Matrix connected components decomposition #19045

sylee957 · 2020-04-01T10:30:02Z

References to other Issues or PRs

Brief description of what is fixed or changed

I see there were some needs of using connected_components.
So I wrap two APIs for this first for finding connected components and second for finding a similar matrix which is block diagonal.

I would also wrap most of the new decomposition methods using the most economical structure possible (with block diagonal matrix and permutation matrix) because

It is the most economical representation.
It loses a lot of usefulness after people start to run some search algorithms to find block structures or permutation from the explicit result, and it poorly utilizes some intermediate results. (I saw this was done poorly in jordan-decomposition related methods)

But someone would find these stuff unintuitive or difficult to use since I don't think that matrix expressions API is more famous than Matrix.
In the examples, I have just used as_explicit because it prints block diagonal matrix into some awkward forms.

Other comments

Release Notes

matrices
- Added connected_components and connected_components_decomposition for matrix which decomposes a matrix into a block diagonal form.
- Added todok function to find dictionary of keys format from any dense or sparse matrices.
- Added BlockDiagMatrix.get_diag_blocks to provide an user API to get diagonal blocks from the matrix.

sympy-bot · 2020-04-01T10:30:04Z

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matrices
- Added connected_components and connected_components_decomposition for matrix which decomposes a matrix into a block diagonal form. (#19045 by @sylee957)
- Added todok function to find dictionary of keys format from any dense or sparse matrices. (#19045 by @sylee957)
- Added BlockDiagMatrix.get_diag_blocks to provide an user API to get diagonal blocks from the matrix. (#19045 by @sylee957)

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#16207

#### Brief description of what is fixed or changed

I see there were some needs of using `connected_components`.
So I wrap two APIs for this first for finding connected components and second for finding a similar matrix which is block diagonal.

I would also wrap most of the new decomposition methods using the most economical structure possible (with block diagonal matrix and permutation matrix) because
1. It is the most economical representation.
2. It loses a lot of usefulness after people start to run some search algorithms to find block structures or permutation from the explicit result, and it poorly utilizes some intermediate results. (I saw this was done poorly in jordan-decomposition related methods)

But someone would find these stuff unintuitive or difficult to use since I don't think that matrix expressions API is more famous than `Matrix`.
In the examples, I have just used `as_explicit` because it prints block diagonal matrix into some awkward forms.

#### Other comments

#### Release Notes

<!-- Write the release notes for this release below. See
https://github.com/sympy/sympy/wiki/Writing-Release-Notes for more information
on how to write release notes. The bot will check your release notes
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<!-- BEGIN RELEASE NOTES -->

- matrices
  - Added `connected_components` and `connected_components_decomposition` for matrix which decomposes a matrix into a block diagonal form.
  - Added `todok` function to find dictionary of keys format from any dense or sparse matrices.
  - Added `BlockDiagMatrix.get_diag_blocks` to provide an user API to get diagonal blocks from the matrix.
<!-- END RELEASE NOTES -->

Update

The release notes on the wiki have been updated.

sympy-bot · 2020-04-01T10:30:05Z

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codecov · 2020-04-01T11:53:55Z

Codecov Report

Merging #19045 into master will increase coverage by 0.049%.
The diff coverage is 95.238%.

@@              Coverage Diff              @@
##            master    #19045       +/-   ##
=============================================
+ Coverage   75.737%   75.787%   +0.049%     
=============================================
  Files          648       649        +1     
  Lines       168798    168852       +54     
  Branches     39786     39798       +12     
=============================================
+ Hits        127844    127969      +125     
+ Misses       35387     35319       -68     
+ Partials      5567      5564        -3

sympy/matrices/graph.py

oscarbenjamin · 2020-04-01T13:01:10Z

sympy/matrices/graph.py

+            if M[i, i] is M.zero:
+                E.append((i, i))
+
+    return strongly_connected_components((V, E))


This should probably be connected_components in which case duplicate edges don't need to be added here.

With todok this can just be:

return connected_components((range(N), M.todok()))

Now, the test fails in python3.5 probably because of dictionary ordering. Does connected_components make unique results? If not, I'm thinking of sorting the output.

The results returned from connected_components can be in any order. The strongly_connected_components function returns components in topological order which is also not unique.

oscarbenjamin · 2020-04-01T13:07:53Z

What is the expected way to get the blocks from a block matrix. I see I can use args but I expected blocks to do it except:

In [27]: A                                                                                                                        
Out[27]: 
⎡a  0  b  0⎤
⎢          ⎥
⎢0  e  0  f⎥
⎢          ⎥
⎢c  0  d  0⎥
⎢          ⎥
⎣0  g  0  h⎦

In [28]: P, B = A.connected_components_decomposition()                                                                            

In [29]: B                                                                                                                        
Out[29]: 
⎡⎡a  b⎤        ⎤
⎢⎢    ⎥    𝟘   ⎥
⎢⎣c  d⎦        ⎥
⎢              ⎥
⎢        ⎡e  f⎤⎥
⎢  𝟘     ⎢    ⎥⎥
⎣        ⎣g  h⎦⎦

In [30]: B.blocks                                                                                                                 
Out[30]: 
⎡⎡a  b⎤        ⎤
⎢⎢    ⎥    𝟘   ⎥
⎢⎣c  d⎦        ⎥
⎢              ⎥
⎢        ⎡e  f⎤⎥
⎢  𝟘     ⎢    ⎥⎥
⎣        ⎣g  h⎦⎦

In [31]: B.args                                                                                                                   
Out[31]: 
⎛⎡a  b⎤  ⎡e  f⎤⎞
⎜⎢    ⎥, ⎢    ⎥⎟
⎝⎣c  d⎦  ⎣g  h⎦⎠

There also seems to be a bug here:

In [33]: P*B                                                                                                                      
---------------------------------------------------------------------------
AttributeError
...
~/current/sympy/sympy/sympy/matrices/expressions/blockmatrix.py in <listcomp>(.0)
    331 
    332     def _eval_inverse(self, expand='ignored'):
--> 333         return BlockDiagMatrix(*[mat.inverse() for mat in self.args])
    334 
    335     def _eval_transpose(self):

AttributeError: 'MutableDenseMatrix' object has no attribute 'inverse'
...

I guess that should be inv rather than inverse.

sylee957 · 2020-04-01T13:40:18Z

I see blocks was reserved to align with the same properties BlockMatrix, so it always returns some storage of immutable matrix.

sympy/sympy/matrices/expressions/blockmatrix.py

Lines 306 to 312 in 08534bb

    
           def blocks(self): 
        
               from sympy.matrices.immutable import ImmutableDenseMatrix 
        
               mats = self.args 
        
               data = [[mats[i] if i == j else ZeroMatrix(mats[i].rows, mats[j].cols) 
        
                               for j in range(len(mats))] 
        
                               for i in range(len(mats))] 
        
               return ImmutableDenseMatrix(data, evaluate=False)

I think that the only thing api that came before to serve the purpose is get_diag_blocks so if we don't want users to use args, it can be made for generic matrix expression API.

smichr · 2020-04-03T16:59:55Z

This is looking good. I hope some linear algebra users will smile after seeing this. I'll leave you and Oscar to figure out when it is ready.

oscarbenjamin · 2020-04-03T17:29:24Z

The reason I ask about args is because looking at this the first thing that comes to my mind is how do I actually get the blocks if I want to use them for something. I think that the doc should explain how to use this. The purpose of getting the blocks is so that you can compute eigenvalues etc so you need the block as a matrix on its own. Putting it altogether in a blockmatrix as a decomposition is nice but in practice not what is wanted in a situation where you would want to use this.

oscarbenjamin · 2020-04-03T21:42:38Z

Looks good.

Add connected_components for matrix API

f1f4226

sylee957 added the matrices label Apr 1, 2020

oscarbenjamin reviewed Apr 1, 2020

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oscarbenjamin reviewed Apr 1, 2020

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oscarbenjamin mentioned this pull request Apr 1, 2020

Support for reducible Markov chains #19011

Open

sylee957 added 5 commits April 2, 2020 05:45

Add todok helper

a20947b

Fix typo

c8a806b

Fix flake8

fe479c1

Simplify code and test case

795314f

Sort dict before passing to connected_components

3f391af

sylee957 added 2 commits April 4, 2020 03:13

Add get_diag_blocks for BlockDiagMatrix

5782bd9

Improve docs for connected_components

de4d830

oscarbenjamin merged commit 7be4fe6 into sympy:master Apr 3, 2020

sylee957 deleted the matrix_connected_components branch April 4, 2020 10:47

oscarbenjamin mentioned this pull request Apr 28, 2020

Improve 2x2 block matrix inversion with off-diagonal invertible blocks #19215

Merged

sylee957 mentioned this pull request May 14, 2021

Use strongly_connected_components for block triangulation of matrix #21470

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Matrix connected components decomposition #19045

Matrix connected components decomposition #19045

sylee957 commented Apr 1, 2020 •

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oscarbenjamin Apr 1, 2020

oscarbenjamin Apr 1, 2020

sylee957 Apr 2, 2020

oscarbenjamin Apr 2, 2020

oscarbenjamin commented Apr 1, 2020

sylee957 commented Apr 1, 2020

smichr commented Apr 3, 2020

oscarbenjamin commented Apr 3, 2020

oscarbenjamin commented Apr 3, 2020

Matrix connected components decomposition #19045

Matrix connected components decomposition #19045

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sylee957 commented Apr 1, 2020 • edited

References to other Issues or PRs

Brief description of what is fixed or changed

Other comments

Release Notes

sympy-bot commented Apr 1, 2020 • edited

sympy-bot commented Apr 1, 2020 • edited

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codecov bot commented Apr 1, 2020 • edited

Codecov Report

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sylee957 commented Apr 1, 2020 •

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