X
|
Data point, X ∈
|
n
|
Number of data points |
Ys
|
Label for one of the t classification tasks, Ys ∈ {1,...,ks} |
t
|
Number of tasks |
Y
|
Vector of task labels Y = [Y1, Y2,...,Yt]T
|
r
|
Cardinality of the output space, r = || |
G
task
|
Task structure graph |
|
Output space of allowable task labels defined by Gtask, Y ∈
|
|
Distribution from which we assume (X, Y) data points are sampled i.i.d. |
si
|
Weak supervision source, a function mapping X to a label vector |
|
Label vector ∈ output by the ith source for X
|
m
|
Number of sources |
λ
|
m × t matrix of labels output by the m sources for X
|
0
|
Source output space, which is augmented to include elements set to zero |
τi
|
Coverage set of - the tasks si gives non-zero labels to; for convenience, τ0 = {1,…,t} |
τi
|
The output space for given coverage set τi
|
|
The output space with all but the first value, for defining a minimal set of statistics |
G
source
|
Source dependency graph, Gsource = (V, E), V = {Y, ,...,} |
|
Cliqueset (maximal and non-maximal) of Gsource
|
|
The maximal cliques (nodes) and separator sets of the junction tree over Gsource
|
Ψ(C, yC) |
The indicator variable for the variables in clique C ∈ taking on values yC, (yC)i ∈ τi
|
μ
|
The parameters of our label model we aim to estimate;
|
O
|
The set of observable cliques, i.e. those corresponding to cliques without Y
|
Σ |
Generalized covariance matrix of O ⊆ , Σ ≡ Cov [Ψ(O ⊆ )] |
K
|
The inverse generalized covariance matrix K = Σ−1
|
dO, d
|
The dimensions of O and respectively |
G
aug
|
The augmented source dependencies graph Gaug = (Ψ, Eaug) |
Ω |
The edge set of the inverse graph of Gaug
|
P
|
Diagonal matrix of class prior probabilities, P(Y) |
Pμ (Y, λ) |
The label model parameterized by μ
|
|
The probabilistic training label, i.e. Pμ(Y|λ) |
fw (X) |
The end model trained using (X, ) |