In this post, I will continue to demonstrate that my own machine learning algorithm (LSRDRs) behaves mathematically by analyzing the spectra of superoperators obtained by training from the Okubo algebra. The LSRDRs are linear machine learning models, but I know how to generalize LSRDRs in such a way so that they perform like neural networks while remaining mathematical, and I am increasing their performance. One cannot therefore discount LSRDRs because they are linear since they are toy models of higher performance algorithms.
This post will be mathematical and contain a lot of linear algebra. In this post, we shall go over a bunch of numerical quantities (multisets of complex numbers (spectra)). The thing to take away from this analysis is that most eigenvalues in the spectra are algebraic numbers of low degree and the spectra are indications that the LSRDRs behave mathematically at least for the Okubo algebra, but I have done plenty of other experiments to conclude that the LSRDRs behave mathematically in general. It seems like these mathematical machine learning models are just what we need to develop inherently interpretable high performance machine learning that we need for AI safety.
I originally produced this machine learning algorithm to analyze block ciphers for cryptocurrency technologies. Please do not talk about this here. If you really want to talk to me about this, please contact me off this site and please sign all your statements using your digital signature. Here, you should only be talking about the topic at hand.
Mathematical definitions: Suppose that are -matrices, and are -matrices. Define a superoperator where is the underlying field by . Define .
Define the -spectral radius similarity between and by
.
Set . We say that -matrices form a real (complex,real symmetric, complex Hermitian, etc.) -spectral radius dimensionality reduction of -matrices if is locally maximized among real (complex,real symmetric, complex Hermitian, etc.) matrices.
The underlying set of the Okubo algebra consists of all 3 by 3 traceless complex Hermitian matrices. The Okubo algebra is endowed with a bilinear operation
The Okubo algebra is endowed with a bilinear operation defined by setting where . The Okubo algebra satisfies the identity where is the Frobenius norm.
Let be a linear isometry (one could use the Gell-Mann basis for this isometry but it really does not matter) and define a bilinear operation on by setting . For each , let be the matrix where . Let be the standard orthonormal basis for real Euclidean space. Let for .
In this post, I will mention the spectra of various LSRDRs of . But for this post, there are a lot of non-unique spectra, so it is harder to comment on that. The spectrum of the LSRDR is generally unique up to a complex scalar, but since the Okubo algebra has a lot of symmetry, we lose some uniqueness of the LSRDR.
Suppose now that is a complex -SRDR of and let
be the spectrum of multiplied by the scalar multiple so that the dominant value of is .
be the spectrum of multiplied by the scalar multiple so that the dominant value of is .
If is a multiset, then let denote the multiplicity of the element in
Experimental results: For the remainder of this post, we shall investigate for in order of the complexity of the spectra. Here the spectra get increasingly complicated as the value increases but since the dimensionality reduction does nothing when , it gets simple again at . These results are obtained experimentally. I do not have a mathematical proof that these results correct nor do I have a mathematical proof that these are the only spectra.
The multiset has an easy description. for .
The multiset has just the eigenvalue with multiplicity .
The multiset has eigenvalues where for all and where .
The multiset has eigenvalues where has multiplicity 4 and all other eigenvalues have multiplicity 1. Take note that both have the same minimal polynomial but only one of the roots is in the spectrum .
has eigenvalues for and for and for . The eigenvalues have the following multiplicities:
and .
The multiset has eigenvalues , for , , where
which is a root of the quartic polynomial
and where which is a root of the quartic polynomial
and . The eigenvalues have the following multiplicities:
The multiset has eigenvalues 0,1, for ,, for . The eigenvalues have the following multiplicities:
The multisets have the same values but they have different multiplicities.
If we do not count for multiplicity, the sets both have 16 distinct elements.
The real eigenvalues of are . Let .
Some imaginary eigenvalues include for and . The eigenvalues also include for some real number and a pair of miscellaneous conjugate eigenvalues .
.
These are all the eigenvalues and their multiplicities.
Since the spectra behave mathematically, we can conclude that the LSRDRs behave mathematically. This shows us that LSRDRs (and likely similar algorithms that I have created) behave mathematically when they are trained on mathematical data as long as the data is fed to the model that is compatible with the inner workings of the model. This is the case with LSRDRs.
In this post, I will continue to demonstrate that my own machine learning algorithm (LSRDRs) behaves mathematically by analyzing the spectra of superoperators obtained by training from the Okubo algebra. The LSRDRs are linear machine learning models, but I know how to generalize LSRDRs in such a way so that they perform like neural networks while remaining mathematical, and I am increasing their performance. One cannot therefore discount LSRDRs because they are linear since they are toy models of higher performance algorithms.
This post will be mathematical and contain a lot of linear algebra. In this post, we shall go over a bunch of numerical quantities (multisets of complex numbers (spectra)). The thing to take away from this analysis is that most eigenvalues in the spectra are algebraic numbers of low degree and the spectra are indications that the LSRDRs behave mathematically at least for the Okubo algebra, but I have done plenty of other experiments to conclude that the LSRDRs behave mathematically in general. It seems like these mathematical machine learning models are just what we need to develop inherently interpretable high performance machine learning that we need for AI safety.
I originally produced this machine learning algorithm to analyze block ciphers for cryptocurrency technologies. Please do not talk about this here. If you really want to talk to me about this, please contact me off this site and please sign all your statements using your digital signature. Here, you should only be talking about the topic at hand.
Mathematical definitions: Suppose that
Define the
Set
The underlying set
The Okubo algebra is endowed with a bilinear operation
Let
In this post, I will mention the spectra of various LSRDRs of
Suppose now that
If
Experimental results: For the remainder of this post, we shall investigate
The multiset
The multiset
The multiset
The multiset
The multiset
and
The multiset
The multisets
If we do not count for multiplicity, the sets
The real eigenvalues of
Some imaginary eigenvalues include
These are all the eigenvalues and their multiplicities.
Since the spectra behave mathematically, we can conclude that the LSRDRs behave mathematically. This shows us that LSRDRs (and likely similar algorithms that I have created) behave mathematically when they are trained on mathematical data as long as the data is fed to the model that is compatible with the inner workings of the model. This is the case with LSRDRs.