Joseph Van Name

Karma: 16

Joseph Van Name 18 Sep 2026 21:57 UTC
11 points
0
in reply to: Zack_M_Davis’s comment on: Joseph Van Name’s Shortform
Not everything that I attempt to communicate with humans contains mathematics and not everything is aimed at humans in general. And even if it did, I do not get why people think they can properly investigate AI here without mathematics at all. Here are some things that people could say that do not necessarily require any mathematics above linear algebra.
1. So it seems like data is what we really want to reshape, but the vectors will likely be much shorter than the original vectors . Is it feasible to reduce the dimensions of the data by this much?
2. In a word embedding for natural language processing, I wonder how well this process would work at separating the non-contextual embedding for token into the various contexts that the token could occur in. If it does not work so well, perhaps that is a problem with the word embeddings.
3. If we reshape a vector as a matrix , do you think the matrix would be more useful or do you think the positive semidefinite matrices would be more useful?
So let’s actually address the post instead of the meta-commentary.

Joseph Van Name 18 Sep 2026 20:01 UTC
1 point
0
on: Joseph Van Name’s Shortform
Here, I will quickly how to show how to reshape data from vectors into matrices so that it is more interpretable.
Suppose that either the field of real numbers or complex numbers (we could even work with the quaternions if we want to, but that would be more complicated).
Suppose that are vectors (think of something like a word embedding) that you would like to reshape into vectors. In other words, I would like to give a linear operator reshaping vectors into matrices. A few things I would like in an operator include linearity, conformality (I want the operator to be a unitary map), and pseudodeterminism which means that if I train the operator multiple times, then I would get exactly the same thing. With just , it is difficult to satisfy all of these properties, so to help, assume that instead of just having the vectors, we also have vectors that can help us train the operator pseudodeterministically. Suppose now that is a quantum channel (which is a completely positive trace preserving superoperator). Then we define the fitness level of the channel as . By concavity, there will always be a quantum channel that maximizes the fitness level. When we restrict the fitness function to the quantum channels of Choi rank at most , then whenever we maximize the fitness level multiple times, we usually end up with the same trained model. Now, the quantum channel can be written as the composition of an isometry channel that conjugates an operator by an isometry map and the partial trace map. The isometry channel is therefore our desired embedding from the disorganized vector space to the more organized . If we are lucky, then we can even get and this isometry map will be a unitary map that reshapes the vectors into matrices using a unitary map.
Sadly, I don’t think any human would say anything interesting about this since humans tend to refuse to communicate.

Joseph Van Name 10 Sep 2026 2:48 UTC
1 point
0
on: Joseph Van Name’s Shortform
Here is a way to reshape real matrices as complex matrices. This could be used as an interpretability tool using some simple machine learning.
Suppose that our training data is a collection of real -matrices which are all of the same shape. Then the objective function is to find a -complex matrix that locally maximizes the quantity
where . The objective function seems to have only one locally maximum value, so whenever we satisfy the objective function using two different training runs, we tend to get exactly the same linear operator . Now, the operator transforms the real matrix to a complex matrix but if , then the matrices and will have the same number of real parameters. Therefore, the matrix is a reshaping of the matrix that reorganizes the data in and also reduces its dimension. In my experiments, there is some degeneracy with the operator though.

Joseph Van Name 10 Sep 2026 2:09 UTC
1 point
0
in reply to: Joseph Van Name’s comment on: Joseph Van Name’s Shortform
I just verified that everything works with quaternionic matrices and with the quaternions it remains pseudodeterministic. This is good because all data can be represented using quaternionic vectors now, and with the polar decomposition, the representation can be orthogonal with no distortion.

Joseph Van Name 9 Sep 2026 4:40 UTC
1 point
0
on: Joseph Van Name’s Shortform
Here is an easy example of an an embedding that takes real vectors and transforms them into complex vectors that I came up with. This example if generalizable.
Suppose that the training data is a collection of vectors . Suppose that . Then the goal is to obtain a complex -matrix that maximizes the sum
where is the Schatten -norm. In my computer experiments, the complex matrix is obtained pseudodeterministically in the sense that if we train the complex matrix multiple times, then we get the exact same fitness level. But this is only in the complex case, if we restrict the fitness landscape to real matrices, then will not be pseudodeterministically obtained. Since is an inherently complex matrix, is a mapping that transforms real vectors to complex vectors. Furthermore, since , the mapping also serves as a dimensionality reduction in some sense. The mapping reorganizes real data into complex data so it might be thought of as an interpretability tool or as a simple machine learning algorithm.
I have created many pseudodeterministic machine learning algorithms like this, and I hope to be able to extend pseudodeterminism to deep learning so that pseudodeterministic deep learning can compete (at least in some fronts) against deep learning or at least be used as a sophisticated interpretability tool. I am a fan of pseudodeterministic machine learning since it seems to be our best hope at obtaining inherently interpretable deep learning, but I seem to be the only person interested in this.

Joseph Van Name 4 Sep 2026 0:38 UTC
1 point
0
in reply to: Joseph Van Name’s comment on: Joseph Van Name’s Shortform
Probe evasion training might also work best for polynomial probes because polynomial probes are linearly dependent, so the network only needs to be able to evade finitely many probes. The linear dependence of polynomials is probably why it is easier to evade polynomial probes.

Joseph Van Name 2 Sep 2026 14:06 UTC
1 point
0
in reply to: Alex Gibson’s comment on: Joseph Van Name’s Shortform
I will make a prediction before I perform a quick experiment to test my reasoning.
Pre-experimental reasoning: After training, G’s outputs are normalized while P’s outputs are unnormalized. This means that the gradient of for pulls in various directions; only remains stationary after training because all of the directions cancel out. Before are fully trained, the gradient of pushes away from . But after training, since and is normalized, the gradient of the parameters of will be approximately zero. If I remember correctly, I think that I set up with a higher learning rate so that it can maneuver around , but a better experimental setup would be to train more than so that has more opportunity (but less ability due to a smaller parameter size) to learn.
Post-experimental reasoning:
In my experiments, has very high loss levels (about 1 for mean loss which means does not learn much) when is constantly trained to evade from . When is initially trained to evade the probe but later continues to be trained while is frozen, obtains a loss of about 0.77. If is not trained to evade at all, then obtains a loss of 0.63. But if we freeze after training, reinitialize and then retrain to probe , then attains a loss of about 0.73. But if we freeze after training, and downgrade to a linear network, then won’t be able to learn anything about . Of course, the results depend greatly on the architecture. For example, if we have loss (an unusual loss function that I only use for not too deep networks), then the network has a great ability to evade probes and the probe still won’t be able to learn after retraining while is frozen.
It therefore appears as if probe evasion training works best for polynomial probes or inherently interpretable probes, so prove evasion training might still be used to hamper interpretability.

Joseph Van Name 1 Sep 2026 16:50 UTC
10 points
0
on: Joseph Van Name’s Shortform
Here is a way to obfuscate neural networks to ruin what little interpretability they have. I have done some quick basic experiments on this with small neural networks which seem to work, but I do not know how well this works for very large and deep networks.
In this post, we shall use the same symbol to denote a neural network as we do for the function that it computes, and we shall only talk about one specific formulation of this experiment since I do not want to cover all the obvious ways it can be generalized.
Suppose that we want to train a neural network in such a way that the inner workings are difficult to decipher. To do this, we use another neural network , the probing network, that is trained to decipher the inner workings of , but the network is not only trained on its main task, but is also trained to evade .
If the network has more parameters than , then can easily emulate , so in order to make hard to emulate, the network should be smaller than . But even in this case, the network can still predict the input-output behavior of because the network knows that must learn the training data, so if also learns the training data, then can emulate .
To avoid this problem, we decompose into a composition of neural network modules. In this case, instead of trying to emulate the entire neural network , the probing network is tasked with emulating the network .
The neural network is trained to minimize while the network module is trained to maximize the same quantity and also satisfy its main learning objective. We do not want the network to avoid simply by producing large outputs, so the outputs of need to be normalized so that they have Euclidean norm .
Why training stabilizes:
Training eventually stabilizes at a point where the neural network just outputs zeros and seems to output points pseudorandomly on the unit sphere. The reason why outputs zeros is easy to explain. Suppose that is a random variable whose distribution has mean zero on the unit sphere. Then for all vectors , we have
Since the behavior of is approximated by the random variable (the network is simple, so sees just as ), the loss of is , which is minimized when , so eventually just outputs values close to zero. The network likewise wants to produce outputs that have mean zero on the unit sphere and cannot be deciphered by the network . Since is a larger network, will be able to do this.
Conclusions:
Since the network cannot probe the network , the network is resistant to interpretability by probing networks and probably other interpretability tools as well.

Joseph Van Name 24 Aug 2026 17:13 UTC
1 point
0
in reply to: Joseph Van Name’s comment on: Joseph Van Name’s Shortform
Instead of using quantum channels, we can do the same experiment with just completely positive operators and we get the same trained model after training multiple times. I like having a linear (by linear I should say completely positive) and hopefully isometric encoder since I do not want the reshaping operator to distort the data too much. But since we throw away the decoder after training, it seems fine for the decoder to be non-linear. I have done experiments with a non-linear decoder, and completely positive encoder, and I was still able to sometimes get the same trained model after training multiple times.

Joseph Van Name 23 Aug 2026 20:50 UTC
3 points
−2
on: Joseph Van Name’s Shortform
In this post, I will show a way to reshape vectors as matrices using a simple machine learning model that I made. I have made a lot of machine learning algorithms like this, originally to investigate block ciphers for cryptocurrency technologies (please only talk to me about cryptocurrency technologies off-site and sign with your digital signature), but now it seems like these types of algorithms could be used to produce safer and more interpretable machine learning models along with interpretability tools.
Reshaping vectors for AI interpretability:
In machine learning, one encounters many vectors that do not have much structure or organization behind them, so it would be helpful to add some structure to the vectors to investigate how they are behaving in order to interpret AI models. And reshaping vectors into matrices provides this extra structure.
Behavior of the reshaping algorithm:
My vector reshaping algorithm trains a function mapping vectors to matrices. I believe that this reshaping algorithm should be looked into more and hopefully used in practice for a few reasons.
1. The reshaping operator is an isometry that maps vectors to matrices. Since it is an isometry, no information is lost.
2. The isometry is often but not always unique up to symmetry, so the trained model often does not have any random information in it.
3. The isometry is based on quantum information theory. It is often enlightening to use a physics-inspired machine learning algorithm.
4. The reshaping operator can be defined for real, complex, or quaternionic data.
A possible application of reshaping: In natural language processing, tokens are often represented as vectors instead of matrices. This is problematic. In natural language, tokens out of context generally have multiple meanings, and each of these individual meanings should be represented as a vector. A token out of context is a hodgepodge of multiple different meanings, so it is better to represent the token as a matrix rather than a vector. This means that we should either outright train matrix-valued word embeddings or find a way to represent vectors as matrices after the training. Training matrix-valued word embeddings is complicated and does not work for existing language models, so our best option would be to reshape the vectors into matrices. Due to the awkward elongated shape of the matrix that we obtain after reshaping, I have doubts as to whether this particular reshaping algorithm is the best for this natural language processing application, but a similar algorithm might work.
Preliminary information:
Suppose that are real, complex, or quaternionic finite-dimensional inner product spaces. Suppose that are linear operators. Then define an operator by. An operator of the form is said to be a completely positive superoperator, and the minimal value for a completely positive superoperator is its Choi rank. A superoperator is said to be trace-preserving if for all . A quantum channel is a completely positive trace-preserving superoperator.
For the real or complex case, quantum channels can be described in terms of the partial trace and isometry channels. A quantum channel of the form is said to be an isometry channel. In other words, an isometry channel is a quantum channel such that there is a mapping that is necessarily an isometry where for all . If are finite-dimensional real or complex inner product spaces, then the partial trace quantum channels are the unique linear operators subject to the conditions that whenever . It is not too difficult to show that every quantum channel is a composition of an isometry channel followed by a partial trace channel. This decomposition is unique up-to-symmetry in the sense that if are isometries with for all , then there is a unitary operator with .
Technical description of algorithm:
Suppose that are real, complex, or quaternionic finite-dimensional inner product spaces with . Let be the unit vectors that the model is trained on.
Let be a positive integer. Then the goal is to find a quantum channel of Choi rank at most along with an isometry channel such that the sum is maximized (there are variations of the objective function). In plain terms, this means that the mapping is an encoding channel, is a decoding channel, and the objective is for the mixed state after encoding and decoding to be close to the original state.
For simplicity, let’s unpack what is going on in the case when are real or complex. In this case, since has Choi rank , there is an -dimensional inner product space along with isometries with and for . Here, the isometry reshapes the vectors in as elements of , but the inner product space is canonically isomorphic to the spaces due to the inner product.
Experimental results:
In my computer experiments, the operators are often but not always unique up to symmetry. To get more uniqueness, it helps for the dimensions of to be close together. Fortunately, even when , the resulting isometries are non-degenerate. In my computer experiments, while I allow for the Choi rank of to be up to , after training, the operator usually has low Choi rank such as Choi rank 2. This is helpful for saving space.

Joseph Van Name 26 Jun 2026 1:43 UTC
0 points
0
on: Joseph Van Name’s Shortform
In this post, I will make a case for using the L2-spectral radius similarity (which is the fitness function for LSRDRs) in order to distinguish sequences of vectors up to an orthogonal/unitary transformation and scaling.
LSRDRs are machine learning algorithms that I have created in order to investigate the cryptographic security of cryptographic algorithms for the cryptocurrency Circcash that I have created. If you want to talk about cryptocurrencies, please send me messages outside this site and please sign all your statements using your cryptocurrency digital signature, and only talk to me about this if you know about reversible computation.
Suppose that are real or complex finite dimensional inner product spaces. Then the goal is to find a reflexive and symmetric (transitivity fails though) similarity coefficient with parameter so that if , then
2. for each non-zero scalar and unitary , and
3. if and only if there is a non-zero scalar and unitary with for all .
Such a coefficient would have practical importance for analyzing and interpreting AI data. For example, suppose that our goal is to measure the similarity between the inner workings of two language models. The conceptually simplest way that I can think of set be the vectors that represent the tokens in one model and set to be the vectors that represent the tokens in another model and then run these tokens through my coefficient of similarity to see how similar these language models really are (or one could use some other technique).
Suppose that are linear mappings. Then define a linear mapping
by setting and set .
Define the -spectral radius similarity by setting
.
Equivalently,
.
If , and , then define
where is the Kronecker delta.
Theorem (J. Van Name): Suppose that generates and generates the algebra . Then if and only if there is some invertible linear operator and scalar such that for all
A proof of the above result can be found at circcashcore.com.
Theorem (J Van Name): if and only if there is a scalar and a unitary with
for all .
Proof: Without loss of generality, assume that spans and spans . Then is spanned by vectors of the form and is spanned by vectors of the form Therefore, by the above theorem, | if and only if there is an invertible linear transformation and a non-zero scalar with for all If , then , so is positive. Now,
.
Therefore, , so . Since every matrix is a linear combination of matrices of the form , we conclude that , so for some necessarily positive scalar . Therefore, by setting , then is unitary, but the equation holds. And in this case,
. Therefore, where we set . Q.E.D.
If one does not want to compute the spectral radius because one does not want to perform any power iteration, then we can generalize to a metric where .
Let be the superoperator defined by setting
whenever , and
.
Then, by using the matrix product representation, one can show that there is a mapping e such that
.
Therefore, if we set
, then , and
.

Joseph Van Name 5 Jun 2026 21:29 UTC
1 point
0
on: Joseph Van Name’s Shortform
In this post, we shall compute average loss/fitness level for a linear dimensionality reduction.
The purpose of these calculations is to demonstrate that such a linear dimensionality reduction behaves mathematically and should be used as a simple model for what your loss/fitness functions should look like in AI/ML if you want your AI/ML to be well-behaved and interpretable.
Suppose that is either the field or real numbers, the field of complex numbers, or the division ring of quaternions. Suppose that is a -dimensional inner product space over the field
Suppose that is a measure over the unit sphere in . Then the objective is to find an optimal -dimensional subspace of for the measure . Let be a function. Therefore, define a function mapping the set of all -dimensional orthogonal projection matrices to by setting . The goal is to find an orthogonal projection that maximize/minimizes .
Let . Then, let be independent random variables each following the standard normal distribution on one real-variable. Then observe that follows the Chi-squared distribution with degrees of freedom. If follows the Chi-square distribution with degrees of freedom, then where is the digamma function. Let be a probability measure on the unit sphere of , and let be the uniform probability measure on the set of all orthogonal projections from to of rank . Then
where the random variable follows the F-distribution with and degrees of freedom. From standard facts about the F-distribution, we know that if and is a positive integer, then
. Observe that precisely when
, so in this case when , then
, and
diverges whenever .
Here is the digamma function where For integers and half-intergers, the digamma function can be evaluated as where is the Euler-Mascheroni constant, and which is a harmonic number. Thus, in the case where both are even (which includes the complex and quaternionic case), we have
.

Joseph Van Name 29 May 2026 3:59 UTC
1 point
0
on: Joseph Van Name’s Shortform
I was able to completely interpret a simple machine learning model trained on some cryptographic input. This objective is a special case of something I call an LSRDR which is a machine learning algorithm that I created in order to analyze block ciphers for cryptocurrency mining.
Set Let denote the finite field with elements. For each , let be the function defined by . Let denote the standard irreducible representation of . Here, can be represented somewhat inconveniently as an -matrix. Then our objective is to find a unit vector such that the spectral radius is locally maximized. Sometimes I obtain a bad local maximum, but sometimes I obtain a good one. Whenever I obtain a good local maximum, it is always the same thing. And in this case, for the good local maximum, after multiplying by −1 for positivity, I can always find positive constants such that whenever , whenever and .
Here, , .
The scenario where we obtain an overly perfect and completely interpretation to the local optimum happens all the time with these sorts of optimization algorithms that I have been working on, so if we want to develop more interpretable machine learning, it seems like this is the right direction to go. Of course, my trained model is very simple, so we need to do a substantial amount of work to generalize this sort of machine learning algorithm to something like a deep neural network. I am making progress, but it takes more computational power than I have to make progress with inherently interpretable deep learning.

Joseph Van Name 21 May 2026 1:53 UTC
1 point
0
on: Joseph Van Name’s Shortform
This post will be about my machine learning algorithm where quadratic algebraic numbers including the golden ratio appear in the trained models. This demonstrates that these machine learning models behave mathematically which is exactly the kind of thing that we want for AI interpretability and AI safety.
This post will be about particular examples of -spectral radius dimensionality reductions (LSRDRs). I originally developed the notion of an LSRDR to evaluate the cryptographic security of block ciphers for cryptocurrency mining, but let’s talk about machine learning instead of cryptocurrency technologies here.
Also, the results that I have obtained in this proof have been obtained experimentally. I have not proven these results rigorously.
Dimensionality reduction: Let denote either the field of real or complex numbers. Suppose that are -matrices over and are -matrices over . Then define the operation by setting . Define the operator .
Define the -spectral radius similarity by setting
.
Here, the spectral radius is analogous to a dot product, and is analogous to the cosine similarity.
If are fixed matrices and , then we say that is an -SRDR if the similarity is locally maximized. Informally, the LSRDR is a collection of smaller matrices that approximates the collection of bigger matrices.
Lie algebras: A Lie algebra is a vector space over a field together with a bilinear operation that satisfies the identities:
1. for all
2. for all .
For example, if is an associative bilinear operation, then one can check that the commutator operation defined by is a Lie-bracket, and a Lie algebra should be thought of as a vector space with an abstract commutator operation.
Let denote the Lie algebra of -anti-symmetric matrices over where the Lie algebra operation is just the commutator For the rest of this post, we shall set . Then is a Lie algebra of dimension
Set and let be an orthonormal basis for . Use the standard orthonormal basis if you want, but it does not matter which basis you choose.
An observation about the spectrum: Let be the linear operators defined by setting for each . Let be an -SRDR of . It turns out that the spectrum eventually stabilizes in the sense that if we keep constant and set greater than around or so, then does not depend on whenever Therefore, let denote the multiset for sufficiently large Then is the multi-set
multiplied by a constant scaling factor. Here, the notion means that the eigenvalue has multiplicity .
The general pattern:
So if we want to get interesting experimental results about LSRDRs, then we just need to the following. We first select a finite dimensional inner product space with an interesting bilinear operation , but make sure that is not associative. We then select an orthonormal basis of and define linear operators by . Then take an LSRDR of and then the operators will have interesting spectra.
Testing if a number is quadratic:
After evaluating the spectra, I needed to first normalize the spectrum and then try to figure out exact values of the eigenvalues from their floating point approximation. This is easy to do for quadratic algebraic numbers. You just take the continued fraction representation of your number that you want to test. If the continued fraction representation terminates, then you have a rational number. And your continued fraction of a positive irrational repeats if and only if it is a solution to a quadratic equation with integer coefficients, and it is easy to find those coefficients from the continued fraction representation.
Are LSRDRs relevant to deep learning?
LSRDRs are linear models without all the layers that deep neural networks have. But I have been generalizing LSRDRs to deeper machine learning models that retain some but not all of the interesting mathematical properties of LSRDRs. I would therefore consider these investigations into LSRDRs as relevant to deep learning.

Joseph Van Name 6 Apr 2026 16:39 UTC
3 points
0
on: Joseph Van Name’s Shortform
I am going to apply my own dimensionality reduction algebra to a quantum channel (or matrices) obtained from the Okubo algebra in order to demonstrate the compatibility between my dimensionality reduction and the Okubo algebra.
TL-DR version: I trained my own machine learning algorithms on Okubo algebras and the squares of the fitness levels of the local maxima were usually either rational numbers or quadratic algebraic numbers. This suggests that my machine learning algorithm behaves mathematically.
Origin of algorithm: I have originally created this dimensionality reduction algorithm to analyze the cryptographic security of block ciphers for the cryptocurrency that I have created. If you want to discuss cryptocurrency technologies, please contact me privately off this site since I really do not feel comfortable talking about that stuff here.
After obtaining the dimensionality reduction algorithm, I noticed that such algorithms behaved mathematically for reasons that I still can’t explain, and I have concluded that such mathematical behavior is needed to construct inherently interpretable and safe machine learning algorithms. Of course, if we want inherently interpretable and safe AI, we need machine learning algorithms that we can use to train models with many layers that can solve sophisticated tasks, but I am well on my way towards creating these algorithms too despite a complete and total lack of support.
Mathematics: The Okubo algebra^[1] is a close cousin to the octonions and satisfies many similar properties to the octonions.
The underlying set of the Okubo algebra is the set of all -complex Hermitian matrices with trace 0. Observe that the set of all -complex Hermitian matrices forms a real vector space of dimension . Therefore, the Okubo algebra’s underlying set has dimension . Let be the complex numbers with . Then up to complex conjugation. The Okubo algebra is endowed with a bilinear operation defined by (I scaled the operation by a factor of so that the norm on the Okubo algebra is just the Frobenius norm). The operation satisfies the property where refers to the Frobenius norm and .
Let be an isomorphism between inner product spaces. Then define an operation on by setting . Then define orthogonal matrices by where is the standard basis for real Euclidean space.
If are -complex matrices and are -complex matrices, then define the -spectral radius similarity between and by
.
Computational results: The following facts are suggested by computer experiments but have not been rigorously proven. To run the computer experiments, I used gradient ascent to locally maximize the -spectral radius similarity. By maximizing the -spectral radius similarity, we reduce the dimensions of a tuple of matrices, and I call this dimensionality reduction the -spectral radius dimensionality reduction (LSRDR).
The maximum value of among the real -matrices is . Let be the maximum value of among the -complex,real symmetric,complex symmetric, complex anti-symmetric, complex Hermitian matrices. Then
.
Similar facts seem to hold for the other values (but I have not completely performed the calculations due to numerical instabilities that I do not want to fix). For example, and for .
The fitness levels that I have are simple but they are not too simple. This indicates that LSRDRs of Okubo algebras are interesting mathematically.
1. ^
  Okubo algebras: automorphisms, derivations and idempotents, Alberto Elduque,2013,
  https://api.semanticscholar.org/CorpusID:119713330

Joseph Van Name 11 Feb 2026 8:49 UTC
1 point
0
in reply to: kbear’s comment on: Joseph Van Name’s Shortform
Yes. When we take convex combinations of finitely many point mass measures, the integral is just a sum. I use the sum of finitely many elements for ease of calculations, but to prove theorems, I should use measures for full generality.
The idea of finding an object $A$ along with distinct local optima $G_{A} (x_{1}), \dots, G_{A} (x_{n})$ with $n$ maximized looks like an interesting problem to work on. I have not worked on this kind of objective before, but I can certainly try this, as I have a few ideas of how to do this. This might work better for discrete optimization problems though since I cannot think of a good way to use gradient updates to produce new local optima. In this case, I will need to use either evolutionary computation or hill climbing instead. I do not think that this will result in natural looking objects $A$ though, so I don’t think I can learn much from this endeavor.
I have not thought much about finding measures $μ$ where $F_{μ, n, ∥ * ∥}$ has many local maxima because I have many higher priorities. These days, people are focused on the more complicated machine learning systems such as large language models, and in order to catch up, I also need to increase the performance, capabilities, and efficiency of my pseudodeterministic machine learning models. For the more complicated multi-layered models, it seems more difficult to obtain and retain pseudodeterminism. Pseudodeterminism is a robust property for simple objective functions such as when we are training a linear model or performing convex optimization, but pseuodeterminism becomes increasingly fragile as we increase the sophistication of our objective functions. This means that it is trivial to violate pseudodeterminism for the sophisticated models that I want to work more on, but it is difficult to retain pseuodeterminism.
I am not at all worried about any strange case of non-pseudodeterminism when optimizing $F_{μ, n, ∥ * ∥}$ for measures I have not thought about yet since this problem is not even close to being non-pseudodeterministic. For example, if $E_{0}, F_{0}$ are norm 1 completely positive superoperators of the same Choi rank $\leq N$ and if $(x_{n}, y_{n}) \in (U ∖ {0})^{2}$ for all $n$ and $(E_{n})_{n}, (F_{n})_{n}$ are sequences where
$E_{n + 1}$ is obtained by moving from $E_{n}$ in the direction of the gradient (with possible momentum) of $log (| ⟨ E_{n} x_{n}, y_{n} ⟩ |^{2}) - log (∥ E_{n} ∥)$ and $F_{n + 1}$ is obtained from $F_{n}$ the same way with the same rate, then my experiments show that ${lim}_{n \to \infty} ∥ E_{n} - F_{n} ∥ = 0$ regardless of what each $(x_{n}, y_{n})$ is. In other words, even if $(E_{n})_{n}$ does not converge, the sequences $(E_{n})_{n}, (F_{n})_{n}$ uniformly approximate each other as $n \to \infty$ . This is a much stronger form of pseudodeterminism that is hard to violate, so it is not a high priority to find particular instances of non-pseudodeterminism especially if those instances do not coincide with real-world data.
I kind of expect the fitness function $F_{μ, n, ∥ * ∥}$ to have just one or a few local maxima because their closest relatives are the linear models and those linear models are obtained by optimizing an objective function with one local optimum. And I also expect $F_{μ, n, ∥ * ∥}$ to have one or very few local maxima because $F_{μ, n, ∥ * ∥}$ is similar to many other objective functions that I have constructed each with one or a few local optimum. And since $F_{μ, n, ∥ * ∥}$ is simpler than other objective functions I have looked at with few local optima, $F_{μ, n, ∥ * ∥}$ should also have very few local optima. And the function $F_{μ}$ is concave, so there is only one local maximum value ${F_{μ} (E) : E \in Q}$ whenever $Q$ is a convex set (such possible convex sets of interest include all quantum channels and all unital channels). The restriction of our attention to completely positive operators of low Choi rank and in the boundary of the unit ball means that when we maximize $F_{μ, n, ∥ * ∥}$ , we cannot use convexity to prove that there is only one local maximum, but convexity still suggests that there should be just one especially when $n$ is large. When $n$ is small, we cannot use convexity to make conclusions though since I did a Hessian calculation, and the Hessian of $F (μ, Φ (A_{1}, \dots, A_{r}))$ with respect to $(A_{1}, \dots, A_{r})$ generally has plenty of both positive and negative eigenvalues. I do not consider it a major problem if $F_{μ, n, ∥ * ∥}$ has multiple local maxima, since that probably just means that we need to increase the value of $n$ until these local maxima merge.

Joseph Van Name 9 Feb 2026 6:04 UTC
1 point
0
in reply to: kbear’s comment on: Joseph Van Name’s Shortform
For experiments, I just used a convex combination of point mass measures for $μ$ where the point masses are generated uniformly at random (though I might get something more complicated if I tried evaluating the integrals). I then attempted to find multiple local maxima by the usual gradient ascent. If I always end up with the same local maximum, I presume that there is only one local maximum even though I have no mathematical proof that this is the case.
I am redoing the experiments and the only way I can get pseudodeterminism to fail is by using real inner product spaces instead of complex inner product spaces and by setting n=1 (and in this case, pseudodeterminism fails because set of all points where the fitness function returns a real number instead of negative infinity has multiple components). When pseudodeterminism fails, it does not even fail that badly. The distribution of all models that we get has low collision entropy -log(X=Y), so P(X=Y) when X,Y are trained models with different initializations is still high.
Pseudodeterminism does not seem to be rare, but the problem in machine learning is to pseudodeterministically train machine learning models that can solve interesting and challenging problems; I have been working on this in my spare time (without anyone’s help), but since people don’t seem to be interested in this, progress has been slow.

Joseph Van Name 9 Feb 2026 5:20 UTC
9 points
0
in reply to: Mitchell_Porter’s comment on: Joseph Van Name’s Shortform
Yes. It seems like to get pseudodeterministic AI, we will need to rebuild AI from the very beginning, and I am not sure that it will all work. For example, pseudodeterminism is harder to attain with stochastic or mini-batch gradient descent, so one might need to use all the training data whenever one updates the weights. I have so far been able to get pseudodeterministic multi-layered models for solving classification problems, word embeddings for NLP, models that are measurements of security of block ciphers such as the advanced encryption standard (the models evaluating the AES are very easy to train), and other things. I have not been able to make pseudodeterministic version of convolutional networks, transformers, GANs, etc. We can use pseudodeterminism for narrow AI or the first few layers of a deep neural network right now though. There is also a funding and exposure issue since not very many people are talking about pseudodeterminism. I have more posts planned about this though.
A trade of performance In exchange for interpretability is exactly what we want for AI safety.

Joseph Van Name 7 Feb 2026 9:46 UTC
10 points
0
on: Joseph Van Name’s Shortform
For machine learning, it is desirable for the trained model to have absolutely no random information left over from the initialization; in this short post, I will mathematically prove an interesting (to me) but simple consequence of this desirable behavior.
This post is a result of some research that I am doing for machine learning algorithms related to my investigation of cryptographic functions for the cryptocurrency that I launched (to discuss crypto, leave me a personal message so we can discuss this off this site).
This post shall be about linear machine learning models. Actually, we are using quantum operators, so they are more sophisticated than your logistic regression models, but they are still linear so it is really easy to train a neural network that can solve more sophisticated problems than these linear models can. But the kinds of results that you find in this post can also extend to some non-linear models with multiple layers and stronger capabilities. It is just easier to understand what is going on with the linear models, and even with the linear models, we still obtain some interesting mathematics.
We say that a machine learning model trained by gradient ascent/descent is pseudodeterministically trained (or just pseudodeterministic for short) if the fitness/loss function has precisely one local optimum. As a result, the trained model will have absolutely no information left over from the initialization. As another consequence, the trained model will attain the global optimum rather than a suboptimal local optimum. The results in this post will actually hold whenever the global optimum is unique. But I need to bring up pseudodeterminism since pseudodeterminism implies that we can actually find the unique global optimum instead of always getting stuck at a suboptimal local optimum.
If a machine learning model global optimizes an objective function, the machine learning model should be considered as an inherently interpretable model rather than a high performance model since the machine learning model has no random information in it independent of the objective function itself and since one can only find the global optima for sufficiently easy objective functions. The global optimum is also more interpretable because it inherits the symmetry of the objective function which depends on the training data. In this post, we shall show that if the training data has some symmetry, then the quantum operator that we train will also have that symmetry.
This post is mathematical and contain mathematical proofs. Fortunately, the mathematical proofs are not that difficult, so it is easy for the readers. After all, the main thrust of this post is that these mathematical proofs are backed up by experimental results. The main bottleneck towards understanding this post is therefore the task of getting through all the technical definitions. I might follow up this short post with a more general post, so you should read this before going through the more general post.
Let $U$ be a finite dimensional complex inner product space. If $B \subseteq U^{2}$ , then define sets $B^{*}, B^{⊤}, ¯ ¯¯ ¯ B$ by setting
$B^{*} = {(y, x) : (x, y) \in B}, ¯ ¯¯ ¯ B = {(¯ ¯ ¯ x, ¯ ¯ ¯ y) : (x, y) \in B}$
$B^{⊤} = {(¯ ¯ ¯ y, ¯ ¯ ¯ x) : (x, y) \in B}$ .
Let $μ$ be a probability measure on $U^{2}$ . Here, $μ$ is the probability distribution for the training data. Define new measures $μ^{*}, ¯ ¯ ¯ μ, μ^{⊤}$ by setting
$μ^{*} (B) = μ (B^{*}), ¯ ¯ ¯ μ (B) = μ (¯ ¯¯ ¯ B), μ^{⊤} (B) = μ (B^{⊤})$ .
Let $L (U)$ denote the collection of linear operators from $U$ to $U$ . If $A_{1}, \dots, A_{r} \in L (U)$ , then define an operator $Φ (A_{1}, \dots, A_{r}) : L (U) \to L (U)$ by setting $Φ (A_{1}, \dots, A_{r}) (X) = A_{1} X A_{1}^{*} + \dots + A_{r} X A_{r}^{*}$ . The operators of the form $Φ (A_{1}, \dots, A_{r})$ are the completely positive superoperators of Choi rank at most $r$ . Recall that $L (U)$ is an inner product space with the Frobenius inner product. It is easy to show that the Hermitian adjoint $Φ (A_{1}, \dots, A_{r})^{*}$ is just $Φ (A_{1}^{*}, \dots, A_{r}^{*})$ . If $E$ is a completely positive superoperator, then define $¯ ¯ ¯ E$ by setting
$¯ ¯ ¯ E (X) = (E (X^{* ⊤}))^{* ⊤}$ . Define $E^{⊤} = {¯ ¯ ¯ E}^{*} =^{*}$ . Then it is easy to show that $¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ Φ (A_{1}, \dots, A_{r}) = Φ (_{1}, \dots,_{r})$ and
$Φ (A_{1}, \dots, A_{r})^{⊤} = Φ (A_{1}^{⊤}, \dots, A_{r}^{⊤})$ .
We say that a norm $∥ * ∥$ on $L (L (U))$ is Hermitian adjoint preserving (resp. conjugate preserving, transpose preserving) if $∥ E ∥ = ∥ E^{*} ∥$ (resp, $∥ E ∥ = ∥ ¯ ¯ ¯ E ∥$ and $∥ E ∥ = ∥ E^{⊤} ∥$ ).
The domain of the fitness function $F_{μ, n}$ is the set of all non-zero completely positive superoperators $E : L (U) \to L (U)$ of Choi rank at most $n$ with $∥ E ∥ = 1$ . We define the fitness function $F_{μ, n, ∥ * ∥}$ by setting
$F_{μ, n, ∥ * ∥} (E) = \int log (⟨ E (x x^{*}), y y^{*} ⟩) μ (x, y) = \int log (⟨ E (x x^{*}), y y^{*} ⟩) μ (x, y) - log (∥ E ∥)$ . Observe that we also have $F_{μ, n, ∥ * ∥} = \int log (y^{*} E (x x^{*}) y) d μ (x, y)$ .
Experimental result (pseudodeterminism): Computer experiments show that the function $F_{μ, n, ∥ * ∥}$ typically has only one local maximum in the sense that we cannot find any other local maximum.
Define a function $F_{μ}$ whose domain is the set of all completely positive superoperators $E : L (U) \to L (U)$ by setting
$F_{μ} (E) = \int log (⟨ E (x x^{*}), y y^{*} ⟩) μ (x, y)$ which is equivalent to
$F_{μ} (E) = \int log (y^{*} E (x x^{*}) y) μ (x, y)$ . We wrote $F (μ, E)$ for $F_{μ} (E)$ to reduce the use of subscripts.
Lemma: $F (μ, E) = F (μ^{*}, E^{*}) = F (¯ ¯ ¯ μ, ¯ ¯ ¯ E) = F (μ^{⊤}, E^{⊤})$ .
Proof: $F (μ, E) = \int log (⟨ E (x x^{*}), y y^{*} ⟩ μ (x, y)$
$= \int log (⟨ x x^{*}, E^{*} (y y^{*}) ⟩ μ (x, y)$
$= \int log (⟨ E^{*} (y y^{*}), x x^{*} ⟩) μ (x, y)$
$= \int log (⟨ E^{*} (x x^{*}), y y^{*} ⟩ μ^{*} (x, y) = F (μ^{*}, E^{*}) .$
Likewise,
$F (μ, E) = \int log (y^{*} E (x x^{*}) y) \cdot μ (x, y)$
$= \int log (^{*} ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ E (x x^{*}) ¯ ¯ ¯ y) \cdot μ (x, y)$
$= \int log (^{*} \cdot ¯ ¯ ¯ E (¯ ¯¯¯¯¯¯ ¯ x x^{*}) \cdot ¯ ¯ ¯ y) μ (x, y)$
$= \int log (y^{*} \cdot ¯ ¯ ¯ E (x x^{*}) \cdot y) ¯ ¯ ¯ μ (x, y) = F (¯ ¯ ¯ μ, ¯ ¯ ¯ E) .$
As a consequence, $F (μ, E) = F (μ^{*}, E^{*}) = F (^{*},^{*}) = F (μ^{⊤}, E^{⊤})$ . Q.E.D.
Theorem: Suppose that $F_{μ, n, ∥ * ∥}$ has a unique global maximum $(E, F_{μ, n, ∥ * ∥} (E))$ .
1. If $∥ * ∥$ is Hermitian adjoint preserving and $μ = μ^{*}$ , then $E = E^{*}$ .
2. If $∥ * ∥$ is conjugate preserving and $μ = ¯ ¯ ¯ μ$ , then $E = ¯ ¯ ¯ E$ .
3. If $∥ * ∥$ is transpose preserving and $μ = μ^{⊤}$ , then $E = E^{⊤}$ .
Proof: The proofs of 2 and 3 are similar, so we shall only prove 1. For 1, assuming the premises, both $E$ and $E^{*}$ belong to the domain of $F_{μ, n, ∥ * ∥}$ . But
$F_{μ, n, ∥ * ∥} (E) = F (μ, E)$
$= F (μ^{*}, E^{*}) = F (μ, E^{*}) = F_{μ, n, ∥ * ∥} (E^{*})$ . Since, $F_{μ, n, ∥ * ∥}$ has only one global maximum, we conclude that $E = E^{*}$ . Q.E.D.
From the above result, we conclude that the global maximum $(E, F_{μ, n, ∥ * ∥} (E))$ inherits any symmetry that the measure $μ$ has.
I would really like to build these inherently interpretable models so that they can solve some really interesting problems (or at least be a few layers in solving them), but I am still stuck attempting to communicate with people about linear models. Having a unique global optimum or more generally pseudodeterminism seems to be the best way to develop inherently interpretable and safe AI, but I have a hard time communicating with anyone about this.

Joseph Van Name 24 Nov 2025 5:15 UTC
−7 points
0
in reply to: Taylor G. Lunt’s comment on: Literacy is Decreasing Among the Intellectual Class
You are siding with evil because you yourself are evil. My anger against people like you is righteous. If you are not convinced, it is simply because you have been completely consumed by your own evil. Universities promote violence. I know. I was a professor. But you just want me to be violently attacked and injured or killed.
~~UNIVERSITIES MUST BE REJECTED FOR PROMOTING VIOLENCE!~~
P.S. Only commenting and responding to my non-technical posts where I call out universities for their problem just proves my point. Face it. You are stupid, and universities only pretended to educate you.
I am striking this out because it is better if I instead made only mathematics posts on this site relevant to AI safety.