A Tensor-Product-Kernel Framework for Multiscale Neural Activity Decoding and Control

Algorithm 1

Quantized kernel least mean square (QKLMS) algorithm.

Input: {𝑥_𝑛,𝑑_𝑛},𝑛=1,2,…,𝑁

Initialization: initialize the weight vector Ω(1): codebook (set of centers) 𝒞(0)={} and coefficient vector 𝑎(0)=[]

Computation:

For  𝑛=1,2,…,𝑁

(1) compute the output

𝑦_𝑛=⟨Ω(𝑛),𝜙(𝑥_𝑛)⟩=∑^{size(𝒞(𝑛−1))}_𝑗=1𝑎_𝑗(𝑛−1)𝜅(𝑥_𝑛,𝒞_𝑗(𝑛−1))

(2) compute the error, 𝑒_𝑛=𝑑_𝑛−𝑦_𝑛

(3) compute the minimum distance in RKHS between 𝑥_𝑛 and each 𝑥_𝑖∈𝒞(𝑛−1),

𝑑(𝑥_𝑛)=min_𝑗(2−2𝜅(𝑥_𝑛,𝒞_𝑗(𝑛−1)))

(4) if 𝑑(𝑥_𝑛)≤𝜀, then keep the codebook unchanged: 𝒞(𝑛)=𝒞(𝑛−1), and update the coefficient of the center closest to 𝑥_𝑛:

𝑎_𝑘(𝑛)=𝑎_𝑘(𝑛−1)+𝜂𝑒(𝑛), where 𝑘=argmin_𝑗2−2𝜅(𝑥_𝑛,𝒞_𝑗(𝑛−1))

(5) otherwise, store the new center: 𝒞(𝑛)={𝒞(𝑛−1),𝑥_𝑛}, 𝑎(𝑛)=[𝑎(𝑛−1),𝜂𝑒_𝑛]

(6) Ω(𝑛+1)=∑^{size(𝒞(𝑛))}_𝑗=1𝑎_𝑗(𝑛)𝜙(𝒞_𝑗(𝑛))

end