Factorization in N Sample Clauses

Factorization in N. Factorization in N is a little harder to see, but we can make use of the trace operators to see a factorization in N in the bound [25]. We need to rewrite some terms in the bound in order to see the factorization in N : Ψ0 = tr (KFF) = Σ(k(Xi⋅, Xi⋅)) t = Σ( ( )t) ∈ ×i i⋅1 N Ψ Y k X , Z y RM D N i=1 Ψ2 = Σ(k(Z, Xi⋅)k(Xi⋅, Z)t) ∈ RM×M N i=1 KL (q(X)||p(X)) = ΣKL (q(Xi⋅)||p(Xi⋅)) .(if q(Xi⋅) factorizes) Ly This makes the four parts above that factorize in N . We can implement workers to work on sub parts of the data in N . The workers return partial sums of the four parts. The master will then sum those parts together and calculate the partial derivatives for ∂ 3 ∂Ψ1 Y and ∂L3 . The workers can then calculate the full derivatives ∂Ψ for the kernel parameters θ, likelihood precision β and inducing inputs Z. Here, the partial gradients w.r.t. Z factorize in N again and can be computed on the worker nodes. The partial gradients get sent back to the master and the master sums them up, updating the gradients of the global parameters. While the master sums the gradients for the global parameters, the workers can update the gradients for their local parameters M and S, if in a Bayesian GPLVM setting.