Support for Sparse Algebras

In certain high-dimensional applications we only make use of a subspace of grades. For example the Cl(n) Clifford algebra grows in dimensionality by O(2^n) while using only blades of grade k = 1 and k = n - 1 grows linearly by O(2n).

I wrote a quick naive implementation just to test initiation times, the kr argument is used to specify a list of desired grades.

https://github.com/Stefan-Endres/clifford/commit/81c78c3cd36b3d952ec652072167af786d874116

>>> cf.Cl(4, kr=[1, 3])
(Layout([1, 1, 1, 1], [(), (1,), (2,), (3,), (4,), (1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)], 
firstIdx=1, 
names=['', 'e1', 'e2', 'e3', 'e4', 'e123', 'e124', 'e134', 'e234']), 
{'e1': (1^e1), 'e2': (1^e2), 'e3': (1^e3), 'e4': (1^e4), 
'e123': (1^e123), 'e124': (1^e124), 'e134': (1^e134), 'e234': 
(1^e234)})

>>> cf.Cl(4)
(Layout([1, 1, 1, 1], [(), (1,), (2,), (3,), (4,), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4), (1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4), (1, 2, 3, 4)], 
firstIdx=1, 
names=['', 'e1', 'e2', 'e3', 'e4', 'e12', 'e13', 'e14', 'e23', 'e24', 'e34', 'e123', 'e124', 'e134', 'e234', 'e1234']), 
{'e1': (1^e1), 'e2': (1^e2), 'e3': (1^e3), 'e4': (1^e4), 
'e12': (1^e12), 'e13': (1^e13), 'e14': (1^e14), 'e23': (1^e23), 'e24': (1^e24), 'e34': (1^e34),
'e123': (1^e123), 'e124': (1^e124), 'e134': (1^e134), 'e234': (1^e234), 'e1234':
(1^e1234)})

Some basic operations such as mutlivector addition, grade projection etc. are preserved nicely. Others such as dual (in cases where the result is an initiated grade) and reflection are not.

The initiation time growth isn’t as promising as I’d hoped for either, but could possibly be drastically improved with a less naive implementation. In particular 99.1% of the initiation is spent in this loop in _getEvenOdd: https://github.com/pygae/clifford/blob/8793d3e4faf758770ae80d8d9e111574677772bc/clifford/__init__.py#L567

There is a comment in the code that states the loop runs for grade! permutations, however, at the moment it is unclear to me how the rest of the algebra and code will be affected when I change the range of these permutations or if it is even still looping for all the canonical grades at the moment. Especially when supplied with more arbitrary grade ranges.

Initiation times:

cf.Cl(4)
Initiation time: 0.22497177124023438
cf.Cl(4, kr=[1, 3]) 
Initiation time: 0.0009617805480957031
cf.Cl(5)
Initiation time: 0.009763717651367188
cf.Cl(5, kr=[1, 4]) 
Initiation time: 0.0020651817321777344
cf.Cl(6)
Initiation time: 0.0555272102355957
cf.Cl(6, kr=[1, 5]) 
Initiation time: 0.010433673858642578
cf.Cl(7)
Initiation time: 0.41026782989501953
cf.Cl(7, kr=[1, 6]) 
Initiation time: 0.08707642555236816
cf.Cl(8)
Initiation time: 4.242865324020386
cf.Cl(8, kr=[1, 7]) 
Initiation time: 1.0086712837219238
cf.Cl(9)
Initiation time: 61.97481608390808
cf.Cl(9, kr=[1, 8]) 
Initiation time: 15.875498056411743
# cf.Cl(10) did not initiate after several minutes 
f.Cl(10, kr=[1, 9]) 
Initiation time: 329.20880341529846

cProfile for cf.Cl(9, kr=[1, 8]:

Name	Call Count	Time (ms)	Own Time (ms)
init	1	23157	0
Cl	1	23157	0
_genEvenOdd	1	23155	973
modify_idx	362880	20229	11265
typeof	3265940	8964	1721
wrapper	3265942	6188	1311
_typeof_int	3265937	3374	1064
bit_length	3265937	2310	1436

I would be happy to contribute as much as possible under the direction of the developers.

Regards, Stefan Endres

[moble]

I’d just like to suggest that since Cl is pretty much the main callable in this library, it might be helpful to have a more descriptive name for the new argument. Maybe desired_grades or even just grades?

[hugohadfield]

Thanks for the table @enkimute interesting how fast the javascript startup times are! just added to my PR here: https://github.com/pygae/clifford/pull/43 this should bring the start up times down very significantly