Matrix Factorization for Evolution Data

Algorithm 1

MAFED(𝑅,𝑑,𝜂₁,𝜂₂,𝜂₃,𝛼,𝛽,𝜆,𝜖).

Input: matrix 𝑅; number of the latent features 𝑑; learning rates 𝜂₁, 𝜂₂ and 𝜂₃;

   regularization parameters 𝛼, 𝛽 and 𝜆; threshold 𝜖

Output: the estimated matrix 𝑈^𝑇𝑉.

  // Initialize  𝑈  and  𝑉.

(1) Generate random vectors 𝑈₁,𝑈₂,…,𝑈_𝑛,𝑉₁∼𝐍(𝟎,𝐼);

(2) for   𝑗=2; 𝑗≤𝑡; 𝑗++   do

(3)  Generate 𝑍 with 𝐍(𝟎,𝐼)

(4)  Let 𝑉_𝑗=𝑉_𝑗−1+𝑍

(5) end

  // Coordinate descent.

(6) 𝑆₁=∑^{𝑖=𝑛,𝑗=𝑡}_{𝑖=1,𝑗=1}𝑅_𝑖,𝑗−𝑈^𝑡_𝑖𝑉_𝑗²+𝛼∑^𝑛_𝑖=1𝑈_𝑖²₂+𝛽𝑉₁²₂+𝜆∑^𝑡−1_𝑗=1𝑉_𝑗+1−𝑉_𝑗²₂;

(7) 𝑆₂=inf;

(8) while   𝑆₂−𝑆₁>𝜖   do

(9)    𝑆₂=𝑆₁;

(10)   for   𝑖=1,2,…,𝑛   do

(11)    Let 𝑈^new_𝑖=𝑈_𝑖−𝜂₁𝜕𝑆/𝜕𝑈_𝑖;

(12)   end

(13)   𝑉^new₁=𝑉₁−𝜂₂𝜕𝑆𝜕𝑉₁;

(14)   for   𝑗=2,…,𝑡   do

(15)    Let 𝑉^new_𝑗=𝑉_𝑗−𝜂₃𝜕𝑆/𝜕𝑉_𝑗;

(16)   end

(17)  Replace the all 𝑈_𝑖𝑠 with 𝑈^new_𝑖𝑠, and 𝑉_𝑗𝑠 with 𝑉^new_𝑗𝑠, recompute 𝑆₁;

(18) end

(19) return   𝑈^𝑇𝑉;