\to M_m\left(K\right)$ is Hermitian preserving if $E\left(X\right)$ is Hermitian whenever $X$ is Hermitian. Every completely positive operator is Hermitian preserving.

Suppose that $E:M_n\left(K\right)\to M_n\left(K\right)$ is $Z\left(K\right)$-linear. Let $t>0$. Let $P_0\in M_n\left(K\right)$ be a random orthogonal projection matrix of rank $d$. Set $P_{N+1}=f^{\infty }(P_N+t\cdot E(P_N\left)\right)$ for all $N$. Then if everything goes well, the sequence $(P_N{)}_N$ will converge to a projection matrix $P$ of rank $d$, and the projection matrix $P$ will typically be unique in the sense that if we run the experiment again, we will typically obtain the exact same projection matrix $P$. If $E$ is Hermitian preserving, then the projection matrix $P$ will typically be an orthogonal projection. This experiment performs well especially when $E$ is completely positive or at least Hermitian preserving or nearly so. The projection matrix $P$ will satisfy the equation $P\cdot E\left(P\right)=E\left(P\right)\cdot P=P\cdot E\left(P\right)\cdot P$.

In the case when $E$ is a quantum channel, we can easily explain what the projection $P$ does. The operator $P$ is a projection onto a subspace of complex Euclidean space that is particularly well preserved by the channel $E$. In particular, the image $\text{Im}\left(P\right)$ is spanned by the top $d$ eigenvectors of $E\left(P\right)$. This means that if we send the completely mixed state $P/d$ through the quantum channel $E$ and we measure the state $E\left(P/d\right)$ with respect to the projective measurement $(P,I-P)$, then there is an unusually high probability that this measurement will land on $P$ instead of $I-P$.

Let us now use the algorithm that obtains $P$ from $E$ to solve a problem in many cases.

If $x$ is a vector, then let $\text{Diag}\left(x\right)$ denote the diagonal matrix where $x$ is the vector of diagonal entries, and if $X$ is a square matrix, then let $\text{Diag}\left(X\right)$ denote the diagonal of $X$. If $x$ is a length $n$ vector, then $\text{Diag}\left(x\right)$ is an $n\times n$-matrix, and if $X$ is an $n\times n$-matrix, then $\text{Diag}\left(X\right)$ is a length $n$ vector.

Problem Input: An $n\times n$-square matrix $A$ with non-negative real entries and a natural number $d$ with $1\le d<n$.

Objective: Find a subset $B\subseteq \{1,\dots ,n\}$ with $|B|=d$ and where if $x=A\cdot {\chi }_B$, then the $d$ largest entries in $x$ are the values $x\left[b\right]$ for $b\in B$.

Algorithm: Let $E$ be the completely positive operator defined by setting $E\left(X\right)=\text{Diag}(A\cdot \text{Diag}(X\left)\right)$. Then we run the iteration using $E$ to produce an orthogonal projection $P$ with rank $d$. In this case, the projection $P$ will be a diagonal projection matrix with rank $d$ where $\text{diag}\left(P\right)={\chi }_B$ and where $B$ is our desired subset of $\{1,\dots ,n\}$.

While the operator $P$ is just a linear operator, the pseudodeterminism of the algorithm that produces the operator $P$ generalizes to other pseudodeterministic algorithms that return models that are more like deep neural networks.

[Joseph Van Name]

This post gives an example of some calculations that I did using my own machine learning algorithm. These calculations work out nicely which indicates that the machine learning algorithm I am using is interpretable (and it is much more interpretable than any neural network would be). These calculations show that one can begin with old mathematical structures and produce new mathematical structures, and it seems feasible to completely automate this process to continue to produce more mathematical structures. The machine learning models that I use are linear, but it seems like we can get highly non-trivial results simply by iterating the procedure of obtaining new structures from old using machine learning.

I made a similar post to this one about 7 months ago, but I decided to revisit this experiment with more general algorithms and I have obtained experimental results which I think look nice.

To illustrate how this works, we start off with the octonions. The octonions consists of an 8-dimensional inner product space $V$ together with a bilinear operation $\ast$ and a unit $1\in V$ where $1\ast v=v\ast 1=v$ for all $v\in V$ and where $\parallel u\ast v\parallel =\parallel u\parallel \cdot \parallel v\parallel$ for all $u,v\in V$. The octonions are uniquely determined up to isomorphism from these properties. The operation $\ast$ is non-associative, but the $\ast$ is closely related to the quaternions and complex numbers. If we take a single element in $v\in V\setminus \text{Span}\left(1\right)$, then $\{1,v\}$ generates a subalgebra of $(V,\ast )$ isomorphic to the field of complex numbers, and if $u,v\in V$ and $\{1,u,v\}$ are linearly independent, then $\{1,u,v,u\ast v\}$ spans a subalgebra of $V$ isomorphic to the division ring of quaternions. For this reason, one commonly thinks of the octonions as the best way to extend the division ring of quaternions to a larger algebraic structure in the same way that the quaternions extend the field of complex numbers. But since the octonions are non-associative, they cannot be used to construct matrices, so they are not as well-known as the quaternions (and the construction of the octonions is more complicated too)

Suppose now that $\{e_0,e_1,\dots ,e_7\}$ is an orthonormal basis for the division ring of octonions with $e_0=1$. Then define matrices $A_0,\dots ,A_7:V\to V$ by setting $A_{j}v=e_j\ast v$ for all $j$. Our goal is to transform $(A_0,\dots ,A_7)$ into other tuples of matrices that satisfy similar properties.

If $(A_1,\dots ,A_r),(B_1,\dots ,B_r)$ are matrices, then define the $L_2$

-spectral radius similarity between $(A_1,\dots ,A_r)$ and $(B_1,\dots ,B_r)$ as

$\parallel (A_1,\dots ,A_r)\simeq (B_1,\dots ,B_r){\parallel }_2=$

$\frac{\rho (A_1\otimes \overline{B_1}+\cdots +A_r\otimes \overline{B_r})}{\rho (A_1\otimes \overline{A_1}+\cdots +A_r\otimes \overline{A_r}{)}^{1/2}\cdot \rho (B_1\otimes \overline{B_1}+\cdots +B_r\otimes \overline{B_r}{)}^{1/2}}$

where $\rho$ denotes the spectral radius, $\otimes$ is the tensor product, and $\overline{X}$ is the complex conjugate of $X$ applied elementwise.

Let $d\in \{1,\dots ,8\}$, and let $F_d,G_d,H_d$ denote the maximum value of the fitness level $8\cdot \parallel (A_0,\dots ,A_7)\simeq (X_0,\dots ,X_7){\parallel }^2$ such that each $X_j$ is a complex $d\times d$ anti-symmetric matrix ($X=-X^T$), a complex $d\times d$ symmetric matrix ($X=X^T$), and a complex $d\times d$-Hermitian matrix ($X=X^{\ast }$) respectively.

The following calculations were obtained through gradient ascent, so I have no mathematical proof that the values obtained are actually correct.

$G_1=2$, $H_1=1$

$G_2=3$, $H_2=3$

$F_3=1+\sqrt{3}$ , $G_3=3.5$, $H_3=1+2\sqrt{2}$

$F_4=4$ , $G_4=4$, $H_4=1+3\sqrt{2}$

$F_5=(5+\sqrt{13})/2$, $G_5=4.5$, $H_5\approx 5.27155841$

$F_6=5$ , $G_6=5$, $H_6=3+2\sqrt{2}$

$F_7=6$ , $G_7=2+2\sqrt{3}\approx 5.4641$, $H_7=1+2\sqrt{7}$

$F_8=7$ , $G_8=6$, $H_8=7$

Observe that with at most one exception, all of these values $F_d,G_d,H_d$ are algebraic half integers. This indicates that the fitness function that we maximize to produce $F_d,G_d,H_d$ behaves mathematically and can be used to produce new tuples $(X_1,\dots ,X_r)$ from old ones $(A_1,\dots ,A_r)$. Furthermore, an AI can determine whether something notable is going on with the new tuple $(X_1,\dots ,X_r)$ in several ways. For example, if $\parallel (A_1,\dots ,A_r)\simeq (X_1,\dots ,X_r){\parallel }^2$ has low algebraic degree at the local maximum, then $(X_1,\dots ,X_r)$ is likely notable and likely behaves mathematically (and is probably quite interpretable too).

The good behavior of $F_d,G_d,H_d$ demonstrates that the octonions are compatible with the $L_2$-spectral radius similarity. The operators $(A_0,\dots ,A_7)$ are all orthogonal, and one can take the tuple $(A_0,\dots ,A_7)$ as a mixed unitary quantum channel that is very similar to the completely depolarizing channel. The completely depolarizing channel completely mixes every quantum state while the mixture of orthogonal mappings $(A_0,\dots ,A_7)$ completely mixes every real state. The $L_2$-spectral radius similarity works very well with the completely depolarizing channel, so one should expect for the $L_2$-spectral radius similarity to also behave well with the octonions.

[Joseph Van Name]

Since AI interpretability is a big issue for AI safety, let’s completely interpret the results of evolutionary computation.

Disclaimer: This interpretation of the results of AI does not generalize to interpreting deep neural networks. This is a result for interpreting a solution to a very specific problem that is far less complicated than deep learning, and by interpreting, I mean that we iterate a mathematical operation hundreds of times to get an object that is simpler than our original object, so don’t get your hopes up too much.

A basis matroid is a pair $(X,M)$ where $X$ is a finite set, and $M\subseteq P\left(X\right)$ where $M$ denotes the power set of $X$ that satisfies the following two properties:

1. If $A,B\in M$, then $|A|=|B|$.

2. if $A,B\in M,A\ne B,a\in A\setminus B$, then there is some $b\in B\setminus A$ with $(A\setminus \{a\left\}\right)\cup \left\{b\right\}\in M$ (the basis exchange property).

I ran a computer experiment where I obtained a matroid $(X,M)$ where $|X|=9$ $|M|=68$ and where each element in $M$ has size $4$ through evolutionary computation, but the population size was kept so low that this evolutionary computation mimicked hill climbing algorithms. Now we need to interpret the matroid $(X,M)$.

The notion of a matroid has many dualities. Our strategy is to apply one of these dualities to the matroid $(X,M)$ so that the dual object is much smaller than the original object $(X,M)$. One may formulate the notion of a matroid in terms of closure systems (flats),hyperplanes, closure operators, lattices, a rank function, independent sets, bases, and circuits. If these seems to complicated, many of these dualities are special cases of other dualities common with ordered sets. For example, the duality between closure systems, closure operators, and ordered sets applies to contexts unrelated to matroids such as in general and point-free topology. And the duality between the basis, circuit, and the hyperplanes may be characterized in terms of rowmotion.

If $(Z,\le )$ is a partially ordered set, then a subset $A\subseteq Z$ is said to be an antichain if whenever $a,b\in A,a\le b$, then $a=b$. In other words, an antichain is a subset $A$ of $Z$ where the restriction of $\le$ to $A$ is equality. We say that a aubset $L$ of $Z$ is downwards closed if whenever $x\le y$ and $y\in L$, then $x\in L$ as well. If $A\subseteq Z$, then let $L\left(A\right)$ denote the smallest downwards closed subset of $Z$ containing $A$. Suppose that $Z$ is a finite poset. If $A$ is an antichain in $Z$, then let $A^{\prime }$ denote the set of all minimal elements in $Z\setminus L\left(A\right)$. Then $A^{\prime }$ is an antichain as well, and the mapping $A\mapsto A^{\prime }$ is a bijection from the set of all antichains in $Z$ to itself. This means that if $A$ is an antichain, then we may define $A^{\left(n\right)}$ for all integers $n$ by setting $A^{\left(0\right)}=A,A^{(n+1)}=(A^{\left(n\right)}{)}^{\prime }$.

If $(X,M)$ is a basis matroid, then $M$ is an antichain in $P\left(X\right)$, so we may apply rowmotion, so we say that $(X,M^{\left(n\right)})$ is an $n$-matroid. In this case, the $1$

-matroids are the circuit matroids while the $-1$-matroids are the hyperplane matroids. Unfortunately, the $n$-matroids have not been characterized for $|n|>1$. We say that the rowmotion order of $(X,M)$ is the least positive integer $n$ where $M^{\left(n\right)}=M$. If $(X,M)$ is a matroid of order $n$, then my computer experiments indicate that $gcd(|X|+2,n)>1$ whichs lends support to the idea that the rowmotion of a matroid is a sensible mathematical notion that may be satisfied mathematically. The notion of rowmotion of a matroid also appears to be a sensible mathematical notion for other reasons; if we apply iteratively apply a bijective operation $g$ (such as a reversible cellular automaton) to a finite object $x$, then that bijective operation will often increase the entropy in the sense that if $x$ has low entropy, then $g^{\left(n\right)}\left(x\right)$ will typically have a high amount of entropy and look like noise. But this is not the case with matroids as $n$-matroids do not appear substantially more complicated than basis matroids. Until and if there is a mundane explanation for this behavior of the rowmotion of matroids, I must consider the notion of rowmotion of matroids to be a mathematically interesting notion even though it is currently not understood by anyone.

With the matroid $(X,M)$ obtained from evolutionary computation, I found that $(X,M)$ has order $1958$ which factorizes as $1958=2\cdot 79\cdot 11$. Set $X=\{1,\dots ,9\}$. By applying rowmotion to this matroid, I found that $M^{\left(342\right)}$={{1, 8, 9},{2, 3, 6, 8},{2, 3, 7, 9},{4, 5},{4, 6, 9},{4, 7, 8},{5, 6, 9},{5, 7, 8}}. If $(X,M^{\left(m\right)})$ is a basis matroid, then $M^{\left(m\right)}=M$, so the set $M^{\left(342\right)}$ is associated with a unique basis matroid. This is the smallest way to represent $(X,M)$ in terms of rowmotion since if $|M^{\left(n\right)}|\le 8$, then $M^{\left(n\right)}=M^{\left(342\right)}$.

I consider this a somewhat satisfactory interpretation of the matroid $(X,M)$ that I have obtained through evolutionary computation, but there is still work to do because nobody has researched the rowmotion operation on matroids and because it would be better to simplify a matroid without needing to go through hundreds of layers of rowmotion. And even if we were to understand matroid rowmotion better, this would not help us too much with AI safety since here this interpretability of the result of evolutionary computation does not generalize to other AI’s and it certainly does not apply to deep neural networks.

I made a video here where one may see the rowmotion of this matroid and that video is only slightly interpretable.

Deep matroid duality visualization: Rowmotion of a matroid

It turns out that evolutionary computation is not even necessary to construct matroids since Donald Knuth has produced an algorithm that can be used to construct an arbitrary matroid in his 1975 paper on random matroids. But I applied the rowmotion to the matroid in his paper and the resulting 10835-matroid $B^{\left(10835\right)}$={{1, 2, 4, 5},{1, 2, 6, 10},{1, 3, 4, 6},{1, 3, 4, 7, 9},{1, 3, 6, 7, 9},{1, 4, 6, 7},{1, 4, 6, 9},{1, 4, 8, 10},{2, 3, 4, 5, 6, 7, 8, 9, 10}}. It looks like the rowmotion operation is good for simplifying matroids in general. We can uniquely recover the basis matroid from the 10835 matroid since $B^{\left(m\right)}$ is not a basis matroid whenever $0<m\le 10835$.

[Joseph Van Name]

I have originally developed a machine learning notion which I call an LSRDR ($L_{2,d}$

-spectral radius dimensionality reduction), and LSRDRs (and similar machine learning models) behave mathematically and they have a high level of interpretability which should be good for AI safety. Here, I am giving an example of how LSRDRs behave mathematically and how one can get the most out of interpreting an LSRDR.

Suppose that $n$ is a natural number. Let $N$ denote the quantum channel that takes an $n$ qubit quantum state and selects one of those qubits at random and send that qubit through the completely depolarizing channel (the completely depolarizing channel takes a state as input and returns the completely mixed state as an output).

If $A_1,\dots ,A_r,B_1,\dots ,B_r$ are complex matrices, then define superoperators $\Phi (A_1,\dots ,A_r)$ and $\Gamma (A_1,\dots ,A_r;B_1,\dots ,B_r)$ by setting

$\Phi (A_1,\dots ,A_r)\left(X\right)={\sum }_{k=1}^r{A}_{k}X{A}_k^{\ast }$ and $\Gamma (A_1,\dots ,A_r;B_1,\dots ,B_r)={\sum }_{k=1}^r{A}_{k}X{B}_k^{\ast }$ for all $X$.

Given tuples of matrices $(A_1,\dots ,A_r),(B_1,\dots ,B_r)$, define the L_2-spectral radius similarity between these tuples of matrices by setting

$\Vert (A_1,\dots ,A_r)\simeq (B_1,\dots ,B_r){\parallel }_2$

$=\frac{\rho \left(\Gamma \right(A_1,\dots ,A_r;B_1,\dots ,B_r\left)\right)}{\rho \left(\Phi \right(A_1,\dots ,A_r\left){)}^{1/2}\rho \right(\Phi (B_1,\dots ,B_r){)}^{1/2}}$.

Suppose now that $A_1,\dots ,A_{4n}$ are matrices where $N=\Phi (A_1,\dots ,A_{4n})$. Let $1\le d\le n$. We say that a tuple of complex $d$ by $d$ matrices $(X_1,\dots ,X_{4n})$ is an LSRDR of $A_1,\dots ,A_{4n}$ if the quantity $\parallel (A_1,\dots ,A_{4n})\simeq (X_1,\dots ,X_{4n}){\parallel }_2$ is locally maximized.

Suppose now that $X_1,\dots ,X_{4n}$ are complex $2\times 2$-matrices and $(X_1,\dots ,X_{4n})$ is an LSRDR of $A_1,\dots ,A_{4n}$. Then my computer experiments indicate that there will be some constant $\lambda$ where $\lambda \Gamma (A_1,\dots ,A_{4n};X_1,\dots ,X_{4n})$ is similar to a positive semidefinite operator with eigenvalues $\{0,\dots ,n+1\}$ and where the eigenvalue $j$ has multiplicity $3\cdot C(n-1,k)+C(n-1,k-2)$ where $C(\cdot ,\cdot )$ denotes the binomial coefficient. I have not had a chance to try to mathematically prove this. Hooray. We have interpreted the LSRDR $(X_1,\dots ,X_{4n})$ of $(A_1,\dots ,A_{4n})$, and I have plenty of other examples of interpreted LSRDRs.

We also have a similar pattern for the spectrum of $\Phi (A_1,\dots ,A_{4n})$. My computer experiments indicate that there is some constant $\lambda$ where $\lambda \cdot \Phi (A_1,\dots ,A_{4n})$ has spectrum $\{0,\dots ,n\}$ where the eigenvalue $j$ has multiplicity $3^{n-j}\cdot C(n,j)$.

[Joseph Van Name]

The notion of the linear regression is an interesting machine learning algorithm in the sense that it can be studied mathematically, but the notion of a linear regression is a quite limited machine learning algorithm as most relations are non-linear. In particular, the linear regression does not give us any notion of any uncertainty in the output.

One way to extend the notion of the linear regression to encapsulate uncertainty in the outputs is to regress a function not to a linear transformation mapping vectors to vectors, but to regress the function to a transformation that maps vectors to mixed states. And the notion of a quantum channel is an appropriate transformation that maps vectors to mixed states. One can optimize this quantum channel using gradient ascent.

For this post, I will only go through some basic facts about quantum information theory. The reader is referred to the book The Theory of Quantum Information by John Watrous for all the missing details.

Let $V$ be a complex Euclidean space. Let $L\left(V\right)$ denote the vector space of linear operators from $V$ to $V$. Given complex Euclidean spaces $V,W$, we say that a linear operator $E$ from $L\left(V\right)$ to $L\left(W\right)$ is a trace preserving if $\text{Tr}\left(E\right(X\left)\right)=\text{Tr}\left(X\right)$

for all $X$, and we say that $E$ is completely positive if there are linear transformations $A_1,...,A_r$ where $E\left(X\right)=A_{1}X{A}_1^{\ast }+\cdots +A_{r}X{A}_r^{\ast }$ for all $X$; the value $r$ is known as the Choi rank of $E$. A completely positive trace preserving operator is known as a quantum channel.

The collection of quantum channels from $L\left(V\right)$ to $L\left(W\right)$ is compact and convex.

If $W$ is a complex Euclidean space, then let $D_p\left(W\right)$

denote the collection of pure states in $W$. $D_p\left(W\right)$

can be defined either as the set of unit vector in $W$ modulo linear dependence, or $D_p\left(W\right)$

can be also defined as the collection of positive semidefinite rank-$1$ operators on $W$ with trace $1$.

Given complex Euclidean spaces $U,V$ and a (multi) set of $r$ distinct ordered pairs of unit vectors $f=\left\{\right(u_1,v_1),\dots ,(u_n,v_n\left)\right\}\subseteq U\times V$, and given a quantum channel

$E:L\left(U\right)\to L\left(V\right)$, we define the fitness level $F(f,E)={\sum }_{k=1}^r\log \left(E\right(u_k{u}_k^{\ast })v_k,v_k⟩$ and the loss level $L(f,E)={\sum }_{k=1}^r-\log \left(E\right(u_k{u}_k^{\ast })v_k,v_k⟩$.

We may locally optimize $E$ to minimize its loss level using gradient descent, but there is a slight problem. The set of quantum channels spans the $L\left(L\right(U),L(V\left)\right)$ which has dimension $\text{Dim}(U{)}^2\cdot \text{Dim}(V{)}^2$. Due to the large dimension, any locally optimal $E$ will contain $\text{Dim}(U{)}^2\cdot \text{Dim}(V{)}^2$ many parameters, and this is a large quantity of parameters for what is supposed to be just a glorified version of a linear regression. Fortunately, instead of taking all quantum channels into consideration, we can limit the scope the quantum channels of limited Choi rank.

Empirical Observation: Suppose that $U,V$ are complex Euclidean spaces, $f\subseteq U\times V$ is finite and $r$ is a positive integer. Then computer experiments indicate that there is typically only one quantum channel $E:L\left(U\right)\to L\left(V\right)$ of Choi rank at most $r$ where $L(f,E)$ is locally minimized. More formally, if we run the experiment twice and produce two quantum channels $E_1,E_2$ where $L(f,E_j)$ is locally minimized for $j\in \{1,2\}$, then we would typically have $E_1=E_2$. We therefore say that when $L(f,E)$ is minimized, $E$ is the best Choi rank $r$ quantum channel approximation to $f$.

Suppose now that $f=\left\{\right(u_1,v_1),\dots ,(u_n,v_n\left)\right\}\subseteq D_p\left(U\right)\times D_p\left(V\right)$ is a multiset. Then we would ideally like to approximate the function $f$ better by alternating between the best Choi rank r quantum channel approximation and a non-linear mapping. An ideal choice of a non-linear but partial mapping is the function $\text{DE}$ that maps a positive semidefinite matrix $P$ to its (equivalence class of) unit dominant eigenvector.

Empirical observation: If $f=\left\{\right(u_1,v_1),\dots ,(u_n,v_n\left)\right\}\subseteq D_p\left(U\right)\times D_p\left(V\right)$ and $E$ is the best Choi rank $r$ quantum channel approximation to $f$, then let $u_j^{♯}=\text{DE}\left(E\right(u_j{u}_j^{\ast }\left)\right)$ for all $j$, and define $f^{♯}=\left\{\right(u_1^{♯},v_1),\dots ,(u_n^{♯},v_n\left)\right\}$. Let $U$ be a small open neighborhood of $f^{♯}$. Let $g\in U$. Then we typically have $g^{♯♯}=g^{♯}$. More generally, the best Choi rank $r$ quantum channel approximation to $g$ is typically the identity function.

From the above observation, we see that the vector $u_j^{♯}$ is an approximation of $v_j$ that cannot be improved upon. The mapping $\text{DE}\circ E:D_p\left(U\right)\to D_p\left(V\right)$ is therefore a trainable approximation to the mapping $f$ and since $D_p\left(U\right),D_p\left(V\right)$ are not even linear spaces (these are complex projective spaces with non-trivial homology groups), the mapping $\text{DE}\circ E$ is a non-linear model for the function to $f$.

I have been investigating machine learning models similar to $\text{DE}\circ E$ for cryptocurrency research and development as these sorts of machine learning models seem to be useful for evaluating the cryptographic security of some proof-of-work problems and other cryptographic functions like block ciphers and hash functions. I have seen other machine learning models that behave about as mathematically as $\text{DE}\circ E$.

I admit that machine learning models like $\text{DE}\circ E$ are currently far from being as powerful as deep neural networks, but since $\text{DE}\circ E$ behaves mathematically, the model $\text{DE}\circ E$ should be considered as a safer and interpretable AI model. The goal is to therefore develop models that are mathematical like $\text{DE}\circ E$ but which can perform more and more machine learning tasks.

(Edited 8/​14/​2024)

[Joseph Van Name]

Here is an example of what might happen. Suppose that for each $u_j$, we select a orthonormal basis $e_{j,1},\dots ,e_{j,s}$ of unit vectors for $V$. Let $R=\left\{\right(u_j,e_{j,k}):1\le j\le n,1\le k\le s\}$. Then

Then for each quantum channel $E$, by the concavity of the logarithm function (which is the arithmetic-geometric mean inequality), we have

$L(R,E)={\sum }_{j=1}^n{\sum }_{k=1}^n-\log \left(E\right(u_j{u}_j^{\ast })e_{j,k},e_{j,k}⟩)$

$\le {\sum }_{j=1}^n-\log ({\sum }_{k=1}^n⟨E(u_j{u}_j^{\ast })e_{j,k},e_{j,k}⟩)$

$={\sum }_{j=1}^n-\log \left(\text{Tr}\right(E\left)\right)$. Here, equality is reached if and only if

$E\left(u_j{u}_j^{\ast }\right)e_{j,k},e_{j,k}⟩=E\left(u_j{u}_j^{\ast }\right)e_{j,l},e_{j,l}⟩$ for each $j,k,l$, but this equality can be achieved by the channel

defined by $E\left(X\right)=\text{Tr}\left(X\right)\cdot I/s$ which is known as the completely depolarizing channel. This is the channel that always takes a quantum state and returns the completely mixed state. On the other hand, the channel $E$ has maximum Choi rank since the Choi representation of $E$ is just the identity function divided by the rank. This example is not unexpected since for each input of $R$ the possible outputs span the entire space $V$ evenly, so one does not have any information about the output from any particular input except that we know that the output could be anything. This example shows that the channels that locally minimize the loss function $L(R,E)$ are the channels that give us a sort of linear regression of $R$ but where this linear regression takes into consideration uncertainty in the output so the regression of a output of a state is a mixed state rather than a pure state.

[Joseph Van Name]

We can use the $L_2-$spectral radius similarity to measure more complicated similarities between data sets.

Suppose that $A_1,\dots ,A_r$ are $m\times m$-real matrices and $B_1,\dots ,B_r$ are $n\times n$-real matrices. Let $\rho \left(A\right)$ denote the spectral radius of $A$ and let $A\otimes B$ denote the tensor product of $A$ with $B$. Define the $L_2$-spectral radius by setting ${\rho }_2(A_1,\dots ,A_r)=\rho (A_1\otimes A_1+\cdots +A_r\otimes A_r{)}^{1/2}$, Define the $L_2$-spectral radius similarity between $A_1,\dots ,A_r$ and $B_1,\dots ,B_r$ as

$\parallel (A_1,\dots ,A_r)\simeq (B_1,\dots ,B_r){\parallel }_2=\frac{\rho (A_1\otimes B_1+\cdots +A_r\otimes B_r)}{{\rho }_2(A_1,\dots ,A_r){\rho }_2(B_1,\dots ,B_r)}$.

We observe that if $C$ is invertible and $\lambda$ is a constant, then

$\parallel (A_1,\dots ,A_r)\simeq (\lambda C{B}_1{C}^{-1},\dots ,\lambda C{B}_r{C}^{-1}){\parallel }_2=1.$

Therefore, the $L_2$-spectral radius is able to detect and measure symmetry that is normally hidden.

Example: Suppose that $u_1,\dots ,u_r;v_1,\dots ,v_r$ are vectors of possibly different dimensions. Suppose that we would like to determine how close we are to obtaining an affine transformation $T$ with $T\left(u_j\right)=v_j$ for all $j$ (or a slightly different notion of similarity). We first of all should normalize these vectors to obtain vectors $x_1,\dots ,x_r;y_1,\dots ,y_r$ with mean zero and where the covariance matrix is the identity matrix (we may not need to do this depending on our notion of similarity). Then $\parallel (x_1{x}_1^{\ast },\dots ,x_r{x}_r^{\ast })\simeq (y_1{y}_1^{\ast },\dots ,y_r{y}_r^{\ast }){\parallel }_2$ is a measure of low close we are to obtaining such an affine transformation $T$. We may be able to apply this notion to determining the distance between machine learning models. For example, suppose that $M,N$ are both the first few layers in a (typically different) neural network. Suppose that $a_1,\dots ,a_r$ is a set of data points. Then if $u_j=M\left(a_j\right)$ and $v_j=M\left(a_j\right)$, then $\parallel (x_1{x}_1^{\ast },\dots ,x_r{x}_r^{\ast })\simeq (y_1{y}_1^{\ast },\dots ,y_r{y}_r^{\ast }){\parallel }_2$ is a measure of the similarity between $M$ and $N$.

I have actually used this example to see if there is any similarity between two different neural networks trained on the same data set. For my experiment, I chose a random collection of $S\subseteq \{0,1{\}}^{32}\times \{0,1{\}}^{32}$ of ordered pairs and I trained the neural networks $M,N$ to minimize the expected losses $E(\parallel N(a)-b{\parallel }^2:(a,b)\in S),E(\parallel M(a)-b{\parallel }^2:(a,b)\in S)$. In my experiment, each $a_j$ was a random vector of length 32 whose entries were 0′s and 1′s. In my experiment, the similarity $\parallel (x_1{x}_1^{\ast },\dots ,x_r{x}_r^{\ast })\simeq (y_1{y}_1^{\ast },\dots ,y_r{y}_r^{\ast }){\parallel }_2$ was worse than if $x_1,\dots ,x_r,y_1,\dots ,y_r$ were just random vectors.

This simple experiment suggests that trained neural networks retain too much random or pseudorandom data and are way too messy in order for anyone to develop a good understanding or interpretation of these networks. In my personal opinion, neural networks should be avoided in favor of other AI systems, but we need to develop these alternative AI systems so that they eventually outperform neural networks. I have personally used the $L_2$-spectral radius similarity to develop such non-messy AI systems including LSRDRs, but these non-neural non-messy AI systems currently do not perform as well as neural networks for most tasks. For example, I currently cannot train LSRDR-like structures to do any more NLP than just a word embedding, but I can train LSRDRs to do tasks that I have not seen neural networks perform (such as a tensor dimensionality reduction).

[Joseph Van Name]

So in my research into machine learning algorithms that I can use to evaluate small block ciphers for cryptocurrency technologies, I have just stumbled upon a dimensionality reduction for tensors in tensor products of inner product spaces that according to my computer experiments exists, is unique, and which reduces a real tensor to another real tensor even when the underlying field is the field of complex numbers. I would not be too surprised if someone else came up with this tensor dimensionality reduction before since it has a rather simple description and it is in a sense a canonical tensor dimensionality reduction when we consider tensors as homogeneous non-commutative polynomials. But even if this tensor dimensionality reduction is not new, this dimensionality reduction algorithm belongs to a broader class of new algorithms that I have been studying recently such as LSRDRs.

Suppose that $K$ is either the field of real numbers or the field of complex numbers. Let $V_1,\dots ,V_n$ be finite dimensional inner product spaces over $K$ with dimensions $d_1,\dots ,d_n$ respectively. Suppose that $V_i$ has basis $e_{i,1},\dots ,e_{i,d_i}$. Given $v\in V_1\otimes \cdots \otimes V_n$, we would sometimes want to approximate the tensor $v$ with a tensor that has less parameters. Suppose that $(m_0,\dots ,m_n)$ is a sequence of natural numbers with $m_0=m_n=1$. Suppose that $X_{i,j}$ is a $m_{i-1}\times m_i$ matrix over the field $K$ for $1\le i\le n$ and $1\le j\le d_i$. From the system of matrices $(X_{i,j}{)}_{i,j}$, we obtain a tensor $T\left(\right(X_{i,j}{)}_{i,j})={\sum }_{i_1,\dots ,i_n}{e}_{i_1}\otimes \cdots \otimes e_{i_n}\cdot X_{1,i_1}\dots X_{n,i_n}$. If the system of matrices $(X_{i,j}{)}_{i,j}$ locally minimizes the distance $\parallel v-T\left(\right(X_{i,j}{)}_{i,j})\parallel$, then the tensor $T\left(\right(X_{i,j}{)}_{i,j})$ is a dimensionality reduction of $v$ which we shall denote by $u$.

Intuition: One can associate the tensor product $V_1\otimes \cdots \otimes V_n$ with the set of all degree $n$ homogeneous non-commutative polynomials that consist of linear combinations of the monomials of the form $x_{1,i_1}\cdots x_{n,i_n}$. Given, our matrices $X_{i,j}$, we can define a linear functional $\varphi :V_1\otimes \cdots \otimes V_n\to K$ by setting $\varphi \left(p\right)=p\left(\right(X_{i,j}{)}_{i,j})$. But by the Reisz representation theorem, the linear functional $\varphi$ is dual to some tensor in $V_1\otimes \cdots \otimes V_n$. More specifically, $\varphi$ is dual to $T\left(\right(X_{i,j}{)}_{i,j})$. The tensors of the form $T\left(\right(X_{i,j}{)}_{i,j})$ are therefore the

Advantages:

1. In my computer experiments, the reduced dimension tensor $u$ is often (but not always) unique in the sense that if we calculate the tensor $u$ twice, then we will get the same tensor. At least, the distribution of reduced dimension tensors $u$ will have low Renyi entropy. I personally consider the partial uniqueness of the reduced dimension tensor to be advantageous over total uniqueness since this partial uniqueness signals whether one should use this tensor dimensionality reduction in the first place. If the reduced tensor is far from being unique, then one should not use this tensor dimensionality reduction algorithm. If the reduced tensor is unique or at least has low Renyi entropy, then this dimensionality reduction works well for the tensor $v$.

2. This dimensionality reduction does not depend on the choice of orthonormal basis $e_{i,1},\dots ,e_{i,d_i}$. If we chose a different basis for each $V_i$, then the resulting tensor $u$ of reduced dimensionality will remain the same (the proof is given below).

3. If $K$ is the field of complex numbers, but all the entries in the tensor $v$ happen to be real numbers, then all the entries in the tensor $u$ will also be real numbers.

4. This dimensionality reduction algorithm is intuitive when tensors are considered as homogeneous non-commutative polynomials.

Disadvantages:

1. This dimensionality reduction depends on a canonical cyclic ordering the inner product spaces $V_1,\dots ,V_n$.

2. Other notions of dimensionality reduction for tensors such as the CP tensor dimensionality reduction and the Tucker decompositions are more well-established, and they are obviously attempted generalizations of the singular value decomposition to higher dimensions, so they may be more intuitive to some.

3. The tensors of reduced dimensionality $T\left(\right(X_{i,j}{)}_{i,j})$ have a more complicated description than the tensors in the CP tensor dimensionality reduction.

Proposition: The set of tensors of the form ${\sum }_{i_1,\dots ,i_n}{e}_{1,i_1}\otimes \cdots \otimes e_{n,i_n}{X}_{1,i_1}\dots X_{n,i_n}$ does not depend on the choice of bases $(e_{i,1},\dots ,e_{i,d_i}{)}_i$.

Proof: For each $i$, let $f_{i,1},\dots ,f_{i,d_i}$ be an alternative basis for $V_i$. Then suppose that $e_{i,j}={\sum }_k{u}_{i,j,k}{f}_{i,k}$ for each $i,j$. Then

${\sum }_{i_1,\dots ,i_n}{e}_{1,i_1}\otimes \cdots \otimes e_{n,i_n}{X}_{1,i_1}\dots X_{n,i_n}$

$={\sum }_{i_1,\dots ,i_n}{\sum }_{k_1}{u}_{1,i_1,k_1}{f}_{1,i_1}\otimes \cdots \otimes {\sum }_{k_n}{u}_{n,i_n,k_n}{f}_{n,i_n}{X}_{1,i_1}\dots X_{n,i_n}$

$={\sum }_{k_1,\dots ,k_n}{f}_{1,k_1}\otimes \cdots \otimes f_{n,k_n}{\sum }_{i_1,\dots ,i_n}{u}_{1,i_1,k_1}\dots u_{n,i_n,k_n}{X}_{1,i_1}\dots X_{n,i,n}$

$={\sum }_{k_1,\dots ,k_n}{f}_{1,k_1}\otimes \cdots \otimes f_{n,k_n}\left({\sum }_{i_1}{u}_{1,i_1,k_1}{X}_{1,i_1}\right)\dots \left({\sum }_{i_n}{u}_{n,i_n,k_n}{X}_{i_n}\right)$. Q.E.D.

A failed generalization: An astute reader may have observed that if we drop the requirement that $m_n=1$, then we get a linear functional defined by letting

$\varphi \left(p\right)=\text{Tr}\left(p\right(\left(X_{i,j}{)}_{i,j}\right))$. This is indeed a linear functional, and we can try to approximate $v$ using a the dual to $\varphi$, but this approach does not work as well.

[Joseph Van Name]

In this post, we shall describe 3 related fitness functions with discrete domains where the process of maximizing these functions is pseudodeterministic in the sense that if we locally maximize the fitness function multiple times, then we typically attain the same local maximum; this appears to be an important aspect of AI safety. These fitness functions are my own. While these functions are far from deep neural networks, I think they are still related to AI safety since they are closely related to other fitness functions that are locally maximized pseudodeterministically that more closely resemble deep neural networks.

Let $K$ denote a finite dimensional algebra over the field of real numbers together with an adjoint operation $\ast$ (the operation $\ast$ is a linear involution with $(xy{)}^{\ast }=y^{\ast }{x}^{\ast }$). For example, $K$ could be the field of real numbers, complex numbers, quaternions, or a matrix ring over the reals, complex, or quaternions. We can extend the adjoint $\ast$ to the matrix ring $M_r\left(K\right)$ by setting $(x_{i,j}{)}_{i,j}^{\ast }=(x_{j,i}^{\ast }{)}_{i,j}$.

Let $n$ be a natural number. If $A_1,\dots ,A_r\in M_n\left(K\right),X_1,\dots ,X_r\in M_d\left(K\right)$, then define

$\Gamma (A_1,\dots ,A_r;X_1,\dots ,X_r):M_{n,d}\left(K\right)\to M_{n,d}\left(K\right)$ by setting $\Gamma (A_1,\dots ,A_r;X_1,\dots ,X_r)\left(X\right)=A_{1}X{X}_1^{\ast }+\cdots +A_{r}X{X}_r^{\ast }$.

Suppose now that $1\le d<n$. Then let $S_d\subseteq M_{n,n}\left(K\right)$ be the set of all $0,1$-diagonal matrices with $d$ many $1$’s on the diagonal. We observe that each element in $S_d$ is an orthogonal projection. Define fitness functions $F_d,G_d,H_d:S_d\to R$ by setting

$F_d\left(P\right)=\rho \left(\Gamma \right(A_1,\dots ,A_r;P{A}_{1}P,\dots ,P{A}_{r}P\left)\right)$,

$G_d\left(P\right)=\rho \left(\Gamma \right(P{A}_{1}P,\dots ,P{A}_{r}P;P{A}_{1}P,\dots ,P{A}_{r}P\left)\right)$, and

$H_d\left(P\right)=\frac{F_d(P{)}^2}{G_d\left(P\right)}$. Here, $\rho$ denotes the spectral radius.

$F_d\left(P\right)$ is typically slightly larger than $G_d\left(P\right)$, so these three fitness functions are closely related.

If $P,Q\in S_d$, then we say that $Q$ is in the neighborhood of $P$ if $Q$ differs from $P$ by at most 2 entries. If $F$ is a fitness function with domain $S_d$, then we say that $(P,F(P\left)\right)$ is a local maximum of the function $F$ if $F\left(P\right)\ge F\left(Q\right)$ whenever $Q$ is in the neighborhood of $P$.

The path from initialization to a local maximum $(P_s,F(P_s\left)\right)$ for will be a sequence $(P_0,\dots ,P_s)$ where $P_j$ is always in the neighborhood of $P_{j-1}$ and where $F\left(P_j\right)\ge F\left(P_{j-1}\right)$ for all $j$ and the length of the path will be $s$ and where $P_0$ is generated uniformly randomly.

Empirical observation: Suppose that $F\in \{F_d,G_d,H_d\}$. If we compute a path from initialization to local maximum for $F$, then such a path will typically have length less than $n$. Furthermore, if we locally maximize $F$ multiple times, we will typically obtain the same local maximum each time. Moreover, if $P_F,P_G,P_H$ are the computed local maxima of $F_d,G_d,H_d$ respectively, then $P_F,P_G,P_H$ will either be identical or differ by relatively few diagonal entries.

I have not done the experiments yet, but one should be able to generalize the above empirical observation to matroids. Suppose that $M$ is a basis matroid with underlying set $\{1,\dots ,n\}$ and where $|A|=d$ for each $A\in M$. Then one should be able to make the same observation about the fitness functions $F_d{|}_M,G_d{|}_M,H_d{|}_M$ as well.

We observe that the problems of maximizing $F_d,G_d,H_d$ are all NP-complete problems since the clique problems can be reduced to special cases of maximizing $F_d,G_d,H_d$. This means that the problems of maximizing $F_d,G_d,H_d$ can be sophisticated problems, but this also means that we should not expect it to be easy to find the global maxima for $F_d,G_d,H_d$ in some cases.

[Joseph Van Name]

This is a post about some of the machine learning algorithms that I have been doing experiments with. These machine learning models behave quite mathematically which seems to be very helpful for AI interpretability and AI safety.

Sequences of matrices generally cannot be approximated by sequences of Hermitian matrices.

Suppose that $A_1,\dots ,A_r$ are $n\times n$-complex matrices and $X_1,\dots ,X_r$ are $d\times d$-complex matrices. Then define a mapping $\Gamma (A_1,\dots ,A_r;X_1,\dots ,X_r):M_{n,d}\left(C\right)\to M_{n,d}\left(C\right)$ by $\Gamma (A_1,\dots ,A_r;X_1,\dots ,X_r)\left(X\right)=A_{1}X{X}_1^{\ast }+\cdots +A_{r}X{X}_r^{\ast }$ for all $X$. Define

$\Phi (A_1,\dots ,A_r)=\Gamma (A_1,\dots ,A_r;A_1,\dots ,A_r)$. Define the $L_2$

-spectral radius by setting ${\rho }_2(A_1,\dots ,A_r)=\rho \left(\Phi \right(A_1,\dots ,A_r){)}^{1/2}$. Define the $L_2$-spectral radius similarity between $(A_1,\dots ,A_r)$ and $(X_1,\dots ,X_r)$ by

$\parallel (A_1,\dots ,A_r)\simeq (X_1,\dots ,X_r){\parallel }_2$

$=\frac{\rho \left(\Gamma \right(A_1,\dots ,A_r;X_1,\dots ,X_r\left)\right)}{{\rho }_2(A_1,\dots ,A_r){\rho }_2(X_1,\dots ,X_r)}$.

The $L_2$-spectral radius similarity is always in the interval $[0,1]$. if $n=d$, $A_1,\dots ,A_r$ generates the algebra of $n\times n$-complex matrices, and $X_1,\dots ,X_r$ also generates the algebra of $n\times n$-complex matrices, then $\parallel (A_1,\dots ,A_r)\simeq (X_1,\dots ,X_r){\parallel }_2=1$ if and only if there are $C,\lambda$ with $A_j=\lambda C{X}_j{C}^{-1}$ for all $j$.

Define ${\rho }_{2,d}^H(A_1,\dots ,A_r)$ to be the supremum of

${\rho }_2(A_1,\dots ,A_r)\cdot \parallel (A_1,\dots ,A_r)\simeq (X_1,\dots ,X_r){\parallel }_2$

where $X_1,\dots ,X_r$ are $d\times d$-Hermitian matrices.

One can get lower bounds for ${\rho }_{2,d}^H(A_1,\dots ,A_r)$ simply by locally maximizing $\parallel (A_1,\dots ,A_r)\simeq (X_1,\dots ,X_r){\parallel }_2$ using gradient ascent, but if one locally maximizes this quantity twice, one typically gets the same fitness level.

Empirical observation/​conjecture: If $(A_1,\dots ,A_r)$ are $n\times n$-complex matrices, then ${\rho }_{2,n}^H(A_1,\dots ,A_r)={\rho }_{2,d}^H(A_1,\dots ,A_r)$ whenever $d\ge n$.

The above observation means that sequences of $n\times n$-matrices $(A_1,\dots ,A_r)$ are fundamentally non-Hermitian. In this case, we cannot get better models of $(A_1,\dots ,A_r)$ using Hermitian matrices larger than the matrices $(A_1,\dots ,A_r)$ themselves; I kind of want the behavior to be more complex instead of doing the same thing whenever $d\ge n$

, but the purpose of modeling $(A_1,\dots ,A_r)$ as Hermitian matrices is generally to use smaller matrices and not larger matrices.

This means that the function ${\rho }_{2,d}^H$ behaves mathematically.

Now, the model $(X_1,\dots ,X_r)$ is a linear model of $(A_1,\dots ,A_r)$ since the mapping $A_j\mapsto X_j$ is the restriction of a linear mapping, so such a linear model should be good for a limited number of tasks, but the mathematical behavior of the model $(X_1,\dots ,X_r)$ generalizes to multi-layered machine learning models.

[Joseph Van Name]

Here are some observations about the kind of fitness functions that I have been running experiments on for AI interpretability. The phenomena that I state in this post are determined experimentally without a rigorous mathematical proof and they only occur some of the time.

Suppose that $F:X\to [-\infty ,\infty )$ is a continuous fitness function. In an ideal universe, we would like for the function $F$ to have just one local maximum. If $F$ has just one local maximum, we say that $F$ is maximized pseudodeterministically (or simply pseudodeterministic). At the very least, we would like for there to be just one real number of the form $F\left(x\right)$ for local maximum $(x,F(x\left)\right)$. In this case, all local maxima will typically be related by some sort of symmetry. Pseudodeterministic fitness function seem to be quite interpretable to me. If there are many local maximum values and the local maximum value that we attain after training depends on things such as the initialization, then the local maximum will contain random/​pseudorandom information independent of the training data, and the local maximum will be difficult to interpret. A fitness function with a single local maximum value behaves more mathematically than a fitness function with many local maximum values, and such mathematical behavior should help with interpretability; the only reason I have been able to interpret pseudodeterminisitic fitness functions before is that they behave mathematically and have a unique local maximum value.

Set $O=F^{-1}\left[\right(-\infty ,\infty \left)\right]=X\setminus F^{-1}\left[\right\{-\infty \left\}\right]$. If the set $O$ is disconnected (in a topological sense) and if $L$ behaves differently on each of the components of $L$, then we have literally shattered the possibility of having a unique local maximum, but in this post, we shall explore a case where each component of $O$ still has a unique local maximum value.

Let $m_0,\dots ,m_n$ be positive integers with $m_0=m_n=1$ and where $m_1\ge 1,\dots ,m_{n-1}\ge 1$. Let $r_0,\dots ,r_{n-1}$ be other natural numbers. The set $X$ is the collection of all tuples $A=(A_{i,j}{)}_{i,j}$ where each $A_{i,j}$ is a real $m_{i+1}\times m_i$-matrix and where the indices range from $i\in \{0,\dots ,n-1\},j\in \{1,\dots ,r_i\}$ and where $(A_{i,j}{)}_j$ is not identically zero for all $i$.

The training data is a set $\Sigma$ that consists of input/​label pairs $(u,v)$ where $v\in \{-1,1\}$ and where $u=(u_0,\dots ,u_{n-1})$ such that each $u_i$ is a subset of $\{1,\dots ,r_i\}$ for all $i$ (i.e.$\Sigma$ is a binary classifier where $u$ is the encoded network input and $v$ is the label).

Define $W(u,A)=\left({\sum }_{j\in u_{n-1}}{A}_{n-1,j}\right)\dots \left({\sum }_{j\in u_0}{A}_{0,j}\right)$. Now, we define our fitness level by setting

$F\left(A\right)={\sum }_{(u,v)\in \Sigma }\log \left(|W\right(u,A\left)|\right)/|\Sigma |-{\sum }_i\log (\parallel {\sum }_j{A}_{i,j}{A}_{i,j}^{\ast }{\parallel }_p)/2$

$=E\left(\log \right(|W(u,A)|\left)\right)-{\sum }_i\log (\parallel {\sum }_j{A}_{i,j}{A}_{i,j}^{\ast }{\parallel }_p)/2$ where the expected value is with respect to selecting an element $(u,v)\in \Sigma$ uniformly at random. Here, $\parallel \ast {\parallel }_p$ is a Schatten $p$-norm which is just the ${\ell }_p$-norm of the singular values of the matrix. Observe that the fitness function $F$ only depends on the list $(u:(u,v)\in \Sigma )$, so $F$ does not depend on the training data labels.

Observe that $O=X\setminus {\bigcup }_{u\in \Sigma }\{A\in X:W(u,A)=0\}$ which is a disconnected open set. Define a function $f:O\to \{-1,1{\}}^{\Sigma }$ by setting $f\left(A\right)=\left(W\right(u,A\left)/|W\right(u,A)|{)}_{(u,v)\in \Sigma }$. Observe that if $x,y$ belong to the same component of $O$, then $f\left(x\right)=f\left(y\right)$.

While the fitness function $F$ has many local maximum values, the function $F$ seems to typically have at most one local maximum value per component. More specifically, for each $({\alpha }_i{)}_{i\in \Sigma }$, the set $f^{-1}\left[\right\{\left({\alpha }_i{)}_{i\in \Sigma }\right\}]$ seems to typically be a connected open set where $F$ has just one local maximum value (maybe the other local maxima are hard to find, but if thye are hard to find, they are irrelevant).

Let $\Omega =f^{-1}\left[\right\{\left(v{)}_{(u,v)\in \Sigma }\right]$. Then $\Omega$ is a (possibly empty) open subset of $O$, and there tends to be a unique (up-to-symmetry) $A_0\in \Omega$ where $F\left(A_0\right)$ is locally maximized. This unique $A_0$ is the machine learning model that we obtain when training on the data set $\Sigma$. To obtain $A_0$, we first perform an optimization that works well enough to get inside the open set $\Omega$. For example, to get inside $\Omega$, we could try to maximize the fitness function ${\sum }_{(u,v)\in \Sigma }\arctan (v\cdot W(u,A\left)\right)$. We then maximize $F$ inside the open set $\Omega$ to obtain our local maximum.

After training, we obtain a function $f$ defined by $f\left(u\right)=W(u,A_0)$. Observe that the function $f$ is a multi-linear function. The function $f$ is highly regularized, so if we want better performance, we should tone down the amount of regularization, but this can be done without compromising pseudodeterminism. The function $f$ has been trained so that $f\left(u\right)/|f\left(u\right)|=v$ for each $(u,v)\in \Sigma$ but also so that $|f\left(u\right)|$ is large compared to what we might expect whenever $(u,v)\in \Sigma$. In other words, $f$ is helpful in determining whether $(u,v)$ belongs to $\Sigma$ or not since one can examine the magnitude and sign of $f\left(u\right)$.

In order to maximize AI safety, I want to produce inherently interpretable AI algorithms that perform well on difficult tasks. Right now, the function $f$ (and other functions that I have designed) can do some machine learning tasks, but they are not ready to replace neural networks, but I have a few ideas about how to improve my AI algorithms performance without compromising pseudodeterminism. I do not believe that pseudodeterministic machine learning will increase AI risks too much because when designing these pseudodeterministic algorithms, we are trading some (but hopefully not too much) performance for increased interpretability, but this tradeoff is good for safety by increasing interpretability without increasing performance.

[Joseph Van Name]

In this note, I will continue to demonstrate not only the ways in which LSRDRs ($L_{2,d}$-spectral radius dimensionality reduction) are mathematical but also how one can get the most out of LSRDRs. LSRDRs are one of the types of machine learning that I have been working on, and LSRDRs have characteristics that tell us that LSRDRs are often inherently interpretable which should be good for AI safety.

Suppose that $N$ is the quantum channel that maps a $n$ qubit state to a $n$ qubit state where we select one of the 6 qubits at random and send it through the completely depolarizing channel (the completely depolarizing channel takes a state as an input and returns the completely mixed state as an output). Suppose that $A_1,\dots ,A_{4n}$ are $2^n$ by $2^n$ matrices where $N$ has the Kraus representation $N\left(X\right)={\sum }_{k=1}^{4n}{A}_{k}X{A}_k^{\ast }$.

The objective is to locally maximize the fitness level $\rho \left({\sum }_{k=1}^{4n}{z}_k{A}_k\right)/\parallel (z_1,\dots ,z_{4n})\parallel$ where the norm in question is the Euclidean norm and where $\rho$ denotes the spectral radius. This is a 1 dimensional case of an LSRDR of the channel $N$.

Let $A={\sum }_{k=1}^{4n}{z}_k{A}_k$ when $(z_1,\dots ,z_{4n})$ is selected to locally maximize the fitness level. Then my empirical calculations show that there is some $\lambda$ where $\lambda {\sum }_{k=1}^{4n}{z}_k{A}_k$is positive semidefinite with eigenvalues $\{0,\dots ,n\}$ and where the eigenvalue $k$ has multiplicity $(\frac{n}{k})$ which is the binomial coefficient. But these are empirical calculations for select values $\lambda$; I have not been able to mathematically prove that this is always the case for all local maxima for the fitness level (I have not tried to come up with a proof).

Here, we have obtained a complete characterization of $A$ up-to-unitary equivalence due to the spectral theorem, so we are quite close to completely interpreting the local maximum for our fitness function.

I made a few YouTube videos showcasing the process of maximizing the fitness level here.

Spectra of 1 dimensional LSRDRs of 6 qubit noise channel during training

Spectra of 1 dimensional LSRDRs of 7 qubit noise channel during training

Spectra of 1 dimensional LSRDRs of 8 qubit noise channel during training

I will make another post soon about more LSRDRs of a higher dimension of the same channel $N$.

[Joseph Van Name]

I personally like my machine learning algorithms to behave mathematically especially when I give them mathematical data. For example, a fitness function with apparently one local maximum value is a mathematical fitness function. It is even more mathematical if one can prove mathematical theorems about such a fitness function or if one can completely describe the local maxima of such a fitness function. It seems like fitness functions that satisfy these mathematical properties are more interpretable than the fitness functions which do not, so people should investigate such functions for AI safety purposes.

My notion of an LSRDR is a notion that satisfies these mathematical properties. To demonstrate the mathematical behavior of LSRDRs, let’s see what happens when we take an LSRDR of the octonions.

Let $K$ denote either the field of real numbers or the field of complex numbers ($K$

could also be the division ring of quaternions, but for simplicity, let’s not go there). If $A_1,\dots ,A_r$ are $n\times n$-matrices over $K$, then an LSRDR ($L_{2,d}$-spectral radius dimensionality reduction) of $A_1,\dots ,A_r$ is a collection $X_1,\dots ,X_r$ of $d\times d$-matrices that locally maximizes the fitness level

$\frac{\rho (A_1\otimes \overline{X_1}+\cdots +A_r\otimes \overline{X_r})}{\rho (X_1\otimes \overline{X_1}+\cdots +X_r\otimes \overline{X_r}{)}^{1/2}}$. $\rho$ denotes the spectral radius function while $\otimes$ denotes the tensor product and $\overline{Z}$ denotes the matrix obtained from $Z$ by replacing each entry with its complex conjugate. We shall call the maximum fitness level the $L_{2,d}$-spectral radius of $A_1,\dots ,A_r$ over the field $K$, and we shall wrote ${\rho }_{2,d}^K(A_1,\dots ,A_r)$ for this spectral radius.

Define the linear superoperator $\Gamma (A_1,\dots ,A_r;X_1,\dots ,X_r)$ by setting

$\Gamma (A_1,\dots ,A_r;X_1,\dots ,X_r)\left(X\right)=A_{1}X{X}_1^{\ast }+\cdots +A_{r}X{X}_r^{\ast }$ and set $\Phi (X_1,\dots ,X_r)=\Gamma (X_1,\dots ,X_r;X_1,\dots ,X_r)$. Then the fitness level of $X_1,\dots ,X_r$ is $\frac{\rho \left(\Gamma \right(A_1,\dots ,A_r;X_1,\dots ,X_r\left)\right)}{\Phi (X_1,\dots ,X_r{)}^{1/2}}$.

Suppose that $V$ is an $8$-dimensional real inner product space. Then the octonionic multiplication operation is the unique up-to-isomorphism bilinear binary operation $\ast$ on $V$ together with a unit $1$ such that$\parallel x\ast y\parallel =\parallel x\parallel \cdot \parallel y\parallel$ and $1\ast x=x\ast 1=1$ for all x$,y\in V$. If we drop the condition that the octonions have a unit, then we do not quite have this uniqueness result.

We say that an octonion-like algbera is a $8$-dimensional real inner product space $V$ together with a unique up-to-isomorphism bilinear operation $\ast$ such that $\parallel x\ast y\parallel =\parallel x\parallel \cdot \parallel y\parallel$ for all $x,y$.

Let $V$ be a specific octonion-like algebra.

Suppose now that $e_1,\dots ,e_8$ is an orthonormal basis for $V$ (this does not need to be the standard basis). Then for each $j\in \{1,\dots ,8\}$, let $A_j$ be the linear operator from $V$ to $V$ defined by setting $A_{j}v=e_j\ast v$ for all vectors $v$. All non-zero linear combinations of $A_1,\dots ,A_8$ are conformal mappings (this means that they preserve angles). Now that we have turned the octonion-like algebra into matrices, we can take an LSRDR of the octonion-like algebras, but when taking the LSRDR of octonion-like algebras, we should not worry about the choice of orthonormal basis $e_1,\dots ,e_8$ since I could formulate everything in a coordinate-free manner.

Empirical Observation from computer calculations: Suppose that $1\le d\le 8$ and $K$ is the field of real numbers. Then the following are equivalent.

1. The $d\times d$ matrices $X_1,\dots ,X_8$ are a LSRDR of $A_1,\dots ,A_8$ over $K$ where $A_1\otimes X_1+\cdots +A_8\otimes X_8$ has a unique real dominant eigenvalue.

2. There exists matrices $R,S$ where $X_j=R{A}_{j}S$ for all $j$ and where $SR$ is an orthonormal projection matrix.

In this case, ${\rho }_{2,d}^K(A_1,\dots ,A_8)=\sqrt{d}$ and this fitness level is reached by the matrices $X_1,\dots ,X_8$ in the above equivalent statements. Observe that the superoperator $\Gamma (A_1,\dots ,A_8;P{A}_{1}P,\dots ,P{A}_{8}P)$ is similar to a direct sum of $\Gamma (A_1,\dots ,A_r;X_1,\dots ,X_r))$ and a zero matrix. But the projection matrix $P$ is a dominant eigenvector of $\Gamma (A_1,\dots ,A_8;P{A}_{1}P,\dots ,P{A}_{8}P)$ and of$\Phi (P{A}_{1}P,\dots ,P{A}_{8}P)$ as well.

I have no mathematical proof of the above fact though.

Now suppose that $K=C$. Then my computer calculations yield the following complex $L_{2,d}$-spectral radii: $\left({\rho }_{2,j}^K\right(A_1,\dots ,A_8){)}_{j=1}^8$

$=(2,4,2+\sqrt{8},\mathrm{5.4676355784...},\mathrm{6.1977259251...},4+\sqrt{8},\mathrm{7.2628726081...},8)$

Each time that I have trained a complex LSRDR of $A_1,\dots ,A_8$, I was able to find a fitness level that is not just a local optimum but also a global optimum.

In the case of the real LSRDRs, I have a complete description of the LSRDRs of $(A_1,\dots ,A_8)$. This demonstrates that the octonion-like algebras are elegant mathematical structures and that LSRDRs behave mathematically in a manner that is compatible with the structure of the octonion-like algebras.

I have made a few YouTube videos that animate the process of gradient ascent to maximize the fitness level.

Edit: I have made some corrections to this post on 9/​22/​2024.

Fitness levels of complex LSRDRs of the octonions (youtube.com)

[Joseph Van Name]

There are some cases where we have a complete description for the local optima for an optimization problem. This is a case of such an optimization problem.

Such optimization problems are useful for AI safety since a loss/​fitness function where we have a complete description of all local or global optima is a highly interpretable loss/​fitness function, and so one should consider using these loss/​fitness functions to construct AI algorithms.

Theorem: Suppose that $U$ is a real,complex, or quaternionic $n\times n$-matrix that minimizes the quantity $\parallel U{\parallel }_2+\parallel U^{-1}{\parallel }_2$. Then $U$ is unitary.

Proof: The real case is a special case of a complex case, and by representing each $n\times n$-quaternionic matrix as a complex $2n\times 2n$-matrix, we may assume that $U$ is a complex matrix.

By the Schur decomposition, we know that $U=VT{V}^{\ast }$ where $V$ is a unitary matrix and $T$ is upper triangular. But we know that $\parallel U{\parallel }_2=\parallel T{\parallel }_2$. Furthermore, $U^{-1}=V{T}^{-1}{V}^{\ast }$, so $\parallel U^{-1}{\parallel }_2=\parallel T^{-1}{\parallel }_2$. Let $D$ denote the diagonal matrix whose diagonal entries are the same as $T$. Then $\parallel T{\parallel }_2\ge \parallel D{\parallel }_2$ and $\parallel T^{-1}{\parallel }_2\ge \parallel D^{-1}{\parallel }_2$. Furthermore, $\parallel T{\parallel }_2=\parallel D{\parallel }_2$ iff T is diagonal and $\parallel T^{-1}{\parallel }_2=\parallel D^{-1}{\parallel }_2$ iff $D$ is diagonal. Therefore, since $\parallel U{\parallel }_2+\parallel U^{-1}{\parallel }_2=\parallel T{\parallel }_2+\parallel T^{-1}{\parallel }_2$ and $\parallel T{\parallel }_2+\parallel T^{-1}{\parallel }_2$ is minimized, we can conclude that $T=D$, so $T$ is a diagonal matrix. Suppose that $T$ has diagonal entries $(z_1,\dots ,z_n)$. By the arithmetic-geometric mean equality and the Cauchy-Schwarz inequality, we know that $\frac{1}{2}\cdot (\parallel (z_1,\dots ,z_n){\parallel }_2+\parallel (z_1^{-1},\dots ,z_n^{-1}\left){\parallel }_2\right)\ge \parallel (|z_1|,\dots ,|z_n|){\parallel }_2\cdot \parallel (|z_1^{-1}|,\dots ,|z_n^{-1})|{\parallel }_2$

$\ge ⟨(|z_1|,\dots ,|z_n|),(|z_1^{-1}|,\dots ,|z_n^{-1})|⟩=\sqrt{n}.$

Here, the equalities hold if and only if $|z_j|=1$ for all $j$, but this implies that $U$ is unitary. Q.E.D.

[Joseph Van Name]

The $L_2$-spectral radius similarity is not transitive. Suppose that $A_1,\dots ,A_r$ are $m\times m$-matrices and $B_1,\dots ,B_r$ are real $n\times n$-matrices. Then define ${\rho }_2(A_1,\dots ,A_r)=\rho (A_1\otimes A_1+\cdots +A_r\otimes A_r{)}^{1/2}$. Then the generalized Cauchy-Schwarz inequality is satisfied:

$\rho (A_1\otimes B_1+\cdots +A_r\otimes B_r)\le {\rho }_2(A_1,\dots ,A_r){\rho }_2(B_1,\dots ,B_r)$.

We therefore define the $L_{2,d}$-spectral radius similarity between $(A_1,\dots ,A_r)$ and $(B_1,\dots ,B_r)$ as $\parallel (A_1,\dots ,A_r)\simeq (B_1,\dots ,B_r)\parallel =\frac{\rho (A_1\otimes B_1+\cdots +A_r\otimes B_r)}{{\rho }_2(A_1,\dots ,A_r){\rho }_2(B_1,\dots ,B_r)}$. One should think of the $L_2$-spectral radius similarity as a generalization of the cosine similarity $\frac{⟨u,v⟩}{\parallel u\parallel \cdot \parallel v\parallel }$ between vectors $u,v$. I have been using the $L_2$-spectral radius similarity to develop AI systems that seem to be very interpretable. The $L_2$-spectral radius similarity is not transitive.

$\parallel (A_1,\dots ,A_r)\simeq (A_1\oplus B_1,\dots ,A_r\oplus B_r)\parallel =1$ and

$\parallel (B_1,\dots ,B_r)\simeq (A_1\oplus B_1,\dots ,A_r\oplus B_r)\parallel =1$, but $\parallel (A_1,\dots ,A_r)\simeq (B_1,\dots ,B_r)\parallel$ can take any value in the interval $[0,1]$.

We should therefore think of the $L_{2,d}$-spectral radius similarity as a sort of least upper bound of $[0,1]$-valued equivalence relations than a $[0,1]$-valued equivalence relation. We need to consider this as a least upper bound because matrices have multiple dimensions.

Notation: $\rho \left(A\right)={lim}_{n\to \infty }\parallel A^n{\parallel }^{1/n}$ is the spectral radius. The spectral radius $A$ is the largest magnitude of an eigenvalue of the matrix $A$. Here the norm does not matter because we are taking the limit.$A\oplus B$ is the direct sum of matrices while $A\otimes B$ denotes the Kronecker product of matrices.

[Joseph Van Name]

Let’s compute some inner products and gradients.

Set up: Let $K$ denote either the field of real or the field of complex numbers. Suppose that $d_1,\dots ,d_r$ are positive integers. Let $m_0,\dots ,m_n$ be a sequence of positive integers with $m_0=m_n=1$. Suppose that $X_{i,j}$ is an $m_{i-1}\times m_i$-matrix whenever $1\le j\le d_i$. Then from the matrices $X_{i,j}$, we can define a $d_1\times \cdots \times d_r$-tensor $T\left(\right(X_{i,j}{)}_{i,j})=(X_{1,i_1}\dots X_{n,i_n}{)}_{i_1,\dots ,i_n}$. I have been doing computer experiments where I use this tensor to approximate other tensors by minimizing the ${\ell }_2$-distance. I have not seen this tensor approximation algorithm elsewhere, but perhaps someone else has produced this tensor approximation construction before. In previous shortform posts on this site, I have given evidence that the tensor dimensionality reduction behaves well, and in this post, we will focus on ways to compute with the tensors $T\left(\right(X_{i,j}{)}_{i,j})$, namely the inner product of such tensors and the gradient of the inner product with respect to the matrices $(X_{i,j}{)}_{i,j}$.

Notation: If $A_1,\dots ,A_r,B_1,\dots ,B_r$ are matrices, then let $\Gamma (A_1,\dots ,A_r;B_1,\dots ,B_r)$ denote the superoperator defined by letting $\Gamma (A_1,\dots ,A_r;B_1,\dots ,B_r)\left(X\right)=A_{1}X{B}_1^{\ast }+\cdots +A_{r}X{B}_r^{\ast }$. Let $\Phi (A_1,\dots ,A_r)=\Gamma (A_1,\dots ,A_r;A_1,\dots ,A_r)$.

Inner product: Here is the computation of the inner product of our tensors.

$⟨T\left(\right(A_{i,j}{)}_{i,j}),T(\left(B_{i,j}{)}_{i,j}\right)⟩$

$=⟨(A_{1,i_1}\dots A_{n,i_n}{)}_{i_1,\dots ,i_n},(B_{1,i_1}\dots B_{n,i_n}{)}_{i_1,\dots ,i_n}⟩$

$={\sum }_{i_1,\dots ,i_n}{A}_{1,i_1}\dots A_{n,i_n}(B_{1,i_1}\dots B_{n,i_n}{)}^{\ast }$

$={\sum }_{i_1,\dots ,i_n}{A}_{1,i_1}\dots A_{n,i_n}{B}_{n,i_n}^{\ast }\dots B_{1,i_1}^{\ast }$

$=\Gamma (A_{1,1},\dots ,A_{1,d_1};B_{1,1},\dots ,B_{1,d_1})\dots \Gamma (A_{n,1},\dots ,A_{n,d_n};B_{n,1},\dots ,B_{n,d_n})$.

In particular, $\parallel T\left(\right(A_{i,j}{)}_{i,j}){\parallel }^2=\Phi (A_{1,1},\dots ,A_{1,d_1})\dots \Phi (A_{n,1},\dots ,A_{n,d_n})$.

Gradient: Observe that ${\nabla }_X\text{Tr}\left(AX\right)=A^T$. We will see shortly that the cyclicity of the trace is useful for calculating the gradient. And here is my manual calculation of the gradient of the inner product of our tensors.

${\nabla }_{X_{\alpha ,\beta }}⟨T\left(\right(X_{i,j}{)}_{i,j}),T(\left(A_{i,j}{)}_{i,j}\right)⟩$

$={\nabla }_{X_{\alpha ,\beta }}{\sum }_{i_1,\dots ,i_n}{X}_{1,i_1}\dots X_{n,i_n}{A}_{n,i_n}^{\ast }\dots A_{1,i_1}^{\ast }$

$={\nabla }_{X_{\alpha ,\beta }}\text{Tr}({\sum }_{i_1,\dots ,i_n}{X}_{1,i_1}\dots X_{n,i_n}{A}_{n,i_n}^{\ast }\dots A_{1,i_1}^{\ast })$

$={\nabla }_{X_{\alpha ,\beta }}\text{Tr}({\sum }_{i_1,\dots ,i_n}{X}_{\alpha ,i_{\alpha }}\dots X_{n,i_n}{A}_{n,i_n}^{\ast }\dots$

$A_{\alpha +1,i_{\alpha +1}}^{\ast }{A}_{\alpha ,i_{\alpha }}^{\ast }{A}_{\alpha -1,i_{\alpha -1}}^{\ast }\dots A_{1,i_1}^{\ast }{X}_{1,i_1}\dots X_{\alpha -1,i_{\alpha -1}})$

$={\nabla }_{X_{\alpha ,\beta }}\text{Tr}({\sum }_{i_{\alpha +1},\dots ,i_n,i_1,\dots ,i_{\alpha -1}}{X}_{\alpha ,\beta }\dots X_{n,i_n}{A}_{n,i_n}^{\ast }\dots$

$A_{\alpha +1,i_{\alpha +1}}^{\ast }{A}_{\alpha ,\beta }^{\ast }{A}_{\alpha -1,i_{\alpha -1}}^{\ast }\dots A_{1,i_1}^{\ast }{X}_{1,i_1}\dots X_{\alpha -1,i_{\alpha -1}})$

$=({\sum }_{i_{\alpha +1},\dots ,i_n,i_1,\dots ,i_{\alpha -1}}{X}_{\alpha +1,i_{\alpha +1}}\dots X_{n,i_n}{A}_{n,i_n}^{\ast }\dots$

$A_{\alpha +1,i_{\alpha +1}}^{\ast }{A}_{\alpha ,\beta }^{\ast }{A}_{\alpha -1,i_{\alpha -1}}^{\ast }\dots A_{1,i_1}^{\ast }{X}_{1,i_1}\dots X_{\alpha -1,i_{\alpha -1}}{)}^T$

$=({\sum }_{i_{\alpha +1},\dots ,i_n,i_1,\dots ,i_{\alpha -1}}{X}_{\alpha +1,i_{\alpha +1}}\dots X_{n,i_n}$

$A_{n,i_n}^{\ast }\dots A_{\alpha +1,i_{\alpha +1}}^{\ast }{A}_{\alpha ,\beta }^{\ast }{A}_{\alpha -1,i_{\alpha -1}}^{\ast }\dots A_{1,i_1}^{\ast }{X}_{1,i_1}\dots X_{\alpha -1,i_{\alpha -1}}{)}^T$

$=[(\Gamma (X_{\alpha +1,1},\dots ,X_{\alpha +1,d_{\alpha +1}};A_{\alpha +1,1},\dots ,A_{\alpha +1,d_{\alpha +1}})\cdots$

$\Gamma (X_{n,1},\dots ,X_{n,d_n};A_{n,1},\dots ,A_{n,d_n})\left(1\right))$

$A_{\alpha ,\beta }^{\ast }$

$(\left(\Gamma \right(A_{\alpha -1,1}^{\ast },\dots ,A_{\alpha -1,d_{\alpha -1}}^{\ast };X_{\alpha -1,1}^{\ast },\dots ,X_{\alpha -1,d_{\alpha -1}}^{\ast })\dots$

$\Gamma (A_{1,1}^{\ast },\dots ,A_{1,d_1}^{\ast };X_{1,1}^{\ast },\dots ,X_{1,d_1}^{\ast })\left(1\right)){]}^T$.

[Joseph Van Name]

So in my research into machine learning algorithms, I have stumbled upon a dimensionality reduction algorithm for tensors, and my computer experiments have so far yielded interesting results. I am not sure that this dimensionality reduction is new, but I plan on generalizing this dimensionality reduction to more complicated constructions that I am pretty sure are new and am confident would work well.

Suppose that $K$ is either the field of real numbers or the field of complex numbers. Suppose that $d_1,\dots ,d_n$ are positive integers and $(m_0,\dots ,m_n)$ is a sequence of positive integers with $m_0=m_n=1$. Suppose that $X_{i,j}$ is an $m_{i-1}\times m_i$-matrix whenever $1\le j\le d_i$. Then define a tensor $T\left(\right(X_{i,j}\left)\right)=(X_{1,i_1}\dots X_{n,i_n}{)}_{i_1,\dots ,i_n}\in K^{d_1}\otimes \cdots \otimes K^{d_n}$.

If $v\in K^{d_1}\otimes \cdots \otimes K^{d_n}$, and $(X_{i,j}{)}_{i,j}$ is a system of matrices that minimizes the value $\parallel v-T\left(\right(X_{i,j}\left)\right)\parallel$, then $T\left(\right(X_{i,j}{)}_{i,j})$ is a dimensionality reduction of $(X_{i,j}{)}_{i,j}$, and we shall denote let $u$ denote the tensor of reduced dimension $T\left(\right(X_{i,j}{)}_{i,j})$. We shall call $u$ a matrix table to tensor dimensionality reduction of type $(m_0,\dots ,m_n)$.

Observation 1: (Sparsity) If $v$ is sparse in the sense that most entries in the tensor $v$ are zero, then the tensor $u$ will tend to have plenty of zero entries, but as expected, $u$ will be less sparse than $v$.

Observation 2: (Repeated entries) If $v$ is sparse and $v=(x_{i_1,\dots ,i_n}{)}_{i_1,\dots ,i_n}$ and the set $\{x_{i_1,\dots ,i_n}:i_1,\dots ,i_n\}$ has small cardinality, then the tensor $u$ will contain plenty of repeated non-zero entries.

Observation 3: (Tensor decomposition) Let $v$ be a tensor. Then we can often find a matrix table to tensor dimensionality reduction $u$ of type $(m_0,\dots ,m_n)$ so that $v-u$ is its own matrix table to tensor dimensionality reduction.

Observation 4: (Rational reduction) Suppose that $v$ is sparse and the entries in $v$ are all integers. Then the value $\parallel u-v{\parallel }^2$ is often a positive integer in both the case when $u$ has only integer entries and in the case when $u$ has non-integer entries.

Observation 5: (Multiple lines) Let $m$ be a fixed positive even number. Suppose that $v$ is sparse and the entries in $v$ are all of the form $r\cdot e^{2\pi in/m}$ for some integer $n$ and $r\ge 0$. Then the entries in $u$ are often exclusively of the form $r\cdot e^{2\pi in/m}$ as well.

Observation 6: (Rational reductions) I have observed a sparse tensor $v$ all of whose entries are integers along with matrix table to tensor dimensionality reductions $u_1,u_2$ of $v$ where $\parallel v-u_1\parallel =3,\parallel v-u_1\parallel =2,\parallel u_2-u_1\parallel =5$.

This is not an exclusive list of all the observations that I have made about the matrix table to tensor dimensionality reduction.

From these observations, one should conclude that the matrix table to tensor dimensionality reduction is a well-behaved machine learning algorithm. I hope and expect this machine learning algorithm and many similar ones to be used to both interpret the AI models that we have and will have and also to construct more interpretable and safer AI models in the future.

[Joseph Van Name]

Suppose that $q,r,d$ are natural numbers. Let $1<p<\infty$. Let $z_{i,j}$ be a complex number whenever $1\le i\le q,1\le j\le r$. Let $L:M_d(C{)}^r\setminus \{0\}\to [-\infty ,\infty )$ be the fitness function defined by letting $L(X_1,\dots ,X_r)$$=({\sum }_{i=1}^q\log \left(\rho \right({\sum }_{j=1}^r{z}_{i,j}{X}_j\left)\right)/q)-\log (\parallel {\sum }_{j=1}^r{X}_j{X}_j^{\ast }{\parallel }_p)/2$. Here, $\rho \left(X\right)$ denotes the spectral radius of a matrix $X$ while $\parallel X{\parallel }_p$ denotes the Schatten $p$-norm of $X$.

Now suppose that $(A_1,\dots ,A_r)\in M_d(C{)}^r\setminus \{0\}$ is a tuple that maximizes $L(A_1,\dots ,A_r)$. Let $M:C^r\setminus \left\{0\right\}\to [-\infty ,\infty )$ be the fitness function defined by letting $M(w_1,\dots ,w_r)=\log \left(\rho \right(w_1{A}_1+\cdots +w_r{A}_r\left)\right)-\log (\parallel (w_1,\dots ,w_r\left){\parallel }_2\right)$. Then suppose that $(v_1,\dots ,v_r)\in C^r\setminus \left\{0\right\}$ is a tuple that maximizes $M(v_1,\dots ,v_r)$. Then we will likely be able to find an $\ell \in \{1,\dots ,q\}$ and a non-zero complex number $\alpha$ where $(v_1,\dots ,v_r)=\alpha \cdot (x_{\ell ,1},\dots ,x_{\ell ,r})$.

In this case, $(z_{i,j}{)}_{i,j}$ represents the training data while the matrices $A_1,\dots ,A_r$ is our learned machine learning model. In this case, we are able to recover some original data values from the learned machine learning model $A_1,\dots ,A_r$ without any distortion to the data values.

I have just made this observation, so I am still exploring the implications of this observation. But this is an example of how mathematical spectral machine learning algorithms can behave, and more mathematical machine learning models are more likely to be interpretable and they are more likely to have a robust mathematical/​empirical theory behind them.

[Joseph Van Name]

I think that all that happened here was the matrices $A_1,\dots ,A_r$ just ended up being diagonal matrices. This means that this is probably an uninteresting observation in this case, but I need to do more tests before commenting any further.