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Every entry in a matrix counts for the -spectral radius similarity. Suppose that  are real -matrices. Set . Define the -spectral radius similarity between  and  to be the number

. Then the -spectral radius similarity is always a real number in the interval , so one can think of the -spectral radius similarity as a generalization of the value  where  are real or complex vectors. It turns out experimentally that if  are random real matrices, and each  is obtained from  by replacing each entry in  with  with probability , then the -spectral radius similarity between  and  will be about . If , then observe that  as well.

Suppose now that  are random real  matrices and  are the  submatrices of  respectively obtained by only looking at the first  rows and columns of . Then the -spectral radius similarity between  and  will be about . We can therefore conclude that in some sense  is a simplified version of  that more efficiently captures the behavior of  than  does.

If  are independent random matrices with standard Gaussian entries, then the -spectral radius similarity between  and  will be about  with small variance. If  are random Gaussian vectors of length , then  will on average be about  for some constant , but  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.

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?

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

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

Suppose that  are -real matrices and  are -real matrices. Let  denote the spectral radius of  and let  denote the tensor product of  with . Define the -spectral radius by setting , Define the -spectral radius similarity between  and  as

.

We observe that if  is invertible and  is a constant, then

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

Example: Suppose that  are vectors of possibly different dimensions. Suppose that we would like to determine how close we are to obtaining an affine transformation  with  for all  (or a slightly different notion of similarity). We first of all should normalize these vectors to obtain vectors  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  is a measure of low close we are to obtaining such an affine transformation . We may be able to apply this notion to determining the distance between machine learning models. For example, suppose that  are both the first few layers in a (typically different) neural network. Suppose that  is a set of data points. Then if  and , then  is a measure of the similarity between  and .

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  of ordered pairs and I trained the neural networks  to minimize the expected losses . In my experiment, each  was a random vector of length 32 whose entries were 0's and 1's. In my experiment, the similarity  was worse than if  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 -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).

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  is either the field of real numbers or the field of complex numbers. Let  be finite dimensional inner product spaces over  with dimensions  respectively. Suppose that  has basis . Given , we would sometimes want to approximate the tensor  with a tensor that has less parameters. Suppose that  is a sequence of natural numbers with . Suppose that  is a  matrix over the field  for  and . From the system of matrices , we obtain a tensor . If the system of matrices  locally minimizes the distance , then the tensor  is a dimensionality reduction of  which we shall denote by .

Intuition: One can associate the tensor product  with the set of all degree  homogeneous non-commutative polynomials that consist of linear combinations of the monomials of the form . Given, our matrices , we can define a linear functional  by setting . But by the Reisz representation theorem, the linear functional  is dual to some tensor in . More specifically,  is dual to . The tensors of the form  are therefore the

Advantages: 

  1. In my computer experiments, the reduced dimension tensor  is often (but not always) unique in the sense that if we calculate the tensor  twice, then we will get the same tensor. At least, the distribution of reduced dimension tensors  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 .
  2. This dimensionality reduction does not depend on the choice of orthonormal basis . If we chose a different basis for each , then the resulting tensor  of reduced dimensionality will remain the same (the proof is given below).
  3. If  is the field of complex numbers, but all the entries in the tensor  happen to be real numbers, then all the entries in the tensor  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 .
  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  have a more complicated description than the tensors in the CP tensor dimensionality reduction.

Proposition: The set of tensors of the form  does not depend on the choice of bases .

Proof: For each , let  be an alternative basis for . Then suppose that  for each . Then

. Q.E.D.

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

. This is indeed a linear functional, and we can try to approximate  using a the dual to , but this approach does not work as well.

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  is a real,complex, or quaternionic -matrix that minimizes the quantity . Then  is unitary.

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

By the Schur decomposition, we know that  where  is a unitary matrix and  is upper triangular. But we know that . Furthermore, , so . Let  denote the diagonal matrix whose diagonal entries are the same as . Then  and . Furthermore,  iff T is diagonal and  iff  is diagonal. Therefore, since  and  is minimized, we can conclude that , so  is a diagonal matrix. Suppose that  has diagonal entries . By the arithmetic-geometric mean equality and the Cauchy-Schwarz inequality, we know that 

Here, the equalities hold if and only if  for all , but this implies that  is unitary. Q.E.D.

The -spectral radius similarity is not transitive. Suppose that  are -matrices and  are real -matrices. Then define . Then the generalized Cauchy-Schwarz inequality is satisfied:

.

We therefore define the -spectral radius similarity between  and  as . One should think of the -spectral radius similarity as a generalization of the cosine similarity  between vectors . I have been using the -spectral radius similarity to develop AI systems that seem to be very interpretable. The -spectral radius similarity is not transitive.

 and

, but  can take any value in the interval .

We should therefore think of the -spectral radius similarity as a sort of least upper bound of -valued equivalence relations than a -valued equivalence relation. We need to consider this as a least upper bound because matrices have multiple dimensions.

Notation:  is the spectral radius. The spectral radius  is the largest magnitude of an eigenvalue of the matrix . Here the norm does not matter because we are taking the limit.  is the direct sum of matrices while  denotes the Kronecker product of matrices.

Let's compute some inner products and gradients.

Set up: Let  denote either the field of real or the field of complex numbers. Suppose that  are positive integers. Let  be a sequence of positive integers with . Suppose that  is an -matrix whenever . Then from the matrices , we can define a -tensor . I have been doing computer experiments where I use this tensor to approximate other tensors by minimizing the -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 , namely the inner product of such tensors and the gradient of the inner product with respect to the matrices .

Notation: If  are matrices, then let  denote the superoperator defined by letting . Let .

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

.

In particular, .

Gradient: Observe that . 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.

.

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  is either the field of real numbers or the field of complex numbers. Suppose that  are positive integers and  is a sequence of positive integers with . Suppose that  is an -matrix whenever . Then define a tensor 

If , and  is a system of matrices that minimizes the value , then  is a dimensionality reduction of , and we shall denote let  denote the tensor of reduced dimension . We shall call  a matrix table to tensor dimensionality reduction of type .

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

Observation 2: (Repeated entries) If  is sparse and  and the set  has small cardinality, then the tensor  will contain plenty of repeated non-zero entries.

Observation 3: (Tensor decomposition) Let  be a tensor. Then we can often find a matrix table to tensor dimensionality reduction  of type  so that  is its own matrix table to tensor dimensionality reduction.

Observation 4: (Rational reduction) Suppose that  is sparse and the entries in  are all integers. Then the value  is often a positive integer in both the case when  has only integer entries and in the case when  has non-integer entries.

Observation 5: (Multiple lines) Let  be a fixed positive even number. Suppose that  is sparse and the entries in  are all of the form  for some integer  and . Then the entries in  are often exclusively of the form  as well.

Observation 6: (Rational reductions) I have observed a sparse tensor  all of whose entries are integers along with matrix table to tensor dimensionality reductions  of  where .

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.

Suppose that  are natural numbers. Let . Let  be a complex number whenever . Let  be the fitness function defined by letting . Here,  denotes the spectral radius of a matrix  while  denotes the Schatten -norm of .

Now suppose that  is a tuple that maximizes . Let  be the fitness function defined by letting . Then suppose that  is a tuple that maximizes . Then we will likely be able to find an  and a non-zero complex number  where