Slide 21
Slide 21 text
• ཚԽϑʔϦΤಛʢRandom Fourier Features, RFFʣ[5] ɺ͋Δ֬ ͔Βͷ

ݸͷαϯϓϧΛ༻͍ͯΧʔωϧؔ

Λ

ͱۙࣅ͢Δख๏
• ͳ͓ɺ

• ֬

ΧʔωϧؔͷछྨʹΑܾͬͯ·Δ
p(ω)
R′

= R/2 k(xi
, xj
) ̂
k(xi
, xj
) = z(xi
)⊤z(xj
)
z(xi
) = 1/R′

(cos(ω⊤
1
xi
), sin(ω⊤
1
xi
), …, cos(ω⊤
R′

xi
), sin(ω⊤
R′

xi
))
p(ω)

21
ରࡦᶃɿΧʔωϧߦྻͷܭࢉෛՙͷରॲ
[5] Miguel L ́azaro-Gredilla, Joaquin Quinonero-Candela, Carl Edward Rasmussen, and An ́ıbal R Figueiras-Vidal. Sparse spectrum gaussian process regression. The Journal of Machine Learning Research, Vol. 11, pp. 1865– 1881, 2010.

K

K ∈ ℝN×N

k(xi
, xj
)

K

ZZ⊤

≃

K ∈ ℝN×N

Z ∈ ℝN×R

Z⊤Z

Z⊤Z ∈ ℝR×R

⋙
k(xi
, xj
) ≃ z(xi
)⊤z(xj
)
ݸผͷΧʔωϧؔʹରͯ͠ܭࢉίετ͕
૿Ճ͢Δ͕ɺجఈؔͷద༻ͱੵͷࠞ߹ૢ
࡞ͷ݁ՌΛղͨ͠ͱݟΔ͜ͱ͕Ͱ͖Δ
→ ύϥϝʔλͷ࣍ݩΛ

࣍ݩʹݻఆͰ͖Δ
R
ϕ(x) = (ϕ1
(x), …, ϕ∞
(x))⊤ ∈ ℝ∞
ϕ(xi
)⊤ϕ(xj
) = k(xi
, xj
) ≈ z(xi
)⊤z(xj
)