マクマホン立方体を用いたテンソル分解理解支援の教材開発

⇃ᮏ㧗➼ᑓ㛛Ꮫᰯ ◊✲⣖せ ➨ ྕ㸦㸧

࣐ࢡ࣐࣍ࣥ❧᪉యࢆ⏝࠸ࡓࢸࣥࢯࣝศゎ⌮ゎᨭ᥼ࡢᩍᮦ㛤Ⓨ

ᒣᮏ ┤ᶞ

1,*

_{ ▼⏣ ᫂⏨}

2

_{ ኱▼ ಙᘯ}

1

_{ ᮧୖ ⣧}

1

Development of Teaching Materials to Support Understanding of Tensor Decomposition

Using MacMahon’s Coloured Cubes

Naoki Yamamoto1,*_{, Akio Ishida}2_{, Nobuhiro Oishi}1_{, Jun Murakami}1

In this paper, we devised a 3D puzzle using the MacMahon’s cubes and represented this with a 2D map by using the matrix unfolding used to calculate HOSVD. Using this map representation, we developed teaching materials to learn some of calculation processes of HOSVD. In order to investigate the visibility of the map and the difficulty level of the puzzle, the teaching materials are tried by our students and others at the Open Campus Event. As a result, it was confirmed that most users could understand the map and that the puzzle was rather easy to solve. Therefore, it was thought that the users also could understand the matrix unfolding.

࣮࣮࢟࣡ࢻ㸸࣐ࢡ࣐࣍ࣥ❧᪉యࠊ❧యࣃࢬࣝࠊ⾜ิᒎ㛤ࠊHOSVDࠊᩍᮦ㛤Ⓨ

.H\ZRUGV㸸MacMahon’s cubes, 3D puzzle, Matrix unfolding, HOSVD, Development of teaching materials

㸯㸬ࡣࡌࡵ࡟

ࣅࢵࢢࢹ࣮ࢱศᯒ࡛ࡣࠊࢸࣥࢯࣝศゎࡀࡼࡃ⏝࠸ࡽࢀ࡚ ࠸ࡿ(1)_{ࠋࡇࢀࡣࠊࢸࣥࢯ࡛ࣝ⾲ࡉࢀࡓ㧗ḟඖࢹ࣮ࢱࢆࠊ⾜ิ} ࡜ࢸࣥࢯࣝࡢ✚ࡸ࣋ࢡࢺࣝࡢ✚ࡢ࿴࡟ศゎࡍࡿࡶࡢ࡛࠶ ࡿࠋࡓࡔࡋࠊ㧗㝵ࢸࣥࢯࣝࡣከḟඖ㓄ิ࡜ྠᵝࡢព࿡࡛⏝ ࠸ࡿࠋᡃࠎࡣࡇࢀࡲ࡛ࠊࢸࣥࢯࣝศゎࢆ฼⏝ࡋࡓ་⒪ࢹ࣮ ࢱศᯒ(2)_{ࡸࠊ❧యࣃࢬࣝࢆ⏝࠸ࡓࢸࣥࢯࣝศゎᩍ⫱࡟㛵ࡍ} ࡿ◊✲(3)_{࡞࡝ࢆ⾜ࡗ࡚ࡁࡓࠋᚋ⪅࡛ࡣࠊ࣮ࣝࣅࢵࢡ࣮࢟ࣗ} ࣈࡸ࢖ࣥࢫࢱࣥࢺ࣭࢖ࣥࢧࢽࢸ࢕ࢆ❧యࣃࢬࣝ࡜ࡋ࡚ྲྀࡾ ୖࡆ࡚࠸ࡿࡀࠊᐇ㝿ࡢࣃࢬࣝࢆ౛♧ࡍࡿ㝿࡟ࠊ࢖ࣥࢫࢱࣥ ࢺ࣭࢖ࣥࢧࢽࢸ࢕ࡣ㞴ゎ࡛࠶ࡿࡢ࡛ࠊࡼࡾゎࡁ᫆࠸ࣃࢬࣝ ࡢᚲせᛶࢆᐇឤࡋ࡚࠸ࡓࠋ ࡑࡇ࡛ࠊ࣐ࢡ࣐࣍ࣥ❧᪉య(4)_{ࢆ฼⏝ࡋ࡚ᵓᡂࡋࡓ❧యࣃ} ࢬࣝࢆ⏝࠸࡚ࠊࢸࣥࢯࣝศゎࡢ1 ࡘ࡛࠶ࡿ㧗ḟ≉␗್ศゎ 㸦HOSVD㸧(5)_{ࡢィ⟬㐣⛬࡟⌧ࢀࡿQ-࣮ࣔࢻ⾜ิᒎ㛤ࡑࡢ㏫} ࡢ᧯స࡛࠶ࡿ␚ࡳ㎸ࡳࡢཎ⌮ࢆᏛࡤࡏࡿᩍᮦࢆ㛤Ⓨࡋࡓࠋ ᮏㄽᩥ࡛ࡣࠊࡲࡎ࣐ࢡ࣐࣍ࣥ❧᪉య࡜❧యࣃࢬࣝ࡟ࡘ࠸࡚ ㄝ᫂ࡋࠊḟ࡟❧యࣃࢬࣝࡢ㧗㝵ࢸࣥࢯࣝ⾲⌧࠾ࡼࡧࡑࡢ࣐ ࢵࣉ⾲⌧࡟ࡘ࠸࡚㏙࡭ࡿࠋࡑࡋ࡚ࠊࡑࢀࡽࡢ⾲⌧ࢆ⏝࠸࡚ࠊ 㛤Ⓨࡋࡓᩍᮦ࡟ࡘ࠸࡚㏙࡭ࠊ᭱ᚋ࡟ᮏᰯࡢ࣮࢜ࣉࣥ࢟ࣕࣥ ࣃࢫ࡛୰Ꮫ⏕࣭㧗ᑓ⏕ࡽ࡟ヨ⏝ࡋ࡚ࡶࡽ࠸ࠊ࣐ࢵࣉ⾲⌧ࡢ ぢ᫆ࡉࡸ⾜ิᒎ㛤➼ࡢ⌮ゎᗘ࡞࡝࡟ࡘ࠸࡚ศᯒࡋࡓ⤖ᯝࡶ ♧ࡍࠋ

㸰㸬࣐ࢡ࣐࣍ࣥ❧᪉యࢆ⏝࠸ࡓ❧యࣃࢬࣝ

࣐ࢡ࣐࣍ࣥ❧᪉యࡣࠊ❧᪉యࡢྛ㠃ࡀ␗࡞ࡿ6 Ⰽ࡛ᙬⰍ ࡉࢀࡓࡶࡢ࡛ࠊྛ㠃ࡢⰍࡢ⤌ࡳྜࢃࡏ࡟ࡼࡾ 30 ✀㢮ࡢ࢟ ࣮ࣗࣈࡀᚓࡽࢀࡿ(4)_{ࠋᮏㄽᩥ࡛ࡣࡇࡢ30 ✀㢮࠿ࡽ 4 ✀㢮ࡢ} ࣮࢟ࣗࣈࢆ㑅ࡧࠊࡑࢀࡽࢆࣃࢬࣝࣆ࣮ࢫ࡜ࡋ࡚❧యࣃࢬࣝ ࢆᵓᡂࡋࡓࠋᅗ  ࡟ᮌ〇ࡢ❧᪉య࡟Ⰽࢩ࣮ࣝࢆ㈞௜ࡋ࡚స 〇ࡋࡓ4 ✀㢮ࡢ࣮࢟ࣗࣈ࠿ࡽ࡞ࡿ❧యࣃࢬࣝࢆ♧ࡍࠋྛ࢟ ࣮ࣗࣈࡢ㓄Ⰽࡣᅗ  ࡟ᒎ㛤ᅗ࡜ࡋ࡚⾲ࡍࠋ౑⏝ࡋࡓⰍࡣ㉥ (1)ࠊⓑ(2)ࠊ㟷(3)ࠊ⥳(4)ࠊ㯤(5)ࠊ㯮(6) ࡢ 6 Ⰽ࡛ࠊᣓᘼෆࡢ ᩘᏐࡣⰍ␒ྕ࡛࠶ࡿࠋᒎ㛤ᅗ࡛⦪ᶓࡢ୪ࡧࡢࢡࣟࢫࡍࡿ㒊 ศࡀ࣮࢟ࣗࣈࡢୖ㠃㸦㉥㸧࡜࡞ࡿࠋ ࡇࡢ❧యࣃࢬࣝࡣࠊᅗ  ࡛ୖ㠃㸦㉥㸧࠿ࡽぢ࡚ࠊྛ࢟ࣗ ࣮ࣈࢆʹ ൈ ʹ࡟୪࡭࡚ᵓᡂࡋࡓ┤᪉య࡟ࡘ࠸࡚ࠊྛഃ㠃ࡢⰍ ࡀྠⰍ࡜࡞ࡿࡼ࠺࡟ᥞ࠼ࡿࡶࡢ࡛࠶ࡾࠊ࢖ࣥࢫࢱࣥࢺ࣭࢖  1 _{㟁Ꮚ᝟ሗࢩࢫࢸ࣒ᕤᏛ⣔} ࠛ861-1102 ⇃ᮏ┴ྜᚿᕷ㡲ᒇ 2659-2

Faculty of Electronics and Information Systems Engineering, 2659-2 Suya, Koshi-shi, Kumamoto, Japan 861-1102  2 _{ࣜ࣋ࣛࣝ࢔࣮ࢶ⣔}

ࠛ861-1102 ⇃ᮏ┴ྜᚿᕷ㡲ᒇ 2659-2 Faculty of Liberal Arts,

2659-2 Suya, Koshi-shi, Kumamoto, Japan 861-1102  * Corresponding author:

E-mail address: [email protected] (N. Yamamoto).

㏿ ሗ

ᅗ  ⮬సࡋࡓ❧యࣃࢬࣝ



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࣐ࢡ࣐࣍ࣥ❧᪉యࢆ⏝࠸ࡓࢸࣥࢯࣝศゎ⌮ゎᨭ᥼ࡢᩍᮦ㛤Ⓨ㸦ᒣᮏ࣭▼⏣࣭኱▼࣭ᮧୖ㸧

  5HVHDUFK5HSRUWVRI 1,7.XPDPRWR&ROOHJH 9RO  ࣥࢧࢽࢸ࢕࡜ẚ࡭࡚ゎࡃࡢࡀᐜ᫆࡞ࣃࢬࣝ࡜ࡋ࡚ࠊ࣐ࢡ࣐ ࣍ࣥ❧᪉యࡢ࣮࢟ࣗࣈࢆ฼⏝ࡋ࡚⪃᱌ࡋࡓࠋ

㸱㸬❧యࣃࢬࣝࡢ㧗㝵ࢸࣥࢯࣝ⾲⌧࡜࣐ࢵࣉ

❧యࣃࢬࣝࡢ㧗㝵ࢸࣥࢯࣝ⾲⌧ ᪤㏙ࡢࡼ࠺࡟ࠊᮏㄽᩥ࡛ᢅ࠺㧗㝵ࢸࣥࢯࣝ࡜ࡣከḟඖ㓄 ิࡢࡇ࡜࡛ࠊ౛࠼ࡤ1㹼3 㝵ࡢࢸࣥࢯࣝࡣࡑࢀࡒࢀࠊ࣋ࢡࢺ ࣝࠊ⾜ิࠊ3 ḟඖ㓄ิ࡟ᑐᛂࡍࡿࠋࡇࡇ࡛ࡣࠊᅗ  ࡢ❧య ࣃࢬࣝࢆከḟඖ㓄ิ࡛⾲ࡍࡓࡵ࡟ࠊᅗ  ࡢࡼ࠺࡞5 㝵ࢸࣥ ࢯ࡛ࣝࣔࢹࣝ໬ࡋࡓࠋࡍ࡞ࢃࡕࠊᅗ  ࡢͳ ൈ ͳ ൈ ͳࡢྛ࢟ࣗ ࣮ࣈࢆ͵ ൈ ͵ ൈ ͵࡟ᣑ኱ࡋࠊྛ㠃ࡢ୰ኸࡢせ⣲࡟ඖࡢ࣮࢟ࣗ ࣈࡢ㓄Ⰽࢹ࣮ࢱࢆ୚࠼ࡓࠋඖࡢྛ࣮࢟ࣗࣈࡣʹ ൈ ʹ࡟୪࡭ࡽ ࢀࡿࡓࡵࠊ5 㝵ࢸࣥࢯࣝࡢࢧ࢖ࢬࡣ͵ ൈ ͵ ൈ ͵ ൈ ʹ ൈ ʹ࡜࡞ࡿࠋ ࠸ࡲࠊࡇࡢ 5 㝵ࢸࣥࢯࣝࢆऋ࡜ࡋ࡚ࠊࡑࡢሺ݅ǡ ݆ǡ ݇ǡ ݈ǡ ݉ሻせ ⣲ࢆܽ௜௝௞௟௠࡜ࡍࡿ࡜ࠊࡇࡢࢸࣥࢯࣝࡣḟᘧ࡛⾲ࡉࢀࡿࠋ ऋ ൌ ൫ܽ௜௝௞௟௠൯ǡ ሺ݅ǡ ݆ǡ ݇ ൌ ͳǡʹǡ͵Ǣ ݈ǡ ݉ ൌ ͳǡʹሻሺͳሻ ⾲  ࡟ࠊࡇࡢࢸࣥࢯࣝ࡟୚࠼ࡿせ⣲ࡈ࡜ࡢ㓄Ⰽ್ࢆ♧ࡍࠋ ࡓࡔࡋࠊ⾲࡟♧ࡉࢀ࡚࠸࡞࠸せ⣲␒ྕࡢせ⣲್ࡣ඲࡚0 ࡜ ࡍࡿࠋᅗ  ࡟࠾ࡅࡿQ-modeǡ ሺ݊ ൌ ͳǡʹǡ ǥ ǡͷሻ࡜ࡣࠊࢸࣥࢯࣝ ࢹ࣮ࢱࡢ▮༳࡛♧ࡋࡓྛ᪉ྥࢆ⾲ࡋ㸪ࡑࢀࡒࢀࡣࢸࣥࢯࣝ ऋࡢῧᏐ݅ǡ ݆ǡ ݇ǡ ݈ǡ ݉࡟ᑐᛂࡍࡿࠋ ❧యࣃࢬࣝࡢ࣐ࢵࣉ  ࡇࡢ❧యࣃࢬࣝࢆ࣐ࢵࣉ࡛⾲ࡍࡓࡵ࡟ࠊ㧗ḟ≉␗್ศゎ 㸦HOSVD㸧ࡢィ⟬࢔ࣝࢦࣜࢬ࣒࡛౑⏝ࡉࢀࡿ Q-࣮ࣔࢻ⾜ิ ᒎ㛤(5)_{ࢆ⏝࠸ࡿࠋHOSVD ࡣ⾜ิࡢ≉␗್ศゎ㸦SVD㸧ࢆ 3} 㝵௨ୖࡢ㧗㝵ࢸࣥࢯࣝࡢศゎ࡟ᣑᙇࡋࡓࡶࡢ࡛ࠊQ-࣮ࣔࢻ ⾜ิᒎ㛤࡜ࡣ㧗㝵ࢸࣥࢯࣝࢆ⾜ิ࡟ኚ᥮ࡍࡿ᧯సࡢࡇ࡜࡛ ࠶ࡿࠋࡇࡇ࡛ࡣࠊ3 㝵ࢸࣥࢯࣝࡢ 1-࣮ࣔࢻ⾜ิᒎ㛤ࢆ฼⏝ ࡍࡿࡓࡵࠊࡑࡢ࢔ࣝࢦࣜࢬ࣒ࢆḟ࡟♧ࡍࠋ 㹙 㹙㸦࢔ࣝࢦࣜࢬ࣒㸧 㝵㝵ࢸࣥࢯࣝࡢ࣮ࣔࣔࢻ⾜ิᒎ㛤㹛 ධຊ㸸 ࢧ࢖ࢬܫ ൈ ܬ ൈ ܭࡢ 3 㝵ࢸࣥࢯࣝञ  ฟຊ㸸 ࢧ࢖ࢬܫ ൈ ܬ ή ܭࡢ 1-࣮ࣔࢻ⾜ิᒎ㛤ࢀሺଵሻ 㸦 ࢫ ࢸ ࢵ ࣉ 1 㸧 3 㝵 ࢸ ࣥ ࢯ ࣝञ ࠿ ࡽ ࠊ 㒊 ศ ⾜ ิ ࢀ௝ൌ ൫ݐכ௝כ൯ǡ ሺ݆ ൌ ͳǡʹǡ ǥ ǡ ܬሻࢆᢳฟࡋࠊࢀሺଵሻൌ ൫ࢀଵȁࢀଶȁ ڮ ȁࢀ௃൯࡜ᶓ࡟⤖ ྜࡍࡿࠋ࡞࠾ࠊ൫ݐכ௝כ൯ࡣࠊ݆ࢆᅛᐃࡋ࡚݅ ൌ ͳǡʹǡ ǥ ǡ ܫ࠾ࡼࡧ݇ ൌ ͳǡʹǡ ǥ ǡ ܭ࡜ࡋ࡚ᚓࡽࢀࡿ⾜ิ࡛࠶ࡿࠋ 㸦ࢫࢸࢵࣉ2㸧 ࢀሺଵሻࢆ㏉ࡍࠋ 㸦 㸦࢔ࣝࢦࣜࢬ࣒⤊ࢃࡾ㸧 ᅗ  ྛ࣮࢟ࣗࣈࡢ㓄Ⰽࡢᒎ㛤ᅗ ⾲  ᅗ  ࡟࠾ࡅࡿྛ࣮࢟ࣗࣈࡢ㓄Ⰽ᝟ሗ 䜻䝳䞊䝤 ␒ྕ㻌 䝔䞁䝋䝹㻌 せ⣲␒ྕ㻌 䜻䝳䞊䝤㻌 㓄Ⰽ䠄್䠅㻌 䜻䝳䞊䝤 ␒ྕ㻌 䝔䞁䝋䝹㻌 せ⣲␒ྕ㻌 䜻䝳䞊䝤㻌 㓄Ⰽ䠄್䠅㻌 㻝㻌 㻔㻞㻘㻝㻘㻞㻘㻝㻘㻝㻕㻌 ⥳䠄㻠䠅㻌 㻟㻌 㻔㻞㻘㻝㻘㻞㻘㻝㻘㻞㻕㻌 ⥳䠄㻠䠅㻌 㻔㻝㻘㻞㻘㻞㻘㻝㻘㻝㻕㻌 ㉥䠄㻝䠅㻌 㻔㻝㻘㻞㻘㻞㻘㻝㻘㻞㻕㻌 ㉥䠄㻝䠅㻌 㻔㻞㻘㻞㻘㻝㻘㻝㻘㻝㻕㻌 㟷䠄㻟䠅㻌 㻔㻞㻘㻞㻘㻝㻘㻝㻘㻞㻕㻌 㟷䠄㻟䠅㻌 㻔㻟㻘㻞㻘㻞㻘㻝㻘㻝㻕㻌 㯮䠄㻢䠅㻌 㻔㻟㻘㻞㻘㻞㻘㻝㻘㻞㻕㻌 㯮䠄㻢䠅㻌 㻔㻞㻘㻞㻘㻟㻘㻝㻘㻝㻕㻌 㯤䠄㻡䠅㻌 㻔㻞㻘㻞㻘㻟㻘㻝㻘㻞㻕㻌 ⓑ䠄㻞䠅㻌 㻔㻞㻘㻟㻘㻞㻘㻝㻘㻝㻕㻌 ⓑ䠄㻞䠅㻌 㻔㻞㻘㻟㻘㻞㻘㻝㻘㻞㻕㻌 㯤䠄㻡䠅㻌 㻞㻌 㻔㻞㻘㻝㻘㻞㻘㻞㻘㻝㻕㻌 ⓑ䠄㻞䠅㻌 㻠㻌 㻔㻞㻘㻝㻘㻞㻘㻞㻘㻞㻕㻌 㯤䠄㻡䠅㻌 㻔㻝㻘㻞㻘㻞㻘㻞㻘㻝㻕㻌 ㉥䠄㻝䠅㻌 㻔㻝㻘㻞㻘㻞㻘㻞㻘㻞㻕㻌 ㉥䠄㻝䠅㻌 㻔㻞㻘㻞㻘㻝㻘㻞㻘㻝㻕㻌 㟷䠄㻟䠅㻌 㻔㻞㻘㻞㻘㻝㻘㻞㻘㻞㻕㻌 㟷䠄㻟䠅㻌 㻔㻟㻘㻞㻘㻞㻘㻞㻘㻝㻕㻌 㯮䠄㻢䠅㻌 㻔㻟㻘㻞㻘㻞㻘㻞㻘㻞㻕㻌 㯮䠄㻢䠅㻌 㻔㻞㻘㻞㻘㻟㻘㻞㻘㻝㻕㻌 㯤䠄㻡䠅㻌 㻔㻞㻘㻞㻘㻟㻘㻞㻘㻞㻕㻌 ⓑ䠄㻞䠅㻌 㻔㻞㻘㻟㻘㻞㻘㻞㻘㻝㻕㻌 ⥳䠄㻠䠅㻌 㻔㻞㻘㻟㻘㻞㻘㻞㻘㻞㻕㻌 ⥳䠄㻠䠅㻌 ᅗ  ❧యࣃࢬࣝࡢ  㝵ࢸࣥࢯࣝ⾲⌧



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⇃ᮏ㧗➼ᑓ㛛Ꮫᰯ ◊✲⣖せ ➨ ྕ㸦㸧  ḟ࡟ࠊࡇࡢ❧యࣃࢬࣝࡢ࣐ࢵࣉసᡂࡢ࢔ࣝࢦࣜࢬ࣒ࢆ♧ ࡍࠋ 㹙 㹙㸦࢔ࣝࢦࣜࢬ࣒㸧❧యࣃࢬࣝࡢ࣐ࢵࣉసᡂ㹛  ධຊ㸸 ࢧ࢖ࢬ͵ ൈ ͵ ൈ ͵ ൈ ʹ ൈ ʹࡢ 5 㝵ࢸࣥࢯࣝऋ  ฟຊ㸸 ❧యࣃࢬࣝࡢ࣐ࢵࣉ⾜ิ࡭୫ୟ୮ 㸦ࢫࢸࢵࣉ1㸧 ऋ࠿ࡽࠊྛ࣮࢟ࣗࣈࡢ 3 㝵ࢸࣥࢯࣝऋ௟ǡ௠ൌ ሺܽכככ௟௠ሻǡ ሺ݈ǡ ݉ ൌ ͳǡʹሻࢆᢳฟࡍࡿࠋࡇࡇ࡛ࠊሺܽכככ௟௠ሻࡣ݈ǡ ݉ࢆᅛ ᐃࡋࡓ݅ǡ ݆ǡ ݇ ൌ ͳǡʹǡ͵ࡢ 3 㝵ࢸࣥࢯ࡛ࣝ࠶ࡿࠋ 㸦ࢫࢸࢵࣉ 2㸧 ࢫࢸࢵࣉ 1 ࡛ᚓࡽࢀࡓऋ௟ǡ௠ǡ ሺ݈ǡ ݉ ൌ ͳǡʹሻ࡟ ࡑࢀࡒࢀ࢔ࣝࢦࣜࢬ࣒ 1 ࢆ㐺⏝ࡋࠊ1-࣮ࣔࢻ⾜ิᒎ㛤 ࡭௟ǡ௠ሺଵሻǡ ሺ݈ǡ ݉ ൌ ͳǡʹሻࢆồࡵࡿࠋ 㸦ࢫࢸࢵࣉ 3㸧 ࢫࢸࢵࣉ 2 ࡛ᚓࡽࢀࡓ࡭௟ǡ௠ሺଵሻǡ ሺ݈ǡ ݉ ൌ ͳǡʹሻ ࢆḟᘧࡢࡼ࠺࡟୪࡭࡚࣐ࢵࣉ⾜ิ࡭୫ୟ୮ࢆᵓᡂࡍࡿࠋ ࡭୫ୟ୮ൌ ൬ ࡭ଵǡଶሺଵሻ ࡭ଶǡଶሺଵሻ ࡭ଵǡଵሺଵሻ ࡭ଶǡଵሺଵሻ൰ሺʹሻ 㸦ࢫࢸࢵࣉ4㸧 ࡭୫ୟ୮ࢆ㏉ࡍࠋ 㸦 㸦࢔ࣝࢦࣜࢬ࣒⤊ࢃࡾ㸧  ᘧ(2)࡟࠾ࡅࡿࠊ࡭ଵǡଵሺଵሻࠊ࡭ଶǡଵሺଵሻࠊ࡭ଵǡଶሺଵሻࠊ࡭ଶǡଶሺଵሻࡀࡑࢀࡒࢀࠊ ࣮࢟ࣗࣈ1ࠊ2ࠊ3ࠊ4 ࡢ㒊ศⓗ࡞࣐ࢵࣉ⾜ิ࡛࠶ࡿࠋ

㸲㸬❧యࣃࢬࣝࡢ࣐ࢵࣉࢆ฼⏝ࡋࡓᩍᮦ㛤Ⓨ

㛤Ⓨࡋࡓᩍᮦ࡜ࡑࡢ౑⏝ἲ  HOSVD ࡢ࢔ࣝࢦࣜࢬ࣒ࡢ࠺ࡕࠊ㧗㝵ࢸࣥࢯࣝࢆ⾜ิ࡟ ኚ᥮ࡍࡿQ-࣮ࣔࢻ⾜ิᒎ㛤࡜ࠊࡑࡢ㏫ࡢ᧯స࡛࠶ࡿ␚ࡳ㎸ ࡳࡢᴫᛕࡸᡭ㡰ࢆᏛ⩦ࡉࡏࡿ┠ⓗ࡛ࠊᅗ4 ࡢࡼ࠺࡞ᩍᮦࢆ ࣃ࣮࣏࣡࢖ࣥࢺ࡟ࡼࡾ㛤Ⓨࡋࡓࠋ௨ୗ࡟ࠊᩍᮦࡢ౑⏝ᡭ㡰 ࢆ㏙࡭ࡿࠋ ᅗ (a)ࡢࢱ࢖ࢺࣝ⏬㠃ࢆࢹ࢕ࢫࣉࣞ࢖ୖ࡟⾲♧ࡋ࡚ࠊ ࣃࢬࣝࡢゎࡁ᪉ࢆㄝ᫂ࡍࡿࠋ ᅗ (b)ࢆ⾲♧ࡋ࡚࣐ࢵࣉࡢぢ᪉ࢆㄝ᫂ࡍࡿࠋࡑࡢᚋࠊ ࣐ࢵࣉࢆ☜ㄆࡋ࡞ࡀࡽࠊᅗ  ࡟♧ࡋࡓྛ࣮࢟ࣗࣈࢆ㓄⨨ࡍ ࡿ⦎⩦ࢆࡉࡏࡿࠋᅗ  ࡟ࡇࡢᵝᏊࢆ♧ࡍࠋࡓࡔࡋࠊᅗ (b) ࡢ࣐ࢵࣉࡣࠊ3.2 ⠇࡛㏙࡭ࡓᘧ(2)ࡢ࣐ࢵࣉ⾜ิ࡭୫ୟ୮ࡢ㓄Ⰽ ್ࢆᣢࡘせ⣲࡟ࠊⰍ᝟ሗ࡜๓ᚋ࣭ୖୗ࣭ᕥྑࡢ఩⨨᝟ሗࢆ ௜ຍࡋ࡚సᡂࡋࡓࡶࡢ࡛࠶ࡿࠋ ᅗ (c)ࢆ⾲♧ࡋ࡚ࣃࢬࣝࢆ㛤ጞࡋࠊ㸯ศ㛫ࡢไ㝈᫬㛫 ෆ࡟ᥞ࠼ࡉࡏࡿࠋ࣐࢘ࢫࢆࢡࣜࢵࢡࡍࡿ࡜ࢫࢱ࣮ࢺ㡢ࡀ㬆 ࡾࠊ⤒㐣᫬㛫ࡣᅗ (c)ࡢୗഃࡢ㟷Ⰽࡢࣂ࣮ࡀᕥ࠿ࡽྑ࡟ఙ ࡧࡿࡇ࡜࡛▱ࡽࡏࡿࠋ  ᅗ (d)ࢆ⾲♧ࡋ࡚ࠊࡇࡢࣃࢬࣝࡢゎࡢ  ౛ࢆ☜ㄆࡉ ࡏࡿࠋᅗ (b)ࡢ᫬࡜ྠᵝ࡟ࠊ࣐ࢵࣉࢆぢ࡞ࡀࡽᐇ㝿࡟࢟ࣗ ࣮ࣈࢆ㓄⨨ࡋ࡚ࡇࡢゎࢆ☜ㄆࡉࡏࡿࠋ ᮏᩍᮦࡢྛ࣮࢟ࣗࣈࡢ࣐ࢵࣉࡣ 1-࣮ࣔࢻ⾜ิᒎ㛤࠿ࡽస ᡂࡉࢀ࡚࠾ࡾࠊࡇࢀࡽࡢ࣐ࢵࣉࢆ⧞ࡾ㏉ࡋ☜ㄆࡍࡿࡇ࡜࡛ ⾜ิᒎ㛤ࡢᏛ⩦ࡀ࡛ࡁࠊࡉࡽ࡟࣐ࢵࣉࢆ☜ㄆࡋ࡞ࡀࡽᐇ㝿 ࡟࣮࢟ࣗࣈࢆ㓄⨨ࡉࡏࡿࡇ࡜࡛␚ࡳ㎸ࡳࡢᏛ⩦ࡀ࡛ࡁࡿ࡜ ⪃࠼࡚࠸ࡿࠋ ᮏᩍᮦࡢヨ⏝⤖ᯝ ᮏᖺ8 ᭶࡟ᮏᰯ࡛㛤ദࡉࢀࡓ࣮࢜ࣉࣥ࢟ࣕࣥࣃࢫ࡟ゼࢀ ࡓ୰Ꮫ⏕ࡸ㧗ᑓ⏕࡞࡝࡟ᮏᩍᮦࢆ౑⏝ࡋ࡚ࡶࡽࡗࡓࠋ౑⏝ ᅗ  㛤Ⓨࡋࡓᩍᮦࡢ⾲♧౛



D ࣃࢬࣝࡢࢱ࢖ࢺࣝ⾲♧



F ไ㝈᫬㛫ෆ࡟ࣃࢬࣝࢆゎ࠿ࡏࡿタၥ⏬㠃



G ゎ⟅☜ㄆ⏝࣐ࢵࣉࡢ⾲♧



E ⦎⩦⏝࣐ࢵࣉࡢ⾲♧



― 81 ― ⇃ᮏ㧗➼ᑓ㛛Ꮫᰯࠉ◊✲⣖せࠉ➨11ྕ㸦2019㸧

࣐ࢡ࣐࣍ࣥ❧᪉యࢆ⏝࠸ࡓࢸࣥࢯࣝศゎ⌮ゎᨭ᥼ࡢᩍᮦ㛤Ⓨ㸦ᒣᮏ࣭▼⏣࣭኱▼࣭ᮧୖ㸧

  5HVHDUFK5HSRUWVRI 1,7.XPDPRWR&ROOHJH 9RO  ⪅ࡢෆヂࡣࠊ୰Ꮫ⏕22 ྡࠊ㧗ᑓ⏕୍࣭⯡ 8 ྡࡢィ 30 ྡ࡛ ࠶ࡗࡓࠋ4.1 ⠇࡛㏙࡭ࡓࡼ࠺࡟ࠊ฼⏝⪅࡟ᩍᮦࢆ฼⏝ࡋ࡚ࡶ ࡽࡗࡓᚋ࡛ࠊ⡆༢࡞㉁ၥࢆ⾜ࡗࡓࠋᅗ  ࡟ࠊࡑࡢෆᐜ࡜㞟 ィ⤖ᯝࢆ♧ࡍࠋ ㉁ၥ1 ࡣࣃࢬࣝࡢ㞴᫆ᗘࢆᑜࡡࡿࡶࡢ࡛ࠊ฼⏝⪅ࡢ 70% ࡀࣃࢬࣝࢆࠕ㞴ࡋ࠸ࠖ࡜ឤࡌ࡚࠸ࡿࡀࠊᅗ (c)ࡢタၥ࡛ࡣ 30 ྡ୰ 16 ྡࡀไ㝈᫬㛫ෆ࡟ࣃࢬࣝࢆᥞ࠼ࡿࡇ࡜ࡀ࡛ࡁ࡚ ࠾ࡾࠊ඲యࡢ༙ᩘ௨ୖࡀࢡࣜ࢔࡛ࡁ࡚࠸ࡓࠋ ㉁ၥ 2 ࡣ࣐ࢵࣉࡢぢ᫆ࡉ࡟ࡘ࠸࡚࡛ࠊࠕぢ᫆࠸ࠖ࡜ࠕᬑ ㏻ࠖࡢྜィࡀ⣙77%࡛࠶ࡾࠊᮏᩍᮦࡢ࣐ࢵࣉࡣᴫࡡぢ᫆࠸ ࡜ཷࡅྲྀࡽࢀ࡚࠸ࡿࡇ࡜ࡀศ࠿ࡗࡓࠋࡲࡓࠊ౑⏝⪅඲ဨࡀࠊ ᅗ (d)ࡢ࣐ࢵࣉࢆぢ࡚ࣃࢬࣝࡢゎ⟅ࡢ☜ㄆࢆ⾜࠺ࡇ࡜ࡀ࡛ ࡁࡓࡇ࡜࠿ࡽࠊ࣐ࢵࣉࡢぢ᪉ࡣ⌮ゎ࡛ࡁࡓ࡜⪃࠼ࡽࢀࡿࠋ ᅗ  ࡣࠊ౑⏝⪅ࡀࣃࢬࣝࡢゎ⟅ࡢ࣐ࢵࣉࢆぢ࡞ࡀࡽ࢟ࣗ ࣮ࣈࢆ㓄⨨ࡋ࡚⟅࠼ࢆ☜ㄆ࡛ࡁࡿࡲ࡛࡟せࡋࡓ᫬㛫ࡢ⣼✚ ┦ᑐᗘᩘศᕸࢆ♧ࡍࡶࡢ࡛࠶ࡿࠋ☜ㄆ࡟せࡋࡓᖹᆒ᫬㛫ࡣ ⣙ 63 ⛊࡛࠶ࡿࡀࠊࡇࡢ᫬㛫ࡲ࡛࡟⣙ 57%ࡀ⟅࠼ࢆ☜ㄆ࡛ ࡁࠊ80 ⛊ࡲ࡛࡟⣙ 70%ࠊ100 ⛊ࡲ࡛࡟⣙ 93%ࡀ☜ㄆ࡛ࡁ࡚ ࠸ࡿࡇ࡜ࡀศ࠿ࡿࠋ 

㸳㸬ࡲ࡜ࡵ

ᡃࠎࡀᚑ᮶࠿ࡽࢸࣥࢯࣝศゎᩍ⫱ᨭ᥼࡟⏝࠸࡚࠸ࡿ࢖ࣥ ࢫࢱࣥࢺ࣭࢖ࣥࢧࢽࢸ࢕ࡼࡾࡶ㞴᫆ᗘࡢప࠸❧యࣃࢬࣝ࡜ ࡋ࡚ࠊ࣐ࢡ࣐࣍ࣥ❧᪉యࡢ࠺ࡕࡢ࠸ࡃࡘ࠿ࢆ⏝࠸ࡓࣃࢬࣝ ࢆ⪃᱌ࡋࡓࠋࡑࡋ࡚ࠊHOSVD ࡢ࣮ࣔࢻ⾜ิᒎ㛤ࢆᛂ⏝ࡋ ࡚ࡇࡢࣃࢬࣝࢆ  ḟඖࡢ࣐ࢵࣉ࡛⾲⌧ࡋࡓࠋࡇࡢ࣐ࢵࣉࢆ ཧ↷ࡋ࡞ࡀࡽࣃࢬࣝࢆゎࡃࡇ࡜࡛ࠊHOSVD ࡢ⾜ิᒎ㛤࠾ ࡼࡧ␚ࡳ㎸ࡳࢆᏛ⩦ࡍࡿᩍᮦ࡟ࡋࡓࠋ㛤Ⓨࡋࡓᩍᮦࡣࠊ࢜ ࣮ࣉࣥ࢟ࣕࣥࣃࢫ࡛୰Ꮫ⏕ࡸ㧗ᑓ⏕ࡽ࡟౑⏝ࡋ࡚ࡶࡽ࠸ࠊ ࣃࢬࣝࡢ㞴᫆ᗘࡸ࣐ࢵࣉࡢぢ᫆ࡉࢆㄪᰝࡋࡓࠋࡑࡢ⤖ᯝࠊ 㞴᫆ᗘ࡟ࡘ࠸࡚ࡣ㞴ࡋ࠸࡜ࡢᅇ⟅ࡀከ࠸ࡶࡢࡢࠊ༙ᩘ௨ୖ ࡣไ㝈᫬㛫ෆ࡟ṇゎࢆᚓࡿࡇ࡜ࡀ࡛ࡁࡿࡇ࡜ࡀ☜ㄆ࡛ࡁ ࡓࠋࡲࡓ࣐ࢵࣉࡣᴫࡡぢ᫆࠸࡜ࡢホ౯࡛ࠊ඲ဨࡀぢ᪉ࢆ⌮ ゎ࡛ࡁࡓࡇ࡜ࡀศ࠿ࡗࡓࠋᮏᩍᮦࢆ⏝࠸ࡿࡇ࡜࡛ࠊ⾜ิᒎ 㛤࠾ࡼࡧ␚ࡳ㎸ࡳࡢᴫᛕࢆ⌮ゎ࡛ࡁࡿࡶࡢ࡜⪃࠼ࡽࢀࡿࠋ ௒ᚋࡢㄢ㢟࡜ࡋ࡚ࡣࠊᮏᩍᮦࢆᮏᰯ㧗Ꮫᖺࡸᑓᨷ⛉⏕ࡢ HOSVD ࡢᏛ⩦⏝࡟౑⏝ࡋ࡚ᩍ⫱ຠᯝࢆ᳨ドࡍࡿࡇ࡜ࡀᣲ ࡆࡽࢀࡿࠋ ㅰ㎡ ᭱ᚋ࡟ࠊᮏࣃࢬࣝࡢヨ⏝࡟㝿ࡋ࡚ㄝ᫂➼ࡢᑐᛂ࡟ᙜࡓࡗ ࡓᮏᰯᑓᨷ⛉⏕ࡢᒸ⏣኱ᖾྩ࡜஬ᔱ༓㈡ဢྩ࡟ឤㅰࡍࡿࠋ 㸦௧࿴ඖᖺ9 ᭶ 25 ᪥ཷ௜㸧 㸦௧࿴ඖᖺ12 ᭶ 5 ᪥ཷ⌮㸧 ཧ⪃ᩥ⊩

(1) A. Cichoki :”Era of Big Data Processing: A New Approach via Tensor Networks and Tensor Decomposition”, arXiv Preprint, arXiv:1403.2048 (2014)㸬

(2) A. Ishida, K. Kawakami, D. Furushima, N. Yamamoto, and J. Murakami :”Analysis of Relationships between Amount of Physical Activity of Patients in Rehabilitation and Their ADL Scores Using Multidimensional PCA”, Advances in Intelligent Systems and Computing, Vol.690, pp.147-158 (2017)㸬

(3) ᒣᮏ┤ᶞ㸪ᮧୖ⣧㸪▼⏣᫂⏨㸸ࠕࢸࣥࢯࣝศゎࣉࣟࢢࣛ ࣑ࣥࢢࡢ⌮ゎᨭ᥼ࡢࡓࡵࡢ❧యࣃࢬࣝࡢ฼⏝ࠖ㸪ࢯࣇ ࢺ࢙࢘࢔࣭ࢩ࣏ࣥࢪ࣒࢘ 2019 ㄽᩥ㞟㸪pp.114-123 (2019)㸬

(4) J. Köller :”MacMahon's Coloured Cubes”,

http://www.mathematische-basteleien.de/macmahon.htm, Retrieved Aug. 7, 2019㸬

(5) L. De Lathauwer, B. De Moor, and J. Vandewalle :”A Multilinear Singular Value Decomposition”, SIAM Journal on Matrix Analysis and Applications, Vol.21, No.4, pp.1253-1278 (2000)㸬 ᅗ  ࣐ࢵࣉࡢぢ᪉ࡢ⦎⩦ࡢᵝᏊ



ᅗ  ⟅࠼ࡢ☜ㄆ࡟せࡍࡿ᫬㛫ࡢ⣼✚┦ᑐᗘᩘศᕸ



                 ⟅傮僔☜ㄆ僑せ傽僅᫬㛫凚⛊凛 ⣼✚┦ᑐᗘᩘ ᅗ  ㉁ၥࡢ㞟ィ⤖ᯝ



僪傻傽傪 ᬑ㏻ 㞴傽傪 ぢ᫆傪 ᬑ㏻ ぢ㞴傪 ㉁ၥࣃࢬࣝࡢ㞴᫆ᗘ ࡟ࡘ࠸࡚ ㉁ၥ ࣃࢬ࣐ࣝࢵࣉࡢぢ᫆ࡉ ࡟ࡘ࠸࡚ ― 82 ―