Low-Dimensional Matrix Approximation

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Discussion Overview

The discussion revolves around the problem of projecting a 4x4 matrix into a lower-dimensional 2x2 matrix while retaining as much information as possible. Participants explore various methods for achieving this, including Singular Value Decomposition (SVD) and considerations of eigenvalues and eigenvectors, within the context of computational limitations and specific applications.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the nature of the "important information" being retained, noting the differences between 4x4 and 2x2 matrices and suggesting that more context is needed.
  • Another participant emphasizes the need for specificity regarding the application and provides a nonlinear transformation method that preserves all information, although they express doubt that this is what the original poster seeks.
  • A later reply describes the use of SVD for dimensionality reduction, detailing the process of rotating the matrix and projecting out rows corresponding to the largest singular values.
  • One participant suggests considering eigenvalues and eigenvectors as an alternative approach, particularly for Hermitian or real symmetric matrices, and discusses the physical interpretation of partitioning energy in the context of the application.
  • There is mention of the relationship between singular values and eigenvalues, with a note that singular values may be more manageable for non-Hermitian matrices.

Areas of Agreement / Disagreement

Participants express differing views on the methods for projecting the matrix and the nature of the information to be retained. There is no consensus on a single optimal approach, and the discussion remains unresolved regarding the best technique for the specific application.

Contextual Notes

Participants highlight limitations related to computational memory costs and the need for clarity on the specific application of the matrix reduction. The discussion also reflects uncertainty regarding the implications of using different mathematical approaches.

jfy4
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Hi,

Lets say that I have a 4x4 matrix, and am interested in projecting out the most important information in that matrix into a 2x2 matrix. Is there an optimal projection to a lower dimensional matrix where one keeps most of the matrix intact as best as possible? Thanks.
 
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What information are you talking about, exactly? 4x4 and 2x2 matrices are rather different (they're functions on completely different spaces). If there's a particular (say, 2-dimensional) subspace of interest, then the restriction of a 4x4 matrix to the subspace may give you a 2x2 matrix, but we really can't say anything specific without more information.
 
I think you should be more specifc. What do you want it for? Do you have an example? Etc.

I can easily think of a (nonlinear) way to transform a ##4\times 4## into a ##2\times 2## such that all information is preserved, but I doubt you're looking for this.
 
good questions. True be told I'm not entirely sure... The matrices house local configurations on a lattice, and I can't keep them all due to computer memory cost, so I want to truncate the matrices and keep as much as I can. The only way I have been doing it is with SVD. Once I preform the SVD I rotate the matrix using the V ( from U λ V^{\dagger}) and then I project out the largest rows corresponding to the largest singular values.

Im interested in a nonlinear way to keep all the information too though lol :)
 
jfy4 said:
The only way I have been doing it is with SVD. Once I preform the SVD I rotate the matrix using the V ( from U λ V^{\dagger}) and then I project out the largest rows corresponding to the largest singular values.

You might like to think about that idea using the eigenvalues and vectors of the matrix rather than the singular values, to see what it means "physically" for your application.

For Hermitian (or real symmetric) matrices, you can interpret it in terms of partitioning the "energy" of the system and then throwing way the "least important" components.

Of course the SVs and EVs are closely related, and for arbitrary non-hermitian matrices the SVs might be easier to work with.
 

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