Slide 14
Slide 14 text
14
Choice of the Tanimoto kernel (similarity between molecules and : )
kM
𝔪
𝔪
′

kM
(
𝔪
,
𝔪
′

)
Step1 : Embeddings for protein and molecule
ψP
ψM
Derived from kernel theory
kM
(
𝔪
,
𝔪
′

)
molecules in the training set
𝔪
𝔪
′

Nystrom approximation [Scholkopf et al, 1999]
Choice of landmarks molecules in the training set
mM
̂
𝔪
Compute the small kernel where
̂
KM
∈ ℝmM
×mM ( ̂
KM
)ℓ,t
:= kM
( ̂
𝔪
ℓ
, ̂
𝔪
t
)
From the SVD of ,
Compute the extrapolation matrix where
̂
KM
= U diag(σ)U⊤
E ∈ ℝmM
×dM E := U[: , : dM
]diag(σ−1/2
s
)dM
s=1
and dimension reduction
ψM
(
𝔪
) := (
mM
∑
ℓ=1
Eℓ,s
kM
( ̂
𝔪
ℓ
,
𝔪
))
dM
s=1
∈ ℝdM
𝔪
Molecule embedding
kM
( ̂
𝔪
ℓ
, ̂
𝔪
t
) = ⟨ψM
( ̂
𝔪
ℓ
), ψM
( ̂
𝔪
t
)⟩
kM
(
𝔪
, ̂
𝔪
t
) ≈ ⟨ψM
(
𝔪
), ψM
( ̂
𝔪
t
)⟩
If ,
dM
= mM