We introduce several problems related to simulations optimization of code and factor graph properties

1. Codes on the Graph related problems
Author: V.S. Usatyuk
Usatyuk.Vasily@huawei.com
Coding Competence Center,
Moscow Research Center, Huawei
Huawei Workshop
Steklov Institute of Mathematics at Saint Petersburg
October 16, 2017
V.S. Usatyuk,
Usatyuk.Vasily@huawei.com Saint-Petersburg, 2017 Page 1 from 9

2. Problem 1: Code optimization
Let an (n − k) × n matrix H over Fq define linear map
Fn
q → Fn−k
q , such that x → Hx. By
ker (H) = c ∈ Fn
q Hc = 0 denote the kernel of the
map.
Consider W (H) = {A0, A1, . . . , An}, where Ai is a
number of c ∈ ker (H) with Hamming weight i.
Define f (z) =
n
i=0
Aizi
the generating function.
How to calculate f (z) with reasonable time?
V.S. Usatyuk,
Usatyuk.Vasily@huawei.com Saint-Petersburg, 2017 Page 2 from 9

3. Problem 2: Cycles optimization
The matrix H defines the factor graph G.
How to enumerate all cycles in G with “reasonable”
complexity?
“reasonable” - O(N), O(N log N)
V.S. Usatyuk,
Usatyuk.Vasily@huawei.com Saint-Petersburg, 2017 Page 3 from 9

4. Problem 3: Graph optimization
The matrix H defines the factor graph G.
How to find “bad connected” cycles with reasonable time?
“bad connected”- ’local’ extrema of certain factor.
e.g. Trapping sets, number of odd degree induced factors;
’local’ - under single element removal, addition, or swap.
V.S. Usatyuk,
Usatyuk.Vasily@huawei.com Saint-Petersburg, 2017 Page 4 from 9

5. Problem 4: Factor graph optimization
Let Fn−k
q → Rn
, where C(Λ, R) = (Λ + t) ∩ R, Λ – lattice
The matrix H and APP sample λT
under some statistical
model (e.g., AWGN) define Marginal polytope M,
M = c : min λT
c, Hc = 0 .
How to enumerate Euclidean weight of all vector
c ∈ M ker (H) or its parts, when LP relaxation finds a
non-integer solution with minimal number of samples?
’non-integer solution’ - URL: ML certificate, ’LP relaxation’- URL:
Tree-based relaxation for factor graphs with cycles
V.S. Usatyuk,
Usatyuk.Vasily@huawei.com Saint-Petersburg, 2017 Page 5 from 9

6. Problem 5: Factor graph optimization
The matrix H defines the normal factor graph Gnormal.
How to obtain optimal cut-set conditioning and graph
traversal (scheduler) which meet the condition of
’parallelism’ level P with reasonable time?
’parallelism’ - message process between layer of normal graph could
be done in P parallel operation
V.S. Usatyuk,
Usatyuk.Vasily@huawei.com Saint-Petersburg, 2017 Page 6 from 9

7. Problem 6: Probabilistic Graphical Models
The dense matrix H defines the normal factor graph
Gnormal.
Can there exist a method* to solve statistical inference
problem under Gnormal with linear complexity?
*except cut-set conditioning(Polar codes/Polar subcodes) and
junction tree algorithm with pruning (NB-LDPC/GLDPC)
V.S. Usatyuk,
Usatyuk.Vasily@huawei.com Saint-Petersburg, 2017 Page 7 from 9

8. Problem 7: Probability code optimization
We have Oracle (Genie) who with probability 1
2 + ε
provide whole or part of f (z) and some optimization
method which improve f (z).
How many number of Oracle’s request require to construct
fgood(z) ≈ fbest(z)?
’best’ - based on Singleton and Union Bounds
V.S. Usatyuk,
Usatyuk.Vasily@huawei.com Saint-Petersburg, 2017 Page 8 from 9

9. Thank you for you attention !
V.S. Usatyuk,
Usatyuk.Vasily@huawei.com Saint-Petersburg, 2017 Page 9 from 9