Message passing between Ptensors¶
Hands et al. [] extended the notion of message passing between permutation equivariant tensors
(Ptensors) to the case when the reference domains of the source and destination tensors
are not necessarily the same. In this case the number of possible linearly independent
linear maps increases because for each summation or broadcast operation we can consider summing/broadcasting
over all elements of reference domain or only the intersection of the reference domains of the sending
and receiving P-tensors.
These linear maps in ptens are called gather operations.
Gather maps between Ptensors¶
gather0¶
Similarly to linmaps0, the gather0 function passes equivariant linear messages to a zeroth order Ptensor.
In contrast to linmaps0, however, the reference domain of the output must be specified explicitly.
The only possible equivariant map from a zeroth order Ptensor to a zeroth order Ptensor is a multiple of the identity:
>> A=ptens.ptensor0.sequential([1,2,3],5)
>> print(A)
Ptensor0(1,2,3):
[ 0 1 2 3 4 ]
>> B=ptens.linmaps0(A)
>> print(B)
Ptensor0(1,2,3):
[ 0 1 2 3 4 ]
A first order Ptensor can gather information to a zeroth order Ptensor by extracting its slice corresponding to the reference domain of the latter:
>> A=ptens.ptensor1.sequential([1,2,3],3)
>> A
Ptensor1 [1,2,3]:
[ 0 1 2 ]
[ 3 4 5 ]
[ 6 7 8 ]
>> B=ptens.gather0(A,[2])
>> B
Ptensor0 [2]:
[ 3 4 5 ]
A second order Ptensor can gather information to a zeroth order Ptensor either by summing the entire block corresponding to the intersection of their reference domains, or just its diagonal:
>> A=ptens.ptensor2.sequential([1,2,3],3)
>> A
Ptensor2(1,2,3):
channel 0:
[ 0 3 6 ]
[ 9 12 15 ]
[ 18 21 24 ]
channel 1:
[ 1 4 7 ]
[ 10 13 16 ]
[ 19 22 25 ]
channel 2:
[ 2 5 8 ]
[ 11 14 17 ]
[ 20 23 26 ]
>> B=ptens.gather0(A,[2])
>> B
Ptensor0(2):
[ 12 13 14 12 13 14 ]
gather1¶
When a message is gatherred from a zeroth order Ptensor to a first order Ptensor, effectively it is just copied into the row corresponding to the intersection of the reference domains:
>> A=ptens.ptensor0.sequential([2],3)
>> A
Ptensor0(2):
[ 0 1 2 ]
>> B=ptens.gather1(A,[2,3])
>> B
Ptensor1(2,3):
[ 0 1 2 ]
[ 0 0 0 ]
A message from a first order Ptensor to a first order Ptensor consists of the concatenation of two maps: copying to the intersection and broadcasting the sum over the elements of the intersection:
>> A=ptens.ptensor1.sequential([1,2,3],3)
>> A
Ptensor1(1,2,3):
[ 0 1 2 ]
[ 3 4 5 ]
[ 6 7 8 ]
>> B=ptens.gather1(A,[2,3,5])
>> B
Ptensor1 [2,3,5]:
[ 9 11 13 3 4 5 ]
[ 9 11 13 6 7 8 ]
[ 0 0 0 0 0 0 ]
When a message is passed from a second order Ptensor to a first order Ptensor we have 5 possible linear maps, hence the number of channels is multiplied by five.
>> A=ptens.ptensor2.sequential([1,2,3],3)
>> A
Ptensor2 [1,2,3]:
channel 0:
[ 0 3 6 ]
[ 9 12 15 ]
[ 18 21 24 ]
channel 1:
[ 1 4 7 ]
[ 10 13 16 ]
[ 19 22 25 ]
channel 2:
[ 2 5 8 ]
[ 11 14 17 ]
[ 20 23 26 ]
>> B=ptens.gather1(A,[2,3,5])
>> B
Ptensor1 [2,3,5]:
[ 72 76 80 36 38 40 33 35 37 27 29 31 12 13 14 ]
[ 72 76 80 36 38 40 39 41 43 45 47 49 24 25 26 ]
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]
gather2¶
Similarly to linmaps, the number of possible gathers maps from zeroth, first and second order Ptensors to second order Ptensors is 2,5 and 15, respectively:
>> A=ptens.ptensor0.sequential([2],3)
>> A
Ptensor0 [2]:
[ 0 1 2 ]
>> B=ptens.gather2(A,[2,3,5])
>> B
Ptensor2 [2,3,5]:
channel 0:
[ 0 0 0 ]
[ 0 0 0 ]
[ 0 0 0 ]
channel 1:
[ 1 0 0 ]
[ 0 0 0 ]
[ 0 0 0 ]
channel 2:
[ 2 0 0 ]
[ 0 0 0 ]
[ 0 0 0 ]
channel 3:
[ 0 0 0 ]
[ 0 0 0 ]
[ 0 0 0 ]
channel 4:
[ 1 0 0 ]
[ 0 0 0 ]
[ 0 0 0 ]
channel 5:
[ 2 0 0 ]
[ 0 0 0 ]
[ 0 0 0 ]
>> A=ptens.ptensor1.sequential([1,2,3],3)
>> A
Ptensor1 [1,2,3]:
[ 0 1 2 ]
[ 3 4 5 ]
[ 6 7 8 ]
>> B=ptens.gather2(A,[2,3,5])
>> B
Ptensor2 [2,3,5]:
channel 0:
[ 9 9 0 ]
[ 9 9 0 ]
[ 0 0 0 ]
channel 1:
[ 11 11 0 ]
[ 11 11 0 ]
[ 0 0 0 ]
channel 2:
[ 13 13 0 ]
[ 13 13 0 ]
[ 0 0 0 ]
channel 3:
[ 9 0 0 ]
[ 0 9 0 ]
[ 0 0 0 ]
channel 4:
[ 11 0 0 ]
[ 0 11 0 ]
[ 0 0 0 ]
channel 5:
[ 13 0 0 ]
[ 0 13 0 ]
[ 0 0 0 ]
channel 6:
[ 3 6 0 ]
[ 3 6 0 ]
[ 0 0 0 ]
channel 7:
[ 4 7 0 ]
[ 4 7 0 ]
[ 0 0 0 ]
...
>> A=ptens.ptensor2.sequential([1,2,3],3)
>> A
Ptensor2 [1,2,3]:
channel 0:
[ 0 3 6 ]
[ 9 12 15 ]
[ 18 21 24 ]
channel 1:
[ 1 4 7 ]
[ 10 13 16 ]
[ 19 22 25 ]
channel 2:
[ 2 5 8 ]
[ 11 14 17 ]
[ 20 23 26 ]
>> B=ptens.gather2(A,[2,3,5])
>> B
Ptensor2 [2,3,5]:
channel 0:
[ 72 72 0 ]
[ 72 72 0 ]
[ 0 0 0 ]
channel 1:
[ 76 76 0 ]
[ 76 76 0 ]
[ 0 0 0 ]
channel 2:
[ 80 80 0 ]
[ 80 80 0 ]
[ 0 0 0 ]
channel 3:
[ 36 36 0 ]
[ 36 36 0 ]
[ 0 0 0 ]
channel 4:
[ 38 38 0 ]
[ 38 38 0 ]
[ 0 0 0 ]
channel 5:
[ 40 40 0 ]
[ 40 40 0 ]
[ 0 0 0 ]
channel 6:
[ 72 0 0 ]
[ 0 72 0 ]
[ 0 0 0 ]
channel 7:
[ 76 0 0 ]
[ 0 76 0 ]
[ 0 0 0 ]