疎行列圧縮フォーマットの紹介

CompML勉強会での発表資料．
- 成分のほとんどがゼロであるような疎行列の圧縮フォーマットの紹介．
- 主要な圧縮フォーマット(CSR)を用いたSparse Matrix Vector Multiplicationの紹介．

Transcript

1. CompML Sparse Matrix Formats ૄߦྻѹॖϑΥʔϚοτͷ঺հ Ryohei Izawa (@zawatsky_r)
2. CompML 5-%3 • ੒෼ͷ΄ͱΜͲ͕θϩͰ͋ΔΑ͏ͳૄߦྻͷѹॖϑΥʔϚοτͷ঺հɽ • ओཁͳѹॖϑΥʔϚοτ $43 Λ༻͍ͨ4QBSTF .BUSJY
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3. CompML ૄߦྻͷ֓ཁ • ੒෼ͷଟ͕͘ඇθϩͰ͋ΔΑ͏ͳߦྻΛີߦྻɼ੒෼ͷଟ͕͘θϩͰ͋ΔΑ͏ͳߦྻΛૄߦྻͱ͍͏ɽ • ີߦྻͰ͸ɼߦྻͷ֤஋͕ଟ࣍ݩ഑ྻʹ֨ೲ͞ΕΔɽ • ૄߦྻͷ৔߹ɼඇθϩͷ஋ͱͦͷߦྻ಺Ͱͷ࠲ඪ৘ใΛ֨ೲ͢Δ͜ͱͰɼσʔλΛѹॖ͢Δ͜ͱ͕Ͱ ͖ɼϝϞϦͷઅ໿΍ߦྻԋࢉͷߴ଎Խ͕ՄೳͱͳΔɽ 0 0 0 0 2 5 8 0 0 0 0 0 3 0 0 0 6 0 0 1 3 2 7 7 1 5 8 4 6 4 3 4 3 1 2 2 6 7 9 1 ີߦྻ ૄߦྻ ѹॖ ʁ 2
4. CompML ૄߦྻѹॖϑΥʔϚοτ
5. CompML ૄߦྻѹॖϑΥʔϚοτ ෳ਺͋ΔૄߦྻͷѹॖϑΥʔϚοτͷ͏ͪɼҎԼͷख๏Λ঺հɽ • Dense • DIA • COO • CSR • CSC • ELL • Hybrid • BSR • BSRX • CSB 4
6. CompML %FOTF ີߦྻΛ֨ೲ͢Δͱ͖ͷϑΥʔϚοτɽߦྻ಺ͷશͯͷ஋͕֨ೲ͞ΕΔɽ 5 𝑑𝑎𝑡𝑎 = 0 0 0 0 2 5 8 0 0 0 … 0 6 0 0 1 0 0 0 0 2 5 8 0 0 0 0 0 3 0 0 0 6 0 0 1
7. CompML %*"
   %JBHPOBM ඇθϩཁૉ͕ɼগ਺ͷର֯ઢ্ʹ͋Δ৔߹ʹద͍ͯ͠ΔϑΥʔϚοτɽ σʔλ഑ྻͱΦϑηοτ഑ྻ͔Βߏ੒͞ΕΔɽσʔλ഑ྻʹ͸ඇθϩཁૉͷ஋ɼΦϑηοτ഑ྻʹ͸த ԝͷର֯ઢ͔ΒͷΦϑηοτ͕֨ೲ͞ΕΔɽ 6 4 0 0 0 0 0 8 0 5 0 0 0 3 0 9 6 0 0 7 0 𝑑𝑎𝑡𝑎 = 6 ∗ ∗ ∗ 4 8 3 7 ∗ 5 9 ∗ 𝑜𝑓𝑓𝑠𝑒𝑡 = −3 0 2 PGGTFU
    PGGTFU
    PGGTFU
   
8. CompML %*"͕ෆద੾ͱͳΔέʔε ҎԼͷΑ͏ͳέʔεͰ͸ɼ%*"͸ෆద੾ͱͳΔɽ 7 0 0 0 0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 𝑑𝑎𝑡𝑎 = ∗ 4 ∗ ∗ ∗ ∗ 3 ∗ ∗ 2 ∗ ∗ 1 ∗ ∗ ∗ 𝑜𝑓𝑓𝑠𝑒𝑡 = −2 0 2 4 0 0 1 0 0 0 0 2 0 5 0 3 0 0 0 6 7 0 4 0 𝑑𝑎𝑡𝑎 = 6 ∗ ∗ ∗ ∗ 7 ∗ ∗ ∗ 3 ∗ ∗ ∗ ∗ ∗ 4 ∗ 2 ∗ ∗ 1 ∗ ∗ ∗ ∗ 5 ∗ ∗ 𝑜𝑓𝑓𝑠𝑒𝑡 = −3 −2 −1 0 1 2 3
9. CompML $00
   $PPSEJOBUF ඇθϩཁૉͷ஋ͷ഑ྻͱɼͦͷཁૉͷߦΠϯσοΫεͱྻΠϯσοΫεͷ഑ྻͰߏ੒͞ΕΔɽ ඇθϩཁૉͷ਺͕ͦͷߦྻͷʹຬͨͳ͍৔߹ʹͷΈɼີߦྻΑΓ΋σʔλྔΛ࡟ݮ͢Δ͜ͱ͕Ͱ͖Δɽ 8 𝑑𝑎𝑡𝑎 = 1 5 8 3 6 1 0 0 0 0 1 5 8 0 0 0 0 0 3 0 0 0 6 0 0 1 𝑟𝑜𝑤_𝑖𝑛𝑑 = 0 1 1 2 3 3 𝑐𝑜𝑙_𝑖𝑛𝑑 = 4 0 1 2 1 4         
10. CompML $43
    $PNQSFTTFE
    4QBSTF
    3PX ඇθϩཁૉͷ஋͕֨ೲ͞Εͨσʔλ഑ྻɼඇθϩཁૉͷྻΠϯσοΫε͕֨ೲ͞ΕͨྻΠϯσοΫε഑ ྻɼ֤ߦͷ࠷ॳʹग़ͯ͘Δඇθϩཁૉ͕ɼඇθϩཁૉશମͷԿ൪໨͔ද͢ΦϑηοτΛ֨ೲͨ͠ߦϙΠ ϯλ഑ྻɼͷͭͷ഑ྻͰߏ੒͞ΕΔɽ 9 0 0 0 0 1 5 8 0 0 0 0 0 3 0 0 0 6 0 0 1 𝑑𝑎𝑡𝑎 = 1 5 8 3 6 1 𝑟𝑜𝑤_𝑝𝑡𝑟 = 0 1 3 4 6 𝑐𝑜𝑙_𝑖𝑛𝑑 = 4 0 1 2 1 4   SPX@QUS  
11. CompML $4$
    $PNQSFTTFE
    4QBSTF
    $PMVNO $43ϑΥʔϚοτͰѹॖ͍ͯͨ͠ߦΠϯσοΫεͷ୅ΘΓʹɼྻΠϯσοΫεΛѹॖ͢Δܗࣜɽ 10 0 0 0 0 1 5 8 0 0 0 0 0 3 0 0 0 6 0 0 1 𝑑𝑎𝑡𝑎 = 5 8 6 3 6 1 1 𝑐𝑜𝑙_𝑝𝑡𝑟 = 0 1 3 4 4 6 𝑟𝑜𝑤_𝑖𝑛𝑑 = 1 1 3 2 0 3   DPM@QUS   
12. CompML &--
    &--1"$, ߦ͝ͱͷඇθϩཁૉͷΈΛ֨ೲͨ͠σʔλ഑ྻͱɼσʔλ഑ྻʹ֨ೲ͞ΕͨඇθϩཁૉͷྻΠϯσοΫ εΛ֨ೲ͢ΔΠϯσοΫε഑ྻͰߏ੒͞ΕΔɽ σʔλ഑ྻͷྻͷ௕͞,͸ɼߦ͝ͱͷඇθϩཁૉ਺ͷ࠷େ਺ͱͳΓɼߦͷ௕͞͸ݩͷߦྻͷߦ਺ͱͳΔɽ 11 0 0 0 0 1 5 8 0 0 0 0 0 3 0 0 0 6 0 0 1 𝑑𝑎𝑡𝑎 = 1 ∗ 5 8 3 ∗ 6 1 𝑖𝑛𝑑𝑖𝑐𝑒𝑠 = 4 ∗ 0 1 2 ∗ 1 4 L  L  L  L  NBY
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13. CompML &--
    &--1"$, ͕ෆద੾ͱͳΔέʔε ߦ͝ͱͷඇθϩཁૉͷ࠷େ਺͕ฏۉ஋ͱେ͖͘ҟͳΔ৔߹͸ɼ&--͸ෆద੾ͳදݱํ๏ͱͳΔɽ ὎ &--ͱ$00ͷ૊Έ߹Θͤͷ)ZCSJEͱ͍͏ख๏ɽ 12 0 0 0 0 1 5 8 3 7 2 0 0 4 9 0 0 6 0 0 0 𝑑𝑎𝑡𝑎 = 1 ∗ ∗ ∗ ∗ 5 8 3 7 2 4 9 ∗ ∗ ∗ 6 ∗ ∗ ∗ ∗ 𝑖𝑛𝑑𝑖𝑐𝑒𝑠 = 4 ∗ ∗ ∗ ∗ 0 1 2 3 4 2 3 ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ L  L  L  L  NBY
    L
    
    ,
    
    
    
    
    
    BWFSBHF
    L
14. CompML ):#
    )ZCSJE ඇθϩཁૉͷେ෦෼Λ&--ʹ֨ೲ͠ɼ࢒ΓΛ$00ʹ֨ೲ͢ΔϋΠϒϦουͳϑΥʔϚοτɽ 13 0 0 0 0 1 5 8 3 7 2 0 0 4 9 0 0 6 0 0 0 𝑑𝑎𝑡𝑎 = 1 ∗ 5 8 4 9 6 ∗ 𝑖𝑛𝑑𝑖𝑐𝑒𝑠 = 4 ∗ 0 1 2 3 1 ∗ L  L  L  L  NBY
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    L ELL COO 𝑑𝑎𝑡𝑎 = 3 7 2 𝑟𝑜𝑤_𝑖𝑛𝑑 = 1 1 1 𝑐𝑜𝑙_𝑖𝑛𝑑 = 2 3 4
15. CompML #43
    #MPDL
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    3PX ߦྻΛαΠζ͕ಉ͡ϒϩοΫʹ෼্ׂͨ͠Ͱɼ$43ϥΠΫͳϑΥʔϚοτʹ֨ೲɽ ඇθϩཁૉͷ஋ؚ͕·ΕΔϒϩοΫ಺ͷશͯͷ஋ΛؚΉσʔλ഑ྻɼඇθϩཁૉؚ͕·ΕΔϒϩοΫͷ ྻΠϯσοΫε͕֨ೲ͞ΕͨྻΠϯσοΫε഑ྻɼ֤ߦͷ࠷ॳʹग़ͯ͘ΔϒϩοΫ͕ɼඇθϩཁૉؚ͕ ·ΕΔϒϩοΫશମͷԿ൪໨͔ද͢ΦϑηοτΛ֨ೲͨ͠ߦϙΠϯλ഑ྻɼͷͭͷ഑ྻͰߏ੒͞ΕΔɽ 14 0 0 0 0 1 0 5 8 0 0 0 2 0 0 3 0 0 0 0 6 0 0 0 0 𝑏𝑙𝑘_𝑟𝑜𝑤_𝑝𝑡𝑟 = 0 2 4 𝑏𝑙𝑘_𝑐𝑜𝑙_𝑖𝑛𝑑 = 0 2 0 1  CML@SPX@QUS  𝑑𝑎𝑡𝑎 = 0 0 5 8 1 0 0 2 0 0 0 6 3 0 0 0
16. CompML #439
    &YUFOEFE
    #43 #43ϑΥʔϚοτͷߦϙΠϯλ഑ྻΛɼͭͷ഑ྻʹ෼ׂ͢Δɽͭ໨ͷ഑ྻʹ͸֤ߦͷ࠷ॳͷඇθϩཁ ૉΛؚΉϒϩοΫͷϙΠϯλɼͭ໨ͷ഑ྻʹ͸֤ߦͷ࠷ޙͷඇθϩཁૉΛؚΉϒϩοΫͷϙΠϯλ͕֨ ೲ͞ΕΔɽ CML@SPX@QSUͱCML@FOE@QUSΛมߋ͢Δ͚ͩͰɼݩͷߦྻͷҰ෦ͷϒϩοΫ͚ͩΛ࢒ͨ͠ߦྻΛ࡞Δ͜ͱ ͕Ͱ͖Δɽ 15 0 0 0 0 1 0 5 8 0 0 0 2 0 0 3 0 0 0 0 6 0 0 0 0 𝑏𝑙𝑘_𝑟𝑜𝑤_𝑝𝑡𝑟 = 0 2 𝑏𝑙𝑘_𝑐𝑜𝑙_𝑖𝑛𝑑 = 0 2 0 1 𝑑𝑎𝑡𝑎 = 0 0 5 8 1 0 0 2 0 0 0 6 3 0 0 0 𝑏𝑙𝑘_𝑒𝑛𝑑_𝑝𝑡𝑟 = 2 4
17. CompML #439
    &YUFOEFE
    #43
    ͷར఺ CML@SPX@QSUͱCML@FOE@QUSΛมߋ͢Δ͚ͩͰɼݩͷߦྻͷҰ෦ͷϒϩοΫ͚ͩΛ࢒ͨ͠ߦྻΛ࡞Δ͜ͱ ͕Ͱ͖Δɽ 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 𝑏𝑙𝑘_𝑟𝑜𝑤_𝑝𝑡𝑟 = 0 2 𝑏𝑙𝑘_𝑐𝑜𝑙_𝑖𝑛𝑑 = 0 2 0 1 𝑑𝑎𝑡𝑎 = 0 0 5 8 1 0 0 2 0 0 0 6 3 0 0 0 𝑏𝑙𝑘_𝑒𝑛𝑑_𝑝𝑡𝑟 = 0 3
18. CompML $4#
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    #MPDLT ߦྻΛϒϩοΫʹ෼ׂɽඇθϩཁૉͷ஋͕֨ೲ͞Εͨσʔλ഑ྻɼϒϩοΫ಺ͰͷඇθϩཁૉͷߦΠϯ σοΫεɾྻΠϯσοΫεΛ֨ೲ͢Δ഑ྻɼ֤ϒϩοΫͷ࠷ॳʹग़ͯ͘Δඇθϩཁૉ͕ɼඇθϩཁૉશ ମͷԿ൪໨͔ද͢ΦϑηοτΛ֨ೲͨ͠ϒϩοΫϙΠϯλ഑ྻͰߏ੒͞ΕΔɽ సஔߦྻ΁ͷԠ༻΋Մೳɽ 17 0 0 0 0 1 0 5 8 0 0 0 2 0 0 3 0 0 0 0 6 0 0 0 0   CML@QUS 𝑑𝑎𝑡𝑎 = 5 8 1 2 6 3 𝑏𝑙𝑘_𝑟𝑜𝑤_𝑖𝑛𝑑 = 1 1 0 1 1 0 𝑏𝑙𝑘_𝑐𝑜𝑙_𝑖𝑛𝑑 = 0 1 0 1 1 0 𝑏𝑙𝑘_𝑝𝑡𝑟 = 0 2 2 4 5 6 6     গͳ͍Ϗοτ਺Ͱ ූ߸Խ
19. CompML CSRでのSpMV
20. CompML ٖࣅίʔυ 19 #VMVÇ
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21. CompML $16্Ͱͷ4Q.7 20  ߦྻΛ$43ϑΥʔϚοτʹม׵  ߦྻͱϕΫτϧͷ৐ࢉΛܭࢉ
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23. CompML ࣮ߦ࣌ؒͷൺֱ 22 CPU GPU 0.008600s 0.000043s $16ஞ࣍ॲཧͱ(16ฒྻॲཧͰɼഒͷ଎౓ࠩɽ ˞4Q.7ɿ ߦྻ 
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24. CompML Reference
25. CompML 3FGFSFODF • #FMM
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