First, it should be determined that a direct division operation is impossible. To obtain the quotient, a multi-step mechanism should be used.
In the integer division scheme, we define this operation as an equation in a multidimensional space H, in which the main goal is to determine the cipher quotient hyperedge ec from the dividend hyperedge ea and the divisor hyperedge eb.
To simplify, we can define division as an operator equation:
F(ec)=Γϵ(Ha)−Γϵ(Hb)⊙ec=0
It is worth clarifying what each element of this equation means:
Γϵ(Ha) and Γϵ(Hb), are incidence matrices with noise;
⊙, a linear transformation that inverts the influence of Γϵ(Hb) on ec, which is constructed using the Moore–Penrose inverse.
To solve the division problem, it is necessary to define the mechanics of range narrowing as:
ec∈R={e∈Rn∣0≤e≤1}
This is done to ensure that the quotient hyperedge ec remains within the ranges of R.
The next step is to initialize the cipher hyperedge and then define the inverse operator for the incidence matrix of the divisor.
Cipher hyperedges with cubic noise elements:
ea=Γϵ(Ha)+ϵa3,eb=Γϵ(Hb)+ϵb3
Where ϵa3 and ϵb3 are the cubic noise vectors added to Ha and Hb.
Also introduce regularization parameters to reduce the noise increase due to small singular values at the points:
Γϵ(Hb)+=(Γϵ(Hb)⊤Γϵ(Hb)+αI)−1Γϵ(Hb)⊤
In this case, Γϵ(Hb)⊤ is a transposition of Γϵ(Hb), I is the identity matrix of the corresponding size.
As a result, the division operation can be expressed as:
ec=Γϵ(Hb)+⊙Γϵ(Ha)
Let's substitute the operator ⊙:
ec=T⋅S⋅Γϵ(Hb)+⋅Γϵ(Ha)
We continue to expand:
ec=T⋅S⋅(Γϵ(Hb)⊤Γϵ(Hb)+αI)−1Γϵ(Hb)⊤Γϵ(Ha)
A further goal is to ensure that ec is within the acceptable range R while minimizing the residual noise. That is, we literally apply the projection operator PR to limit ec:
Hope it's clear why, but if for some reason it is not clear then you can read our article on noise management (there we are talking about noise components after subtraction, but the logic is appropriate).
As was said earlier, to guarantee the stability of the hypergraph structure in the presence of cubic noise, the inequality must be satisfied for good strings with large r (more details were described ):