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次元削減(主成分分析・線形判別分析・カーネル主成分分析)

 次元削減(主成分分析・線形判別分析・カーネル主成分分析)

研究室内発表用の資料です.

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Takahiro Kawashima

June 03, 2020
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  1. ࣍ݩ࡟ݮ M2 ઒ౡوେ June 03, 2020

  2. ໨࣍ 1. ΠϯτϩμΫγϣϯɿ࣍ݩ࡟ݮ 2. ओ੒෼෼ੳ 3. ઢܗ൑ผ෼ੳ 4. Χʔωϧओ੒෼෼ੳ 2

  3. ΠϯτϩμΫγϣϯɿ࣍ݩ࡟ݮ

  4. ߴ࣍ݩσʔλͷ೰ΈͲ͜Ζ σʔλͷ࣍ݩ͕େ͖͘ͳΔͱʜʜ X ղऍੑ͕མͪΔ X σʔλಉ࢜ͷؔ܎ΛՄࢹԽͮ͠Β͍ X ʢ΄΅ʣઢܗैଐͳ͕࣠૿͑Δʢଟॏڞઢੑʣ X ML

    ΞϧΰϦζϜͷ݁Ռ͕ෆ҆ఆੑͷ΋ͱʢLasso ͳͲʣ X ྼܾఆͷΞϧΰϦζϜ͕ग़ͯ͘Δ X ࠷খೋ৐๏͸ σʔλ਺ ≥ ࣍ݩ਺ Ͱͳ͍ͱղ͚ͳ͍ X ͦͷଞ͍Ζ͍Ζ ˠ ಘΒΕͨߴ࣍ݩσʔλͷಛੑΛͳΔ΂͘อ࣋ͨ͠௿࣍ݩදݱΛٻ ΊΒΕΕ͹ղܾʂ 3
  5. ߴ࣍ݩσʔλͷ “ಛੑ”ʁ ߴ࣍ݩσʔλͷͲΜͳ “ಛੑ” Λอ͍͔࣋ͨ͠ʹԠͯ͡ɼ࣍ݩ࡟ ݮͷख๏͸༷ʑɽࠓճѻ͏ͷ͸ ڭࢣͳ͠ɿ X ओ੒෼෼ੳɿσʔλͷ࣠ํ޲ͷ͹Β͖ͭΛ৘ใͷ๛͔ͩ͞ͱ ղऍ͠ɼ࠷େԽ

    X Χʔωϧओ੒෼෼ੳɿओ੒෼෼ੳΛ͞ΒʹඇઢܗԽ ڭࢣ͋Γɿ X ઢܗ൑ผ෼ੳɿ࣍ݩ࡟ݮޙͷۭؒͰͷઢܗ൑ผثͷੑೳΛ࠷ େԽ ଞʹ΋ Isomap, LLE, t-SNE, KDR ͳͲॏཁͳख๏͸͍Ζ͍Ζ 4
  6. ओ੒෼෼ੳ

  7. ओ੒෼෼ੳɿ෼ࢄ࠷େԽͷߟ͑ํ 軸方向の分散を最大化 σʔλͷ͹Β͖ͭ (⇔ ෼ࢄ) ͕େ ͖͘ͳΔ࣠͸৘ใΛଟؚ͘ΜͰ ͍Δ͸ͣ ˣ 1.

    ࠷΋σʔλ෼ࢄͷେ͖͘ͳ Δ࣠ u1 Λ୳͢ 2. u1 ʹ௚ަ͢Δ΋ͷͷ͏ͪ࣍ ʹσʔλ෼ࢄ͕େ͖͘ͳΔ ࣠ u2 Λ୳͢ 3. ಉ༷ʹଓ͚͍ͯ͘ ಘΒΕͨ {ui} ͷ͏ͪɼখ͍͞ i ʹରԠ͢Δ࣠͸σʔλͷ৘ใΛΑ ͘ଊ͍͑ͯΔ͸ͣ ͜ͷํ๏Λओ੒෼෼ੳ (PCA; Principal Component Analysis) ͱ ݺͿ 5
  8. ओ੒෼෼ੳɿ෼ࢄ࠷େԽʹΑΔఆࣜԽ (1) αϯϓϧू߹ɿ{xn}N n=1 , xn ∈ RD ໨తɿ{xn} ͷ෼ࢄΛ࠷΋Α͘ଊ͑Δ࣠

    u1 ∈ RD Λ୳͢ ࣠ํ޲͚ܾͩΊΕ͹Α͍ͷͰɼ∥u1∥ = (u⊺ 1 u1)1/2 = 1ͷ৚݅Λ௥Ճ {xn} ͷฏۉɾڞ෼ࢄߦྻΛఆٛɿ ¯ x ∶= 1 N N ∑ n=1 xn ∈ RD, S ∶= 1 N N ∑ n=1 (xn − ¯ x)(xn − ¯ x)⊺ ∈ RD×D. 6
  9. ࢉ਺ͷ෮शɿϢʔΫϦου಺ੵ ϢʔΫϦου಺ੵʹؔͯ͠ ͱ͘ʹ ∣a∣ = 1 ͷͱ͖ɼb ͷ a ΁

    ͷࣹӨઌͷ஋͸ ∣bcosθ∣ = ∣b∣cosθ = a ⋅ b 7
  10. ओ੒෼෼ੳɿ෼ࢄ࠷େԽʹΑΔఆࣜԽ (2) ∥u1∥ = 1ͳͷͰɼ఺ xn ∈ RD ͷ࣠ u1

    ∈ RD ΁ͷࣹӨઌ͸ u⊺ 1 xn Αͬͯ u1 ΁ࣹӨޙͷσʔλͷ෼ࢄ͸ 1 N N ∑ i=1 (u⊺ 1 xn − u⊺ 1 ¯ x)2 = 1 N N ∑ i=1 (u⊺ 1 xnx⊺ n u1 − 2u⊺ 1 xn ¯ x⊺u1 + u⊺ 1 ¯ x¯ x⊺u1) = u⊺ 1 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 N N ∑ i=1 (xnx⊺ n − 2xn ¯ x⊺ + ¯ x¯ x⊺) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ u1 = u⊺ 1 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 N N ∑ i=1 (xn − ¯ x)(xn − ¯ x)⊺ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ u1 = u⊺ 1 Su1. ࠷ӈล u⊺ 1 Su1 Λ৚݅ u⊺ 1 u1 = 1 ͷ΋ͱ u1 ʹؔͯ͠࠷େԽʂ 8
  11. ओ੒෼෼ੳɿ෼ࢄ࠷େԽʹΑΔఆࣜԽ (3) ղ͖͍ͨ࠷େԽ໰୊͸ҎԼɿ Maximize u⊺ 1 Su1 subject to 1

    − u⊺ 1 u1 = 0 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ͜ΕΛ Lagrange ૒ର໰୊ʹམͱ͠ࠐΉͱ Maximize L(u1,λ1) = u⊺ 1 Su1 + λ1(1 − u⊺ 1 u1). ໨తؔ਺͸ u1 ʹؔͯ͠ 2 ࣍ܗࣜͰತͳͷͰɼͨΜʹۃ஋ΛٻΊ Ε͹࠷େԽͰ͖Δɿ ∂L(u1,λ) ∂u1 = 2Su1 − 2λ1u1 = 0 ⇔ Su1 = λ1u1. 9
  12. ओ੒෼෼ੳɿ෼ࢄ࠷େԽʹΑΔఆࣜԽ (4) Lagrange ૒ର໰୊Λղ͘ͱɼ෼ࢄ࠷େԽ໰୊͸ Su1 = λ1u1 ͷܗʹؼண͞Εͨɽ∥u1∥ = 1

    ͩͬͨ͜ͱΛࢥ͍ग़͢ͱɼ σʔλ {xn} ͷڞ෼ࢄߦྻ S ʹؔ͢Δݻ༗஋໰୊ʹͳͬͯΔʂ ͞Βʹ྆ลʹࠨ͔Β u⊺ 1 Λֻ͚Δͱɼu⊺ 1 u1 = 1 ΑΓ Su1 = λ1u1 ⇔ u⊺ 1 Su1 = λ1 ˠ ࠷େݻ༗஋ λ1 ͸σʔλͷୈҰओ੒෼࣠ u1 ํ޲ͷ෼ࢄʂ Ұൠʹୈ m ओ੒෼࣠ um ͸ S ͷୈ m ݻ༗஋ʹରԠ͢Δݻ༗ϕΫ τϧͱͯ͠ٻ·Δ 10
  13. ओ੒෼෼ੳɿޡࠩ࠷খԽͷߟ͑ํ PCA ͸ۙࣅޡࠩΛ࠷খԽ͢ΔΑ͏ʹ࣠ΛબͿํ਑Ͱ΋ಋग़͞ΕΔ 軸への射影による誤差を最小化 D ࣍ݩσʔλΛ M(< D) ݸͷओ ੒෼࣠ʹࣹӨ

    ˣ ࢒Γͷ D − M ࣍ݩ্ۭؒͷ࢒ࠩ Λ࠷খԽ͢ΔΑ͏ʹਖ਼ن௚ަج ఈ {um}M m=1 ΛͱΔ PCA ͷݟ௨͕͠Α͘ͳΔͷͰ͜Ε΋௥ͬͯΈΔ 11
  14. ओ੒෼෼ੳɿޡࠩ࠷খԽʹΑΔఆࣜԽ (1) D ݸͷਖ਼ن௚ަجఈͷ૊ {um}D m=1 ΛͱΔɿ uT l um

    = δlm = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 (l = m) 0 (l ≠ m). ೚ҙͷσʔλ xn ∈ RD ͸ {um}D m=1 ͷઢܕ݁߹Ͱදͤͯ xn = D ∑ m=1 αnmum, (1) ྆ลʹ u⊺ l (l ≠ m) Λࠨ͔Βॻ͚Δͱ u⊺ l xn = D ∑ m=1 αnmu⊺ l um = D ∑ m=1 αnmδlm = αnl. ͜ΕΛ (1) ʹ୅ೖ͢Δͱ xn = D ∑ m=1 (u⊺ m xn)um. 12
  15. ओ੒෼෼ੳɿޡࠩ࠷খԽʹΑΔఆࣜԽ (2) ͱ͜ΖͰɼσʔλ఺ xn ∈ RD ͷ M ݸͷ࣠Ͱͷۙࣅͨ͠఺ ˜

    xn ͸ ˜ xn = M ∑ m=1 αnmum + D ∑ m=M+1 βmum ͱදͤΔ͸ͣ 2→1次元の 変換 ࣗ༝ύϥϝʔλ͸ {αnm},{βm},{um}. ʢฏۉʣೋ৐ޡࠩ J = 1 N N ∑ n=1 ∥xn − ˜ xn∥2 Λ࠷খԽ͢ΔΑ͏ʹύϥϝʔλ Λܾఆ 13
  16. ओ੒෼෼ੳɿޡࠩ࠷খԽʹΑΔఆࣜԽ (3) J ಺ͷ ˜ xn Λల։ J = 1

    N N ∑ n=1 ∥xn − ˜ xn ∥2 = 1 N N ∑ n=1 [x⊺ n xn − 2x⊺ n ( M ∑ m=1 αnm um + D ∑ m=M+1 βm um ) + M ∑ m=1 α2 nm ] + M ∑ m=M+1 β2 m ͜ͷ J ʹରͯ͠ αnm,βm ʹؔ͢ΔҰ࣍ͷ࠷దੑ৚݅ΛͱΔɿ ∂J ∂αnm = 2 N (−x⊺ n um + αnm) = 0 ⇒ αnm = x⊺ n um ∂J ∂βm = − 2 N N ∑ n=1 x⊺ n um + 2βm = 0 ⇒ βm = ¯ x⊺um ͢ͳΘͪ ˜ xn = ∑M m=1 αnmum + ∑D m=M+1 βmum ͸ ˜ xn = M ∑ m=1 (x⊺ n um)um + D ∑ m=M+1 (¯ x⊺um)um 14
  17. ओ੒෼෼ੳɿޡࠩ࠷খԽʹΑΔఆࣜԽ (4) ֤σʔλ఺Ͱxn = ∑D m=1 (u⊺ m xn)um ͕੒ཱͨ͜͠ͱΛࢥ͍ग़͢

    ઌఔͷ˜ xn = ∑M m=1 (x⊺ n um)um + ∑D m=M+1 (¯ x⊺um)um ͱ͋Θͤͯ xn − ˜ xn = D ∑ m=1 (u⊺ m xn)um − M ∑ m=1 (x⊺ n um)um − D ∑ m=M+1 (¯ x⊺um)um = D ∑ m=M+1 [(xn − ¯ x)⊺um]um ͳͷͰ J = 1 N ∑N n=1 ∥xn − ˜ xn∥2 ͸ J = 1 N N ∑ n=1 ( D ∑ m=M+1 [(xn − ¯ x)⊺um ]um ) ⊺ ( D ∑ m=M+1 [(xn − ¯ x)⊺um ]um ) = 1 N N ∑ n=1 D ∑ m=M+1 u⊺ m (xn − ¯ x)(xn − ¯ x)⊺um = D ∑ m=M+1 u⊺ m 1 N N ∑ n=1 (xn − ¯ x)(xn − ¯ x)⊺um = D ∑ m=M+1 u⊺ m Sum . 15
  18. ओ੒෼෼ੳɿޡࠩ࠷খԽʹΑΔఆࣜԽ (5) D = 2 ͔Β M = 1 ΁ͷ࣍ݩ࡟ݮΛߟ͑Δɽղ͖͍ͨ࠷খԽ໰୊͸

    Minimize J = u⊺ 2 Su2 subject to 1 − u⊺ 2 u2 = 0 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ Ͱ͋ΓɼLagrange ૒ର໰୊͸ Minimize L(u2,λ2) = u⊺ 2 Su2 + λ2(1 − u⊺ 2 u2). ෼ࢄ࠷େԽͷࡍͱಉ༷ʹҰ࣍ͷ࠷దੑ৚݅Λղ͘ͱ Su2 = λ2u2 ݻ༗஋໰୊ʂ ⇒ J = u⊺ 2 Su2 = λ2u⊺ 2 u2 = λ2. ΍͸Γ PCA ͸ S ͷݻ༗஋໰୊ʹؼண͞Εɼ੾Γࣺͯͨ࣠ͷݻ༗ ஋͸௿࣍ݩۙࣅͷೋ৐ޡࠩͰ͋Δ͜ͱ΋Θ͔ͬͨʢ΍ͬͨʔʂʣ 16
  19. ಛҟ஋෼ղ ಛҟ஋෼ղ (SVD; Singular Value Decomposition) ೚ҙͷ X ≠ O

    ∈ RD×D ͸௚ަߦྻ U ∈ RD×D,V ∈ DN×N ͱର֯ ੒෼ͷΈʹ஋Λ΋ͭ Σ ∈ RD×N ʹΑͬͯ X = UΣV ⊺ ͱදͤΔ 1 SVD ͸ݻ༗஋෼ղͷҰൠԽͱଊ͑ΒΕΔ 1rank(X) = K ≤ min{D, N} ͱͯ͠ U ∈ RD×K , V ∈ DN×K , Σ ∈ RK×K ͱͱΔ ྲّྀ΋͋Γɼͪ͜Β΋Α͘࢖͏ 17
  20. ಛҟ஋෼ղͷ֓೦ਤ の場合 U,V ͷ֤ྻϕΫτϧΛࠨ/ӈಛҟϕΫτϧɼΣ ͷର֯੒෼ σm Λಛҟ஋ͱݺͿɽ ಛҟ஋ʹॖୀ͕ͳ͍৔߹ɼσm > σm+1

    ͷ৚݅ͷ΋ͱ৭෇͖ͷ෦෼ ͸ҰҙʹܾΊΒΕΔ 18
  21. ओ੒෼෼ੳɿಛҟ஋෼ղͱͷؔ࿈ (1) ฏۉ 0 ͷσʔλ {xn} ͔Β X = (x1,...,xN

    ) ∈ RD×N Λఆٛ 2 ڞ෼ࢄߦྻ͸ S = N−1 ∑N n=1 xnx⊺ n = N−1XX⊺ʢฏۉ 0 ʹ஫ҙʣ S ͷࣜ಺ͷ X ʹ SVD Λద༻͢ΔͱʢX = UΣV ⊺ʣ S = 1 N UΣV ⊺(UΣV ⊺)⊺ = 1 N UΣ2U⊺ PCA ʹΑΔ S ͷݻ༗஋෼ղΛ S = W ΛW ⊺ ͱ͢Δͱ 3 W ΛW ⊺ = U ⎛ ⎝ 1 N Σ2 ⎞ ⎠ U⊺. ௚ަߦྻ W ↔ U ͱର֯ߦྻ Λ ↔ 1 N Σ2 ͕ରԠʂ 2؆୯ͷͨΊ rank(X) = min{D, N} ͱ͢Δ 3Σ2 = ΣΣ⊺ ∈ RD×D ͱఆٛɽ 19
  22. ओ੒෼෼ੳɿಛҟ஋෼ղͱͷؔ࿈ (2) ࣍ʹɼSVD ͷࣜ X = UΣV ⊺ Λมܗ͢Δͱ X⊺U

    = V Σ⊺ɽ ֤੒෼͝ͱͷදݱʹ͢Δͱ x⊺ n um = σmvmn. PCA Ͱ u⊺ m xn = x⊺ n um ͸σʔλ xn ͷ࣠ um ΁ͷࣹӨͩͬͨ 4 ·ͨಛҟ஋ σm ͕࣠ m ͷεέʔϧΛఆΊ͍ͯΔͷͰɼ V = (vmn) ͸εέʔϧਖ਼نԽ͞ΕࣹͨӨઌͷσʔλදݱ 4p.8 ࢀর 20
  23. ओ੒෼෼ੳɿಛҟ஋෼ղͱͷؔ࿈ (3) ·ͱΊΔͱɼσʔλߦྻ X ∈ RD×N ͷಛҟ஋෼ղ X = UΣV

    ⊺ Λ ߟ͑ͨͱ͖ɼ X ࠨಛҟϕΫτϧ um ͸ PCA ͷओ੒෼࣠ͱରԠͱ X ಛҟ஋ͱ PCA ͷݻ༗஋͸ λm = σ2 m /N ͱରԠ X ӈಛҟϕΫτϧ vn ͸ओ੒෼ۭؒ΁ࣹӨͨ͠σʔλʹରԠ PCA ΑΓ΋ SVD ͷܭࢉΞϧΰϦζϜͷํ͕҆ఆͱݴΘΕ͓ͯΓɼ scikit-learn ͷ PCA ͸಺෦తʹ͸ SVD 21
  24. ओ੒෼෼ੳɿ࣮ફ (1) Wine σʔληοτʹ PCA Λ͔͚ͯΈΔ X Ξϧίʔϧ౓਺ɼϦϯΰࢎͷؚ༗ྔɼ৭૬ͳͲ D =

    13 ࣍ݩ ͷಛ௃ྔ X αϯϓϧ਺͸ N = 178 X ຊདྷ͸ 3 ඼छͷ൑ผ໰୊༻ͷσʔληοτ X ՄࢹԽͷͨΊ 2 ࣍ݩʹ࣍ݩ࡟ݮͯ͠ΈΔ 22
  25. ओ੒෼෼ੳɿ࣮ફ (2) -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 -0.15

    -0.10 -0.05 0.00 0.05 0.10 0.15 Linear PCA (wine) ओ੒෼࣠ 2 ͭͰ΋ͦΕͳΓʹࣝผͰ͖ͦ͏ͳ৘ใྔΛอ͍ͬͯΔ 23
  26. ओ੒෼෼ੳɿ࣮ફ (3) ֤ओ੒෼࣠ͷݻ༗஋ u10 ͘Β͍Ͱ΋໌֬ʹϊΠζͱׂΓ੾ΕΔ͔ඍົͳϥΠϯ 1 2 3 4 5

    6 7 8 9 10 11 12 13 0 1 2 3 4 PCA eigenvalues (wine) PC number lambda 24
  27. ओ੒෼෼ੳɿ࣮ફ (4) cM = ∑M m=1 λm ∑D m=1 λm

    Ͱఆٛ͞ΕΔྦྷੵد༩཰΋࣍ݩ਺ΛܾΊΔॏཁͳࢦඪ 1 2 3 4 5 6 7 8 9 10 11 12 13 0.4 0.5 0.6 0.7 0.8 0.9 1.0 PCA cumulative contribution rates PC number c u10 Ͱ 95%௒ 25
  28. ओ੒෼෼ੳɿ·ͱΊͱ஫ҙ ·ͱΊ X ෼ࢄ࠷େԽͱޡࠩ࠷খԽͷ؍఺͔ΒͦΕͧΕ PCA Λಋग़ͨ͠ X PCA ͷݻ༗஋͸࣠ํ޲ͷ෼ࢄ΍੾Γࣺͯͨ࣠ํ޲ͷೋ৐ޡࠩ Λҙຯ

    X ࣮͸ SVD ͱ౳Ձͳखଓ͖ X ݻ༗஋ΛݟΕ͹Կ൪໨ͷओ੒෼࣠·Ͱ৘ใΛ࣋ͬͯͦ͏͔େ ମΘ͔Δ X ఆྔతʹ࣠਺ΛܾΊΔํ๏࿦΋͋Δʢ৘ใྔج४ͳͲʣ ஫ҙ X લॲཧͱͯ͠σʔλͷਖ਼نԽ͸΄΅ඞਢ X ࣠ͷεέʔϧ͕େ͖͚Ε͹෼ࢄ΍ޡࠩ΋େ͖͘ͳͬͯ͠·͏ 26
  29. ઢܗ൑ผ෼ੳ

  30. ઢܗ൑ผ෼ੳɿ͍ΜͱΖ 2 ࣍ݩ্ۭؒͰ͸ઢ ܗ൑ผͰ͖ͦ͏ ˣ 1 ࣍ݩͰ΋͍͚Δʁ ઢܗ൑ผ෼ੳ (LDA; Linear

    Discriminant Analysis) ͸ઢܗ൑ผੑΛ ͳΔ΂͘อͭ௿࣍ݩͷ࣠Λݟ͚ͭΔڭࢣ͋Γ࣍ݩ࡟ݮ๏ FLDA (Fisher’s LDA) ΍ FDA ͳͲݺͼํ͸৭ʑ͋Δ 5 5ͨΜʹ LDA ͱݺͿͱ Latent Dirichlet Allocation ͱ΍΍͍͜͠ 27
  31. ઢܗ൑ผ෼ੳɿ ʮฏۉͷࠩʯ࠷େԽʁ(1) σʔλɿ{(yn,xn)}N n=1 , yn ∈ {0,1}, xn ∈

    RD ൑ผؔ਺ɿf(x) = w⊺x f(x) ≥ w0 ͳΒΫϥε 0, f(x) < w0 ͳΒΫϥε 1 ͱ༧ଌ͢Δ ֤Ϋϥεͷఴࣈू߹ Ck = {n ∈ {1,...,N}∣yn = k} ΋ఆٛ (k = 0,1) ·ࣹͣӨޙͷ 2 Ϋϥεؒͷฏۉͷࠩͷ࠷େԽΛߟ͑ͯΈΔ ∥w∥2 = 1,mk = ∑n∈Ck xk ʢΫϥε಺ฏۉʣͱͯ͠ɼ࣠ w ΁ͷࣹӨ ޙͷ֤Ϋϥε಺ฏۉ m0,m1 ͷࠩ͸ m1 − m0 = w⊺(m1 − m0). 28
  32. ઢܗ൑ผ෼ੳɿ ʮฏۉͷࠩʯ࠷େԽʁ(2) લϖʔδͷ m1 − m0 = w⊺(m1 − m0)

    ͷ࠷େԽ͸ Lagrange ͷະఆ৐਺๏ΑΓҎԼʹؼணɿ Maximize L(w,λ) = w⊺(m1 − m0) + λ(1 − w⊺w) w ʹ͍ͭͯҰ࣍ͷ࠷దੑ৚݅ΛٻΊΔͱ ∂L ∂w = (m1 − m0) − 2λw = 0 ⇔ w = 1 2λ (m1 − m0). w ͸࣠ͷํ޲͚ͩఆΊΕ͹͍͍ͷͰɼw ∝ m1 − m0 Ͱ OK 29
  33. ઢܗ൑ผ෼ੳɿ ʮฏۉͷࠩʯ࠷େԽʁ(3) ·ͩμϝʁ ࣹӨޙͷ֤ΫϥεͷώετάϥϜͷʹ͹Β͖͕ͭ࢒͍ͬͯΔͷ͕ Α͘ͳ͍ ˠ͜ͷ͹Β͖ͭΛ཈͑ͭͭʮฏۉͷࠩʯ΋࠷େԽ͠Α͏ 30

  34. ઢܗ൑ผ෼ੳɿ෼ࢄ΋ߟྀͯ͠ΈΑ͏ (1) ࣹӨޙͷΫϥε಺෼ࢄΛ s2 k = ∑ n∈Ck (f(xn) −

    mk)2 = ∑ n∈Ck (w⊺xn − mk)2 ͱͯ͠ɼҎԼͷ໨తؔ਺ 6 Λ࠷େԽ͢Δ J(w) = (m1 − m0)2 s2 0 + s2 1 . ͜ΕͰʮฏۉͷࠩʯͷ࠷େԽͱࣹӨޙͷ֤Ϋϥε಺ͷ෼ࢄΛ࠷খ ԽΛಉ࣌ʹߟ͑ΒΕͦ͏ 6PRML Ͱ͸ Fisher criterion ͱݺ͹Ε͍ͯΔ͕ɼίϯηϯαεͷͱΕͨݺশ͸ ͳͦ͞͏ 31
  35. ઢܗ൑ผ෼ੳɿ෼ࢄ΋ߟྀͯ͠ΈΑ͏ (2) ͜͜ͰҎԼͷΫϥεؒ෼ࢄͱΫϥε಺෼ࢄΛఆٛ 7ɿ SB = (m1 − m0)(m1 −

    m0)⊺ ɿΫϥεؒ෼ࢄ, SW = ∑ k∈{0,1} ∑ n∈Ck (xn − mk)(xn − mk)⊺ ɿΫϥε಺෼ࢄ. ͢Δͱ J(w) = (m1 − m0)2/(s2 0 + s2 1 ) ͷ෼ࢠ͸ (m1 − m0)2 = w⊺(m1 − m0)(m1 − m0)⊺w = w⊺SBw ͱͳΓɼ෼฼͸ s2 0 + s2 1 = ∑ k∈{0,1} ∑ n∈Ck (w⊺xn − mk)2 = w⊺ ∑ k∈{0,1} ∑ n∈Ck (xn − mk)(xn − mk)⊺w = w⊺SW w. 7΋͸΍ڞ෼ࢄߦྻͦͷ΋ͷͰ͸ͳ͍ͷͰɼscatter matrix ͳͲͱ΋ݺ͹ΕΔ 32
  36. ઢܗ൑ผ෼ੳɿ෼ࢄ΋ߟྀͯ͠ΈΑ͏ (3) J(w) = (m1 − m0)2 s2 0 +

    s2 1 = w⊺SBw w⊺SW w ͕Θ͔ͬͨ 8 ͷͰɼ͜ͷ࠷େԽ໰୊Λߟ͑ͯΈΔɽ ద౰ʹ w ΛεέʔϦϯά͢Δ͜ͱʹΑΓɼ෼฼Λ w⊺SW w = 1 ͱ Ͱ͖Δ͸ͣɽΑͬͯҎԼͷ໰୊Λߟ͑ͯΈΔ Maximize w⊺SBw subject to 1 − w⊺SW w = 0 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ Lagrange ؔ਺͸ L(w,λ) = w⊺SBw + λ(1 − w⊺SW w) ͱͳΔɽ 8͜Ε͸ Rayleigh ঎ͱݺ͹ΕΔܗࣜɽ 33
  37. ઢܗ൑ผ෼ੳɿ෼ࢄ΋ߟྀͯ͠ΈΑ͏ (4) L(w,λ) = w⊺SBw + λ(1 − w⊺SW w)

    ʹؔͯ͠ɼྫʹΑͬͯ w ʹ͍ͭͯͷҰ࣍ͷ࠷దੑ৚݅Λղ͘ͱɼ ∂L ∂w = 2SBw − 2λSW w = 0 ⇔ SBw = λSW w ͜Ε͸ҰൠԽݻ༗஋໰୊ͱݺ͹ΕΔ໰୊Ϋϥε ࠓճ͸σʔλͷϥϯΫམ͕ͪͳ͚Ε͹ SW ͸ਖ਼ଇͳͷͰɼ S−1 W SBw = λw ͱ͍͏ී௨ͷݻ༗஋໰୊ʹམͱ͠ࠐΊΔ 34
  38. ઢܗ൑ผ෼ੳɿ෼ࢄ΋ߟྀͯ͠ΈΑ͏ (5) S−1 W SBw = λw ࣮͸ࠓճ͸ݻ༗஋໰୊Λղ͘͜ͱ͢Βෆཁɽ SBw =

    (m1 − m0)(m1 − m0)⊺w = (m1 − m0)(m1 − m0) ΑΓ w = λ−1S−1 W SBw = λ−1(m1 − m0)S−1 W (m1 − m0). w ͷํ޲͚͕ͩඞཁͰεΧϥʔ஋܎਺͸ແࢹͯ͠Α͍ͷͰɼ w ∝ S−1 W (m1 − m0) ͕ٻΊΔ΂͖ࣹӨઌͷ࣠Λද͢ SW ͕౳ํతͳΒʮฏۉͷࠩʯ࠷େԽͱ౳Ձʂ 35
  39. ઢܗ൑ผ෼ੳɿͲ͏ʁ ͑͑ΜͱͪΌ͏ʁ 36

  40. ઢܗ൑ผ෼ੳɿଟ࣍ݩଟΫϥε΁ͷҰൠԽ (1) ͜Ε·Ͱ͸ 2 ஋෼ྨ໰୊ɼRD → R ʹࣹӨ͢Δ໰୊ͷΈߟ͑ͨ ౰વͷཉٻͱͯ͠ɼK Ϋϥε෼ྨ໰୊Ͱ

    RD → RM (M < D) ΁ͷ ࣍ݩ࡟ݮΛߟ͑ͨ͘ͳΔ ͭ·Γ໰୊ઃఆ͸ σʔλɿ{(yn,xn)}N n=1 ,yn ∈ {0,...,K − 1},xn ∈ RD ൑ผؔ਺ɿf(x) = W ⊺x ∈ RM ,W ∈ RD×M . 37
  41. ઢܗ൑ผ෼ੳɿଟ࣍ݩଟΫϥε΁ͷҰൠԽ (2) Ϋϥε k ʹଐ͢Δσʔλ਺Λ Nk ɼΫϥε಺ɾΫϥεؒ෼ࢄΛ SB = K−1

    ∑ k=0 Nk(mk − m)(mk − m)⊺, SW = K−1 ∑ k=0 ∑ n∈Ck (xn − mk)(xn − mk)⊺ ͱఆٛ͢Δɽ ͜͜Ͱ mk,m ͸ͦΕͧΕΫϥε಺͓Αͼશαϯϓϧͷฏۉɿ mk = 1 Nk ∑ n∈Ck xk, m = 1 N N ∑ n=1 xn = 1 N K−1 ∑ k=0 Nkmk. 38
  42. ઢܗ൑ผ෼ੳɿଟ࣍ݩଟΫϥε΁ͷҰൠԽ (3) 1 ࣍ݩ΁ͷࣹӨͷ৔߹͸ J(w) = w⊺SBw w⊺SW w ͷ࠷େԽΛߟ͑ͨɽ

    ૉ௚ʹ (W ⊺SBW )(W ⊺SW W )−1 ͱ֦ு͢ΔͱɼεΧϥʔ஋Ͱ ͳ͍ͷͰ௨ৗͷ࠷దԽ໰୊ͷൣᙝΛ௒͑ͯ͠·͏ɽ ͦ͜ͰʮࣹӨޙͷ֤࣠ํ޲ͷΫϥεؒ෼ࢄΛ࠷େԽ͠ɼ Ϋϥε಺෼ࢄΛ࠷খԽ͢Δʯͱ͍͏؍఺͔Β J(W ) = tr(W ⊺SBW ) tr(W ⊺SW W ) Λ໨తؔ਺ͱ͢Ε͹͍͍͔΋ʂ 39
  43. ઢܗ൑ผ෼ੳɿଟ࣍ݩଟΫϥε΁ͷҰൠԽ (3) J(W ) = tr(W ⊺SBW ) tr(W ⊺SW

    W ) ͱ͍͏ LDA ͷ໨తؔ਺͸ trace ratio ܗࣜͱݺ͹Ε͍ͯΔ [1]ɽ ͔͠͠ trace ratio ܗࣜͷ࠷దԽ໰୊͸ඇತͰ͋ΔͨΊɼҎԼͷ ratio trace ܗࣜͱݺ͹ΕΔತ؇࿨໰୊͕ϝδϟʔͰ͋Δɿ J(W ) = tr{(W ⊺SBW )(W ⊺SW W )−1}. ratio trace ܗࣜ͸ҎԼͷ໨తؔ਺ͱ౳ՁʹͳΔɿ J(W ) = det(W ⊺SBW ) det(W ⊺SW W ) . 40
  44. ઢܗ൑ผ෼ੳɿଟ࣍ݩଟΫϥε΁ͷҰൠԽ (4) ࠓճ͸ ratio trace ܗࣜ J(W ) = tr{(W

    ⊺SBW )(W ⊺SW W )−1}. ͷ࠷େԽΛߟ͑Δɽͨͩࣗ͠༝౓Λམͱͨ͢Ίʹ W ⊺SW W = I ͷ੍໿Λ՝͢ɽ ͢Δͱ tr(⋅) ಺ͷ W ⊺SW W ΋୯ҐߦྻʹͳΔͷͰɼओ໰୊͸ Maximize tr(W ⊺SBW ) subject to I − W ⊺SW W = O ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ͱͳΔɽLagrange ؔ਺͸ ⟨⋅,⋅⟩ Λߦྻͷ಺ੵͱͯ͠ɼ L(W ,Λ) = tr(W ⊺SBW ) + ⟨Λ,I − W ⊺SW W ⟩. 41
  45. ઢܗ൑ผ෼ੳɿଟ࣍ݩଟΫϥε΁ͷҰൠԽ (5) Lagrange ؔ਺Λղ͘ͷʹඞཁͳߦྻܭࢉͷ४උΛ͓ͯ͘͠ɽ ·ͣߦྻͷ಺ੵ͸୯ʹཁૉ͝ͱͷੵͷ࿨Ͱɼ ⟨A,B⟩ = ∑ i ∑

    j AijBij ͱఆٛ͞Εɼ ⟨A,B⟩ = tr(A⊺B) ͕੒Γཱͭɽ ࣍ʹೋ࣍ܗࣜͷτϨʔεͷඍ෼ެࣜͱͯ͠ ∂ ∂X tr(CX⊺BX) = BXC + B⊺XC⊺ ͕੒Γཱͭ 9ɽ 9Matrix Cookbook [2] ࣜ (117) 42
  46. ઢܗ൑ผ෼ੳɿଟ࣍ݩଟΫϥε΁ͷҰൠԽ (6) ಺ੵͱτϨʔεͷؔ܎ΑΓϥάϥϯδϡؔ਺͸ L(W ,Λ) = tr(W ⊺SBW ) +

    ⟨Λ,I − W ⊺SW W ⟩ = tr(W ⊺SBW ) + tr(Λ) − tr(ΛW ⊺SW W ) ͱͳΔͷͰɼW ʹؔͯ͠ภඍ෼͢Δͱɼ ∂L ∂W = SBW + S⊺ B W − SW W Λ − S⊺ W W Λ⊺ = 2SBW − 2SW W Λ = O ⇔ SBW = SW W Λ ͱͳΔɽΛ = diag(λm),W = (wm) ͱ͢Δͱɼ SBwm = λmSW wm ͱ΍͸ΓҰൠԽݻ༗஋໰୊ʹؼண͞Εͨʂ 43
  47. ઢܗ൑ผ෼ੳɿ·ͱΊͱ஫ҙ ·ͱΊ X LDA ͱͦͷଟ࣍ݩଟΫϥε΁ͷҰൠԽΛಋग़ͨ͠ X LDA ͕ҰൠԽݻ༗஋໰୊ʹམͱ͠ࠐΊΔ͜ͱΛࣔͨ͠ ஫ҙ X

    LDA ͸֤܈ͷਖ਼نੑͱ౳෼ࢄੑΛ҉໧తʹԾఆ͍ͯ͠Δ X ౳෼ࢄੑΛԾఆ͠ͳ͍ҰൠԽͱͯ͠ QDA (Quadratic Discriminant Analysis) ͕͋Δ 44
  48. Χʔωϧओ੒෼෼ੳ

  49. Χʔωϧओ੒෼෼ੳɿඇઢܗ࣍ݩ࡟ݮͷಈػ PCA ͕ྑ͍ͷ͸Θ͔ͬͨɽ͔͠͠σʔλ͸ߴ࣍ݩۭؒͰڧ͍ඇઢ ܗੑΛ͍࣋ͬͯΔ͔΋ -4 -2 0 2 4 -4

    -2 0 2 4 R^2 data Ͳ͏࣍ݩ࡟ݮ͢Δʁ 45
  50. Χʔωϧओ੒෼෼ੳɿඇઢܗ࣍ݩ࡟ݮ΁ͷల๬ -4 -2 0 2 4 -4 -2 0 2

    4 -0.10 -0.05 0.00 0.05 0.10 -0.10 -0.05 0.00 0.05 0.10 PCA 3次元へ 非線形変換 0.04 0.06 0.08 0.10 -0.10 -0.05 0.00 0.05 0.10 PCA ద੾ͳߴ࣍ݩۭؒ΁ͷࣹӨΛఆٛͰ͖Ε͹ʜʜʁ 46
  51. Χʔωϧओ੒෼෼ੳɿΧʔωϧ๏ͷಋೖ (1) ͱ͸ݴͬͯ΋ʮద੾ͳߴ࣍ݩۭؒʯͳΜͯී௨͸Θ͔Βͳ͍ͷͰɼ Ұ୴ߟ͑ͳ͍͜ͱʹ͢Δ ͱΓ͋͑ͣ PCAʢ෼ࢄ࠷େԽʣͷ໨తؔ਺Λݟ௚ͯ͠ΈΔɿ u⊺Su = u⊺ ⎧

    ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 N N ∑ i=1 (xn − ¯ x)(xn − ¯ x)⊺ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ u = 1 N N ∑ i=1 {u⊺(xn − ¯ x)}2 . ۩ମతͳؔ਺ܗ͸ؾʹͤͣɼσʔλ఺Λ x ↦ Φ(x) ∈ H ͱඇઢܗ ม׵͢Δࣸ૾ Φ Λಋೖͯ͠ΈΔɽ Φ ͕ඈ͹͢ઌͷۭؒ H Ͱͷ “࣠” Λ f ∈ H ͱ͓͘ͱɼ໨తؔ਺͸ 1 N N ∑ i=1 ⟨f, ˜ Φ(xn)⟩2, ˜ Φ(x) = Φ(x) − 1 N N ∑ n=1 Φ(xn). ͱͳΔɽ 47
  52. Χʔωϧओ੒෼෼ੳɿΧʔωϧ๏ͷಋೖ (2) ໨తؔ਺͕ 1 N N ∑ i=1 ⟨f, ˜

    Φ(xn)⟩2 ͱͳͬͨ ˠ ͱʹ͔͘಺ੵ͕ॏཁͬΆ͍ ͦ͜ͰɼఱԼΓత͕ͩH ͸಺ੵ͕ఆٛ͞Εͨઢܗۭؒͱ͢Δ ಺ੵ͕ఆٛ͞ΕΔͱϊϧϜ΋ ∥f∥ = ⟨f,f⟩1/2 ͷܗͰఆΊΒΕΔͨ Ίɼ∥f∥ = 1 ͷ৚݅΋༩͑ΒΕΔ ͕ͨͬͯ͠ Maximize 1 N N ∑ i=1 ⟨f, ˜ Φ(xn)⟩2 subject to 1 − ∥f∥ = 0 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Λߟ͑Ε͹ OK 48
  53. Χʔωϧओ੒෼෼ੳɿΧʔωϧ๏ͷಋೖ (3) Maximize 1 N N ∑ i=1 ⟨f, ˜

    Φ(xn)⟩2 subject to 1 − ∥f∥ = 0 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ͱ͍͏໰୊ΛͲ͏ʹ͔ղ͚͹ɼඇઢܗ࣍ݩ࡟ݮΛ࣮ݱͰ͖Δʢ͔ ΋͠Εͳ͍ʣ͜ͱ͕Θ͔ͬͨ ͔͠͠ҰମͲ͏΍ͬͯʜʜʁ ˠ ࣮͸࣠ f ͸ f = N ∑ n=1 an ˜ Φ(xn) ͷܗͰ୳ͤ͹े෼ʂ ͜ͷࣄ࣮Λ͍ࣔͯ͘͠ 49
  54. Χʔωϧओ੒෼෼ੳɿΧʔωϧ๏ͷಋೖ (3) f Λ {˜ Φ(xn)} ͷுΔ্ۭؒͷ੒෼ f0 ͱͦͷ௚ަ੒෼ f

    ʹ෼ղɿ f = f0 + f . ಺ੵԋࢉͷੑ࣭ΑΓ ⟨f, ˜ Φ(xn)⟩ = ⟨f0, ˜ Φ(xn)⟩ + ⟨f , ˜ Φ(xn)⟩ = ⟨f0, ˜ Φ(xn)⟩. f ͸د༩͠ͳ͍ ·ͨ ∥f∥ = 1 ͷ੍໿ʹؔͯ͠͸ɼϐλΰϥεͷఆཧΑΓ ∥f∥2 = ∥f0∥2 + ∥f ∥2 = 1 ͳͷͰɼ∥f ∥ = 0 ͱͯ͠ f0 ੒෼ͷΈΛߟ͑Ε͹ OKʂ10 10͜ͷࣄ࣮ΛҰൠతʹड़΂ͨఆཧΛ Representer ఆཧͱΑͿɽ 50
  55. Χʔωϧओ੒෼෼ੳɿΧʔωϧ๏ͷಋೖ (4) Ҏ্ͷٞ࿦ΑΓɼ໰୊ max∥f∥=1 1 N ∑N n=1 ⟨f, ˜

    Φ(xn)⟩2 ͕ max a1,...,aN 1 N N ∑ n=1 ⟨ N ∑ m=1 am ˜ Φ(xm), ˜ Φ(xn)⟩ 2 = 1 N N ∑ n=1 N ∑ m=1 (am⟨˜ Φ(xm), ˜ Φ(xn)⟩)2 ͱมܗͰ͖Δ͜ͱ͕Θ͔ͬͨɽ த৺ԽάϥϜߦྻ ˜ K ∈ RN×N Λ ˜ Knm = ⟨˜ Φ(xn), ˜ Φ(xm)⟩ ʹΑΓఆٛ͠ɼa = (an)N n=1 ͱ͢Δͱɼ໨తؔ਺͸ max a 1 N a⊺ ˜ K2a. 51
  56. Χʔωϧओ੒෼෼ੳɿΧʔωϧ๏ͷಋೖ (5) ࣍ʹ ∥f∥ = 1 ͷ੍໿ࣜΛվΊͯோΊΔͱɼ ∥f∥2 = ⟨

    N ∑ n=1 an ˜ Φ(xn), N ∑ m=1 am ˜ Φ(xm)⟩ = a⊺ ˜ Ka = 1. ͕ͨͬͯ͠ Maximize a⊺ ˜ K2a subject to 1 − a⊺ ˜ Ka = 0 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ͱ͍͏࠷େԽ໰୊Λղ͚͹ྑ͍͜ͱʹͳͬͨ 11 ͔͠͠·ͩࣸ૾ Φ ΛͲ͏ѻ͏͔ɼͱ͍͏໰୊͕࢒͍ͬͯΔʜʜ 11໨తؔ਺ͷఆ਺܎਺͸লུɽ 52
  57. Χʔωϧओ੒෼෼ੳɿΧʔωϧ๏ͷಋೖ (6) த৺ԽάϥϜߦྻͷఆٛࣜΛల։ͯ͠ΈΔͱ ˜ Knm = ⟨˜ Φ(xn), ˜ Φ(xm)⟩

    = ⟨Φ(xn) − 1 N N ∑ i=1 Φ(xi),Φ(xm) − 1 N N ∑ j=1 Φ(xj)⟩ = ⟨Φ(xn),Φ(xm)⟩ − 1 N N ∑ i=1 ⟨Φ(xi),Φ(xm)⟩ − 1 N N ∑ j=1 ⟨Φ(xn),Φ(xj)⟩ + 1 N2 N ∑ i,j ⟨Φ(xi),Φ(xj)⟩ ͱɼ⟨Φ(xi),Φ(xj)⟩ ͰఆΊΒΕ͍ͯΔ͜ͱ͕Θ͔Δ ͦ͜Ͱࢥ͍੾ͬͯࣸ૾ Φ ͸ཅʹߟ͑ͣ಺ੵ ⟨Φ(xi),Φ(xj)⟩ ͷܭ ࢉํ๏͚ܾͩΊΔ͜ͱʹ͢Δ ͢ͳΘͪΧʔωϧؔ਺ k(xi,xj) = ⟨Φ(xi),Φ(xj)⟩ ͷܗͷΈΛఆٛ 53
  58. Χʔωϧओ੒෼෼ੳɿΧʔωϧ๏ͷಋೖ (7) Χʔωϧ๏ ಺ੵܭࢉ͕ࠜװΛͳ͢ઢܗͳ౷ܭղੳ๏Λɼσʔλ఺ͷඇઢܗ ม׵ޙͷ಺ੵΛΧʔωϧؔ਺ k(⋅,⋅) ʹΑΓఆΊΔ͜ͱͰɼඇઢ ܗʹ֦ு͢Δํࡦ ओ੒෼෼ੳͷΧʔωϧԽɿΧʔωϧओ੒෼෼ੳ (Χʔωϧ

    PCA) Χʔωϧؔ਺ k ͸ͳΜͰ΋ OK ͱ͍͏Θ͚͸ͳ͘ɼάϥϜߦྻ K = ⎛ ⎜ ⎜ ⎝ k(x1,x1) ⋯ k(x1,xN ) ⋮ ⋱ ⋮ k(xN ,x1) ⋯ k(xN ,xN ) ⎞ ⎟ ⎟ ⎠ ͕೚ҙͷ {xn} ʹ͍ͭͯ൒ਖ਼ఆ஋Ͱͳ͚Ε͹ͳΒͳ͍ 12 12͜ΕΛຬͨ͢ k(⋅, ⋅) Λͱ͘ʹਖ਼ఆ஋ΧʔωϧͱΑͿɽ 54
  59. Χʔωϧओ੒෼෼ੳɿΧʔωϧ PCA Λղ͘ (1) Χʔωϧ PCA ͷओ໰୊͸ҎԼͰ༩͑ΒΕͨɿ Maximize a⊺ ˜

    K2a subject to 1 − a⊺ ˜ Ka = 0 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ Lagrange ؔ਺͸ L(a,λ) = a⊺ ˜ K2a + λ(1 − a⊺ ˜ Ka) ͱͳΓɼa ʹ͍ͭͯҰ࣍ͷ࠷దੑ৚݅͸ ∂L(a,λ) ∂a = 2 ˜ K2a − 2λ ˜ Ka = 0 ͳͷͰ ˜ K2a = λ ˜ Ka ͳΔҰൠԽݻ༗஋໰୊ʹؼணͨ͠ɽ 55
  60. Χʔωϧओ੒෼෼ੳɿΧʔωϧ PCA Λղ͘ (2) Χʔωϧ PCA ͷղ ˜ K2a =

    λ ˜ Ka ͸ K ͕ਖ਼ଇͳΒ ˜ Ka = λa ͱ͍͏ݻ༗஋໰୊ʹͰ͖Δ 13 13࣮͸ K ͕൒ਖ਼ఆ஋Ͱ΋ݻ༗஋໰୊ʹམͱ͠ࠐΊΔɽจݙ [3](p.7) ͳͲࢀরɽ 56
  61. Χʔωϧओ੒෼෼ੳɿਖ਼ఆ஋Χʔωϧͷܾఆ (1) ݁ہΧʔωϧ PCA ͷखॱ͸ 1. Χʔωϧؔ਺ k(⋅,⋅) Λద੾ʹఆΊΔ 2.

    ಘΒΕͨσʔλ఺ͱ k(⋅,⋅) ͔Βத৺ԽάϥϜߦྻΛܭࢉ 3. ʢҰൠԽʣݻ༗஋໰୊Λղ͍ͯ࣠Λܾఆ ͱͳΔ ͱ͜ΖͰ݁ہΧʔωϧؔ਺ k(⋅,⋅) ͸Ͳ͏ܾΊΔʁ ˠ େ఍͸طʹੑ࣭͕Α͘஌ΒΕ͍ͯΔΧʔωϧؔ਺Λ༻͍Δ 57
  62. Χʔωϧओ੒෼෼ੳɿਖ਼ఆ஋Χʔωϧͷܾఆ (2) ୅දతͳΧʔωϧؔ਺ ଟ߲ࣜΧʔωϧ k(x,x′) = (c + x⊺x′)d (c

    ≥ 0, d ∈ N) Ψ΢εʢRBFʣΧʔωϧ k(x,x′) = exp(−β∥x − x′∥2) (β > 0) ଟ߲ࣜΧʔωϧɿݩͷσʔλۭؒͷ d ࣍ଟ߲ࣜͰද͞ΕΔۭؒ Ψ΢εΧʔωϧɿແݶ࣍ݩʢʂʣͷۭؒ ΁ͷࣸ૾Λߟ͍͑ͯΔ͜ͱʹͳΔ 14 14௚ײతʹ͸ exp ͕ແݶ࣍ͷଟ߲ࣜʹల։͞ΕΔͨΊ 58
  63. Χʔωϧओ੒෼෼ੳɿਖ਼ఆ஋Χʔωϧͷܾఆ (3) Ψ΢εΧʔωϧͷྫ (x,x′ ∈ Rɼԣ࣠͸ x − x′) -4

    -2 0 2 4 0.00 0.25 0.50 0.75 1.00 Gauss kernel x - x' similarity beta = 1.0 beta = 0.3 beta = 0.2 Χʔωϧ PCA ͷ݁Ռ͸ϋΠύʔύϥϝʔλʹେ͖͘ґଘ 59
  64. Χʔωϧओ੒෼෼ੳɿਖ਼ఆ஋Χʔωϧͷܾఆ (4) Χʔωϧ PCA ͸ڭࢣͳֶ͠शͳͷͰަࠩݕূ͕Ͱ͖ͳ͍ ˠ Χʔωϧύϥϝʔλͷܾఆ͸Ұൠʹ͔ͳΓ೉͍͠ X ࣍ݩ࡟ݮ͕ڭࢣ͋Γֶशͷલॲཧͷ৔߹ɿަࠩݕূ X

    άϥϜߦྻͷ஋͕๫Εͳ͍Α͏ʹ͢Δɿmedian heuristics median heuristics Ψ΢εΧʔωϧͷύϥϝʔλ β Λ β = 1/median{∥xn − xm∥2∣n < m} ͱܾΊΔͱάϥϜߦྻͷ஋͕๫ΕͣɼͦΕͳΓʹྑ͘ͳΔ͜ͱ ͕ଟ͍ 60
  65. Χʔωϧओ੒෼෼ੳɿ࣮ફ (1) Boston Housing Ϙετϯͷ֤஍۠Ͱͷॅ୐Ձ֨ͷσʔληοτ X Ұਓ͋ͨΓͷ൜ࡑ཰ɼҰࢎԽ஠ૉೱ౓ɼฏۉ෦԰਺ͳͲ D = 12

    ࣍ݩͷಛ௃ྔ X ໨తม਺͸ॅ୐Ձ֨ͷΤϦΞ಺தԝ஋ X αϯϓϧ਺͸ N = 506 X ՄࢹԽͷͨΊಛ௃ྔΛ 2 ࣍ݩʹ࣍ݩ࡟ݮͯ͠ΈΔ 61
  66. Χʔωϧओ੒෼෼ੳɿ࣮ફ (2) -0.10 -0.05 0.00 0.05 -0.10 -0.05 0.00 0.05

    0.10 0.15 Linear PCA (Boston) -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 -60 -40 -20 0 20 0 20 40 60 Kernel PCA (Boston, Poly-2) -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.25 0.00 0.25 0.50 Kernel PCA (Boston, Gaussian w/ med. heuristics) -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0 2.5 2 ࣍ଟ߲ࣜΧʔωϧɿߴՁ֨ଳͷ αϯϓϧ͕௵Ε͍ͯΔ Ψ΢εΧʔωϧɿ෼཭͸Ͱ͖ͯ ͍Δ͕͙ʹΌ ͬͱ͍ͯ͠Δ 62
  67. Χʔωϧओ੒෼෼ੳɿ࣮ફ (3) MNIST ͷσʔλΛ 2000 ݸ࢖ͬͯ΍ͬͯΈΔ -0.075 -0.050 -0.025 0.000

    0.025 -0.050 -0.025 0.000 0.025 0.050 0.075 Linear PCA (MNIST) 0 1 2 3 4 5 6 7 8 9 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 Kernel PCA (MNIST, Gauss w/ med. heuristics) 0 1 2 3 4 5 6 7 8 9 “1” ΍ “9” ͸Χʔωϧ PCA ͷํ͕͏·͘ଊ͑ΒΕͯΔ 63
  68. Χʔωϧओ੒෼෼ੳɿ·ͱΊͱ஫ҙ ·ͱΊ X ෼ࢄ࠷େԽʹجͮ͘ PCA ͷ֦ுͱͯ͠Χʔωϧ PCA Λಋग़ X ݁Ռ͸Χʔωϧؔ਺΍ύϥϝʔλʹେ͖͘ґଘ

    X ڭࢣͳ͠ͳͷͰύϥϝʔλΛ͏·ܾ͘ΊΔͷ͸݁ߏ೉͍͠ X ࣍ݩ࡟ݮޙʹڭࢣ͋ΓֶशΛߦ͏ͳΒ CV ͕࢖͑Δ X Ψ΢εΧʔωϧͷύϥϝʔλΛܾΊΔ heuristics Λ঺հ ஫ҙ X ແ೉ͳͷ͸ઢܗ PCA X ඇઢܗ࣍ݩ࡟ݮ΁ͷڧ͍ಈػ͕ͳ͚Ε͹ɼύϥϝʔλϑϦʔ Ͱղऍ΋͠΍͍͢ઢܗ PCA Λ࢖͏΂͖ʢචऀॴײʣ X աֶशʹؾΛ͚ͭΔ X Ψ΢εΧʔωϧͳͲ͸දݱྗ͕ߴ͗͢ΔͷͰɼͱ͘ʹߴ࣍ݩ σʔλʹରͯ͠͸ʮσʔλ఺ͷ͋Δͱ͜Ζ͔͠஋Λ΋ͨͳ͍ʯ Α͏ͳ͜ͱʹͳΓ͕ͪ 64
  69. References i [1] Wang, H., Yan, S., Xu, D., Tang,

    X., and Huang, T. (2007), “Trace Ratio vs. Ratio Trace for Dimensionality Reduction,” in 2007 IEEE Conference on Computer Vision and Pattern Recognition, Minneapolis, MN: IEEE, pp. 18. [2] Petersen, K., and M. S. Pedersen. (2012) “The matrix cookbook,” Version November 15 2012. [3] ෱ਫ݈࣍. (2010) “Χʔωϧ๏ೖ໳ -ਖ਼ఆ஋ΧʔωϧʹΑΔ σʔλղੳ-,” ே૔ॻళ.