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凸最適化を用いた 過負荷MIMO信号検出 林 和則 ,早川 諒 大阪市立大学大学院工学研究科 京都大学大学院情報学研究科

凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

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Page 1: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

凸最適化を用いた過負荷MIMO信号検出

林和則† ,早川 諒‡

†大阪市立大学大学院工学研究科‡京都大学大学院情報学研究科

Page 2: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

2

v過負荷MIMO信号検出のシナリオv線形観測にもとづく連立方程式vスパースベクトルの再構成(圧縮センシング)v離散値ベクトルの再構成(SOAV最適化)v SOAV最適化を解く凸最適化アルゴリズムv SOAV最適化の理論解析v過負荷MIMO信号検出への応用例vまとめ

内容

Page 3: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

3

MIMO信号検出

送信信号ベクトル:受信信号ベクトル:

MIMO信号検出··

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···

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<latexit sha1_base64="DKGc5YOyM889QsEdamFYANN6JF4=">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</latexit>

x<latexit sha1_base64="yfD+14GSJLzDSmo3XBkZj6XUNM8=">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</latexit> y

<latexit sha1_base64="HmLb5oio1cu8WQDUKUc1NkATOvc=">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</latexit>

H<latexit sha1_base64="1asjXLpKsComHuCS5BFBJIb14tQ=">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</latexit>

v<latexit sha1_base64="n153FCYgMkLSmWA0+On01k6etso=">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</latexit>

x̂<latexit sha1_base64="nIE7jl22ReNXi8BwuV7Ty9Sp2lo=">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</latexit>

y = Hx+ v 2 CM<latexit sha1_base64="JKRT7LR5fMw3F7VWl8YB1rUWMaM=">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</latexit>

x 2 CN<latexit sha1_base64="MEWdHNRrPNB9H1kT4fsxPD9J8fo=">AAACp3ichVHLLgRBFD3ae7wGiUhsJiaEzaQaCbGS2Fh5z5jEMOlqhYp+pbtmgjY/4AcsrEgshMQXWNn4AQufIJYkNhbu9HQiCG6nq06duufUvbncs2SgGHus0+obGpuaW1oTbe0dnV3J7p5c4JZ8U2RN13L9PDcCYUlHZJVUlsh7vjBsbok1vjdbvV8rCz+QrrOqDjyxYRs7jtyWpqGIKib7CtwO9ysF6aQKtqF2OQ9nK5vzxWSaZVgUqZ9Aj0EacSy6yVsUsAUXJkqwIeBAEbZgIKBvHToYPOI2EBLnE5LRvUAFCdKWKEtQhkHsHq07dFqPWYfOVc8gUpv0ikW/T8oUhtgDu2Qv7J5dsSf2/qtXGHlUazmgnde0wit2HfevvP2rsmlX2P1U/aHglF2r7JHdUF3Pf/ansI2pqC9JfXoRU+3YrHmUD09eVqaXh8Jhds6eqdczcr0jV6f8al4sieVTJGhY+vfR/AS5sYw+ntGXJtIzU/HYWjCAQYzQbCYxgzksIkvvHuEcV7jWRrUFLafla6laXazpxZfQjA8USp2U</latexit>

ZF:MMSE:

線形MIMO信号検出:N M

<latexit sha1_base64="p4AUhchT8tWG55wJj3j154nVqR0=">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</latexit>

のとき

x̂zf = (HHH)�1

HHy

<latexit sha1_base64="BRpV3rHx9t1aIaCJQQy6EcONHnE=">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</latexit>

x̂mmse = HH

✓HH

H +�2v

�2x

I

◆�1

y

<latexit sha1_base64="XsN2CF4PkCh5fWpTxiClqj9EV6g=">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</latexit>

Page 4: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

4

過負荷MIMO信号検出

MIMO信号検出

···

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H<latexit sha1_base64="1asjXLpKsComHuCS5BFBJIb14tQ=">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</latexit>

v<latexit sha1_base64="n153FCYgMkLSmWA0+On01k6etso=">AAAClXichVFNLwNRFD3GV9VXsSCxaTTEqnmDRCORNCFih1ZLoo3MjIdhvjIznaQm/oCVnWBFYiEWfoGVjT9g0Z8gliQ2Fu58JILgTua9886757x7c2VLUx2XsUaT0NzS2tae6Eh2dnX39Kb6+suOWbMVXlJMzbTXZcnhmmrwkqu6Gl+3bC7pssbX5P254H7N47ajmsaqW7d4VZd2DHVbVSSXqHJF1n3vcDOVYVkWRvonEGOQQRzLZuoOFWzBhIIadHAYcAlrkODQtwERDBZxVfjE2YTU8J7jEEnS1iiLU4ZE7D6tO3TaiFmDzoGnE6oVekWj3yZlGqPskV2zF/bAbtgTe//Vyw89glrqtMuRllubvUdDxbd/VTrtLnY/VX8oZMqOKmuwW6rr+c/+XGwjF/alUp9WyAQdK5GHd3DyUpwpjPpj7JI9U68X5HpProb3qlyt8MI5kjQs8ftofoLyRFaczIorU5l8Lh5bAsMYwTjNZhp5LGIZJXp3D8c4xZkwKMwK88JClCo0xZoBfAlh6QPAyZa9</latexit>

x̂<latexit sha1_base64="nIE7jl22ReNXi8BwuV7Ty9Sp2lo=">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</latexit>

「過負荷」= (送信ストリーム数 > 受信アンテナ)N > M<latexit sha1_base64="VYfDeNTRJr2Z7NJlKBeKaqtrGEA=">AAACknichVG7SgNBFD1ZXzG+4qMQbMSgWIVZFQwpJJLGQiUPo4KK7K5jXLIvdieBGPwBwdYUVgoWYuEXWNn4AxZ+glgq2Fh4d7MgKupddubMmXvO3MtVHUP3BGOPEamtvaOzK9od6+nt6x+IDw6te3bV1XhJsw3b3VQVjxu6xUtCFwbfdFyumKrBN9RK1r/fqHHX021rTdQdvmMqZUvf1zVFEFVcXVjZjSdYkgUx/hPIIUggjJwdv8U29mBDQxUmOCwIwgYUePRtQQaDQ9wOGsS5hPTgnuMIMdJWKYtThkJshdYynbZC1qKz7+kFao1eMeh3STmOSfbArtgLu2fX7Im9/+rVCDz8Wuq0qy0td3YHjkeLb/+qTNoFDj5VfyhUym5V9shuqK7nP/sT2Ecq6EunPp2A8TvWWh61w+ZLMV2YbEyxC/ZMvZ6T6x25WrVX7TLPC2eI0bDk76P5CdZnkvJsUs7PJTKpcGxRjGEC0zSbeWSwhBxK9G4ZJzhFUxqR0tKilG2lSpFQM4wvIS1/AH2qlN8=</latexit>

線形の検出法は特性が大きく劣化

最尤推定に基づく検出法:x̂ml = arg min

s2SN||y �Hs||2

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S<latexit sha1_base64="3Ph320RBndKgWs2nkoWEJKbqPAw=">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</latexit> :送信シンボルのアルファベット

問題のサイズが大きくなると計算量的に破綻

低演算量で送信シンボルの離散性を考慮できる検出法は?

Page 5: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

5

大規模過負荷MIMO信号検出のシナリオIoT端末からのデータ収集:

MIMO信号検出

···

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···

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大規模マルチユーザMIMO:

···

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· · ·<latexit sha1_base64="DKGc5YOyM889QsEdamFYANN6JF4=">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</latexit>

IoTノードクラウド

張り出し局

ファイバ

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6

劣決定線形観測からの信号再構成

:観測ベクトル(既知):未知ベクトル:観測行列(既知)

劣決定線形観測モデル(=過負荷MIMO受信信号):

H<latexit sha1_base64="1asjXLpKsComHuCS5BFBJIb14tQ=">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</latexit>

x<latexit sha1_base64="yfD+14GSJLzDSmo3XBkZj6XUNM8=">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</latexit>y

<latexit sha1_base64="HmLb5oio1cu8WQDUKUc1NkATOvc=">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</latexit>

N<latexit sha1_base64="WghJ/ZIaNqbjew0uknKNO18hG9k=">AAACkHichVG7SgNBFD2u7/iKsRFsgkGxCrMqGGyM2IiFmGhUiCHsrpM4uNlddicBDf6AtoqFlYKFWPgFVjb+gIWfICkj2Fh4s1kQDca77MyZM/ecuZerO6bwJGOvHUpnV3dPb19/aGBwaHgkPBrZ9uyya/CMYZu2u6trHjeFxTNSSJPvOi7XSrrJd/TDlcb9ToW7nrCtLXnk8FxJK1qiIAxNEpVaz4djLM78iLYCNQAxBLFhhx+xh33YMFBGCRwWJGETGjz6slDB4BCXQ5U4l5Dw7zlOECJtmbI4ZWjEHtJapFM2YC06Nzw9X23QKyb9LimjmGIv7I7V2TO7Z2/s80+vqu/RqOWIdr2p5U5+5HR88+NfVYl2iYNvVRuFTtnNyl7ZA9VVa9ufRAEJvy9BfTo+0+jYaHpUji/rm4vpqeo0u2E16vWaXJ/I1aq8G7cpnr5CiIal/h5NK9iejatzcTU1H0smgrH1YQKTmKHZLCCJVWwgQ+9ynOEcF0pESShLynIzVekINGP4EcraFwPDlEA=</latexit>

N<latexit sha1_base64="WghJ/ZIaNqbjew0uknKNO18hG9k=">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</latexit>

M<latexit sha1_base64="uTio7ELmZFm+goIhWRdUC/+TLHo=">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</latexit>

のとき,一般に方程式の解が無限に存在はスパースベクトル

圧縮センシング(2006)

N > M<latexit sha1_base64="VYfDeNTRJr2Z7NJlKBeKaqtrGEA=">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</latexit>

x<latexit sha1_base64="yfD+14GSJLzDSmo3XBkZj6XUNM8=">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</latexit>

は離散値ベクトル

ベイズ圧縮センシング(2012)SOAV最適化(2015)

x<latexit sha1_base64="yfD+14GSJLzDSmo3XBkZj6XUNM8=">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</latexit>

y 2 RM

x 2 RN

H 2 RM⇥N<latexit sha1_base64="2Pyqd0pbbIxs0V6a1SDtl5Pqa0w=">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</latexit>

Page 7: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

7スパースベクトルの再構成:圧縮センシング

劣決定:未知ベクトルはスパース:

観測ベクトル(既知):未知ベクトル:観測行列(既知):

H<latexit sha1_base64="1asjXLpKsComHuCS5BFBJIb14tQ=">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</latexit> x

<latexit sha1_base64="yfD+14GSJLzDSmo3XBkZj6XUNM8=">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</latexit>

y<latexit sha1_base64="HmLb5oio1cu8WQDUKUc1NkATOvc=">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</latexit>

y 2 RM

x 2 RN

H 2 RM⇥N<latexit sha1_base64="2Pyqd0pbbIxs0V6a1SDtl5Pqa0w=">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</latexit>

N > M<latexit sha1_base64="W689PDmHmwy1phMORK7LhGfCxiw=">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</latexit>

||x0|| ⌧ N<latexit sha1_base64="4UvVRUf34jB0So/KnA/8qOWk+nk=">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</latexit>

-線形計画問題に帰着される(実際に計算機で解ける)-ある条件の下では であっても「真の解」が完全に再構成できる理論的な保証がある

- 最適化:�1

N > M<latexit sha1_base64="VYfDeNTRJr2Z7NJlKBeKaqtrGEA=">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</latexit>

x̂`1 = arg minx2RN

||x||1 subject to y = Hx<latexit sha1_base64="LDqxaxFTaTPojVUohp8drsjNwLU=">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</latexit>

Page 8: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

8離散値ベクトルの再構成:SOAV (sum-of-absolute-values)再構成

不良設定:未知ベクトルは離散値:

観測行列(既知):未知ベクトル:観測ベクトル(既知):

H<latexit sha1_base64="1asjXLpKsComHuCS5BFBJIb14tQ=">AAAClXichVE9S8NQFD2NX7V+VR0UXMRScSovKlgEoaBIN7XaVrBFkvjUaL5I0kIN/QNObqJOCg7i4C9wcvEPOPgTxLGCi4O3SUC0qDfkvfPOu+e8e7mypamOy9hzRGhr7+jsinbHenr7+gfig0MFx6zYCs8rpmbam7LkcE01eN5VXY1vWjaXdFnjRflwsXlfrHLbUU1jw61ZvKxLe4a6qyqSS1ShJOtetr4dT7AU82O8FYghSCCMVTN+jxJ2YEJBBTo4DLiENUhw6NuCCAaLuDI84mxCqn/PUUeMtBXK4pQhEXtI6x6dtkLWoHPT0/HVCr2i0W+TchxJ9sRuWIM9slv2wj5+9fJ8j2YtNdrlQMut7YHj0fX3f1U67S72v1R/KGTKDip7ZndU1+uf/bnYRdrvS6U+LZ9pdqwEHtWj08b6fC7pTbIr9kq9XpLrA7ka1Tfleo3nLhCjYYk/R9MKCtMpcSYlrs0mMulwbFGMYQJTNJs5ZJDFKvL07gFOcIZzYURYEJaE5SBViISaYXwLYeUTXLOWjw==</latexit> x

<latexit sha1_base64="yfD+14GSJLzDSmo3XBkZj6XUNM8=">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</latexit>

y<latexit sha1_base64="HmLb5oio1cu8WQDUKUc1NkATOvc=">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</latexit>

-SOAV最適化問題:

M. Nagahara, “Discrete signal reconstruction by sum of absolute values,” IEEE Signal Process. Lett., vol. 22, no. 10, pp. 1575‒1579, Oct. 2015

(離散値ベクトル)ー(ある1つの値だけからなるベクトル)=スパースベクトル

minx2RN

✓1

2||x� 1||1 +

1

2||x+ 1||1

◆subject to y = Hx

<latexit sha1_base64="8yQYJ6TZRA8xvlJuRyT50KFzZoc=">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</latexit>

N > M<latexit sha1_base64="W689PDmHmwy1phMORK7LhGfCxiw=">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</latexit>

y 2 RM<latexit sha1_base64="pRNKs5ccyhTj360h9mM+sMwAUhY=">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</latexit>

H 2 RM⇥N<latexit sha1_base64="CdJ3zBLotKHhLYAwj6PRdUzv8qI=">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</latexit>

x 2 SN<latexit sha1_base64="EoDvwCM3FsJld1QOfAaHcjj6wIw=">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</latexit>

S = {�1, 1}<latexit sha1_base64="dFKyi6qFAerOjSy1pFPD41hYcGw=">AAACpnichVFNSxtRFD2Oto2xrVERBDfBYMmiDXcmIYmCIHTTVVHTGMGEdGb6NEPmi5mXQBziD/APuHBlwUXror+gq276B7rwJxSXFty46J1JSulCe4d5777z7jnvHq7h21Yoia4mlMmpR4+fpKbTM0+fPZ/NzM3vhl4vMEXd9Gwv2DP0UNiWK+rSkrbY8wOhO4YtGkb3dXzf6IsgtDz3nRz4ouXoh651YJm6ZKidWWw6uuyYuh3VhhvN6JX68lhtDtuZHBXWqmWtVM5SgaiiamqcaJVSsZRVGYkjh3FseZmvaOIDPJjowYGAC8m5DR0hf/tQQfAZayFiLODMSu4Fhkgzt8dVgit0Rru8HvJpf4y6fI41w4Rt8is2/wEzs1ilH/SJbug7XdJPurtXK0o04l4GvBsjrvDbsydLtdv/shzeJTp/WQ8wDK4edXZFX7iv6wf9SRygmviy2KefILFjc6TRPzq9qa3vrEYv6CNds9dzVv3Gqm7/l3mxLXbOkOZh/ZlI9v5kVyuoxYK2XcptVsdjS2EZK8jzbCrYxBtsoc7vRjjHZ1wqeeWtUlcao1JlYsxZwD+hvP8NU4ec1Q==</latexit>

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9

SOAV最適化を解くアルゴリズム近接分離に基づく凸最適化手法-Douglas-Rachford 分離-近接勾配法-交互方向乗数法(ADMM)

メッセージ伝搬法に基づく手法-近似メッセージ伝搬法(AMP)に基づく手法(DAMP)

R. Hayakawa and K. Hayashi, "Convex Optimization Based Signal Detection for Massive Overloaded MIMO Systems," IEEE Trans. Wireless Commun., Vol. 16, No. 11, pp. 7080-7091, Nov. 2017.H. Sasahara, K. Hayashi, and M. Nagahara, "Symbol Detection for Faster-than-Nyquist Signaling by Sum-of-Absolute-Values Optimization," IEEE Signal Processing Letters, Vol. 23, No. 12, pp. 1853-1857, Dec. 2016.R. Hayakawa and K. Hayashi, “Reconstruction of Complex Discrete-Valued Vector via Convex Optimization with Sparse Regularizers,” IEEE Access, vol. 6, no. 1, pp. 66499-66512, Dec. 2018.R. Hayakawa and K. Hayashi, "Discreteness-Aware Approximate Message Passing for Discrete-Valued Vector Reconstruction," IEEE Trans. Signal Process., vol. 66, no. 24, pp. 6443-6457, Dec. 2018.

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10

(ただし, はプロパーな閉凸関数,は関数 の実効定義域)

近接分離に基づく凸最適化手法近接写像と呼ばれる操作を利用した最適化アルゴリズム群

パラメータ をもつ近接写像:prox�f (z) = arg min

u2dom(f)

⇢f(u) +

1

2�||u� z||22

<latexit sha1_base64="bv5CETDtZgvY2i2PAHr1WL9IwXo=">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</latexit>

� > 0<latexit sha1_base64="q9o9YRoN3f8wnx+RKuoSoEyR5/o=">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</latexit>

f : RN ! R [ {1}<latexit sha1_base64="oswb+Qa9tnZ1zFj5YnwK2TYO7dw=">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</latexit>

� ! 0<latexit sha1_base64="D8LvA+x5ijGCmaTv+cmv3w82VeU=">AAACpXichVG7SgNBFD2u7/iKj0KwCQYfVZhVQbESbGwEX4mCkTC7jsngvpidRDT4A/6AhY0KFmrhF1jZ+AMWfoJYKthYeLNZEBX1Ljtz5sw9Z+7lWoEjQ83YY4PR2NTc0trWnujo7OruSfb25UK/rGyRtX3HVxsWD4UjPZHVUjtiI1CCu5Yj1q3d+dr9ekWoUPremt4PxJbLi57ckTbXRBWSA/kid12eyitZLGmulL+XYoVkmmVYFKmfwIxBGnEs+clb5LENHzbKcCHgQRN2wBHStwkTDAFxW6gSpwjJ6F7gEAnSlilLUAYndpfWIp02Y9ajc80zjNQ2veLQr0iZwgh7YJfshd2za/bE3n/1qkYetVr2abfqWhEUeo4GV9/+Vbm0a5Q+VX8oLMquV/bIbqiu5z/709jBTNSXpD6DiKl1bNc9KgfHL6uzKyPVUXbOnqnXM3K9I1ev8mpfLIuVEyRoWOb30fwEuYmMOZkxl6fSczPx2NowhGGM02ymMYcFLCFL7x7gFJe4MsaMRWPNyNVTjYZY048vYRQ+ACtsnFU=</latexit>

� ! 1<latexit sha1_base64="oog3BpaMEAqn9/I6F3hktUnbIA8=">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</latexit>

dom(f)<latexit sha1_base64="+CV5EQq1/NT+JgtQMkZca7ff+6g=">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</latexit>

z<latexit sha1_base64="zl0iaOYc5C1kOEid3UtxoJYffN0=">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</latexit>

z<latexit sha1_base64="zl0iaOYc5C1kOEid3UtxoJYffN0=">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</latexit>

最小値射影近接写像

dom(f)<latexit sha1_base64="+CV5EQq1/NT+JgtQMkZca7ff+6g=">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</latexit>

f<latexit sha1_base64="5K40lw/QR8zBJwLL04xdAK1IpjA=">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</latexit>

Page 11: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

11SOAV最適化を解くアルゴリズム:Douglas-Rachford 分離

mins2RN

{�1(s) + �2(s)}<latexit sha1_base64="iAL3S5m+it5nc/9BRvc2PnU+bVo=">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</latexit>

�1(·), �2(·) : RN ! R [ {1}<latexit sha1_base64="61IJWrJvKhe4rJkOcyrY6596BmE=">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</latexit>

はプロパーな閉凸関数

sk = prox��1(rk)

rk+1 = rk + prox��2(2sk � rk)� sk<latexit sha1_base64="lYvzK8Frgm/Fcb7as6G40h0mByg=">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</latexit>

Douglas-Rachfordアルゴリズム:を与えて,以下を繰り返すr0, � > 0

<latexit sha1_base64="FfwzzpmrXUnHYhoTbAT0+TlAaGI=">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</latexit>

近接写像 が閉形式で得られる場合に有効(微分可能でなくて良い. ノルム+指示関数など)

prox��1, prox��2

<latexit sha1_base64="OUg6LnvkBUlnhQav7ys4+gGVnjw=">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</latexit>

`1<latexit sha1_base64="adXE//uOOebTEhZYRsRgHZXpsyg=">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</latexit>

考える最適化問題:

Page 12: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

12SOAV最適化を解くアルゴリズム:近接勾配法 (FISTA)

mins2RN

{�1(s) + �2(s)}<latexit sha1_base64="iAL3S5m+it5nc/9BRvc2PnU+bVo=">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</latexit>

考える最適化問題:�1(·)

<latexit sha1_base64="MWBhrG2VOKh7d12sim03e9O/f4A=">AAACnHichVHLSsNAFD2Nr1pfVTeCIGJRdFMmKlhcFXQhiGBbWwVbShKnOpgmMZkWavEH/IEu3KjgQlz4Ba7c+AMu+gniUsGNC2/TgKioN2TmzJl7ztzL1R1TeJKxZkjp6Ozq7gn3Rvr6BwaHosMjOc+uuAbPGrZpuzu65nFTWDwrhTT5juNyraybfFs/XGndb1e56wnb2pI1hxfK2r4lSsLQJFGFvHMgiups3tiz5VwxGmNx5sfkT6AGIIYgNu3oHfLYgw0DFZTBYUESNqHBo28XKhgc4gqoE+cSEv49xwkipK1QFqcMjdhDWvfptBuwFp1bnp6vNugVk36XlJOYZo/smr2wB3bDntj7r15136NVS412va3lTnHodCzz9q+qTLvEwafqD4VO2e3KmuyW6nr+sz+JEhJ+X4L6dHym1bHR9qgeN14yy+np+gy7ZM/U6wW53pOrVX01rlI8fYYIDUv9PpqfIDcfVxfiamoxlkwEYwtjHFOYpdksIYk1bCJL7x6hgXNcKBPKqrKubLRTlVCgGcWXUHIfCLqY7g==</latexit>

:プロパーな閉凸関数�2(·)

<latexit sha1_base64="6OnyiR9GN9iEDivkU+jmT+960Ok=">AAACnHichVHLSsNAFD3GV62vqhtBELEodVMmVbC4KuhCEMFXq2BLSdKpDqZJTKaFWvwBf8CFGxVciAu/wJUbf8BFP0FcKrhx4W0SEBX1hsycOXPPmXu5umMKTzLWbFPaOzq7uiM90d6+/oHB2NBwzrOrrsGzhm3a7o6uedwUFs9KIU2+47hcq+gm39YPFlv32zXuesK2tmTd4YWKtmeJsjA0SVQh7+yLYiqRN0q2nCnG4izJ/Jj4CdQQxBHGmh27Qx4l2DBQRQUcFiRhExo8+nahgsEhroAGcS4h4d9zHCNK2iplccrQiD2gdY9OuyFr0bnl6flqg14x6XdJOYEp9siu2Qt7YDfsib3/6tXwPVq11GnXAy13ioMno5tv/6oqtEvsf6r+UOiUHVTWZLdU1/Of/UmUkfb7EtSn4zOtjo3Ao3Z0+rK5sDHVmGaX7Jl6vSDXe3K1aq/G1TrfOEOUhqV+H81PkEsl1dmkuj4Xz6TDsUUwhkkkaDbzyGAZa8jSu4c4xTkulHFlSVlRVoNUpS3UjOBLKLkPCu2Y7w==</latexit>

: リプシッツ連続な勾配をもつ微分可能な凸関数

��<latexit sha1_base64="IyJepQgD2huH+CcTGxNt9iz4qMQ=">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</latexit>

近接勾配法:を与えて,以下を繰り返すs0, � 2 (0, 2/�)

<latexit sha1_base64="1SOZ46wm0dYa1e6RR8Xk+iO+/M8=">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</latexit>

sk+1 = prox��1(sk � �r�2(sk))

<latexit sha1_base64="jpFSh+j5t6eJI90Bctk9jBSizfk=">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</latexit>

FISTA: sk+1 = prox��1�1(rk � ��1r�2(rk))

tk+1 =

✓1 +

q4t2k + 1

◆/2

rk+1 = sk +

✓1 +

tk � 1

tk+1

◆(sk+1 � sk)

<latexit sha1_base64="SgLGDgkm2woiSpZM2LGj89o/zR0=">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</latexit>

Page 13: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

13SOAV最適化を解くアルゴリズム:交互方向乗数法 (ADMM)考える最適化問題:

:プロパーな閉凸関数�1(·), �2(·)<latexit sha1_base64="rvbd/RtL5FTWuhe2hRBiwWs8nx8=">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</latexit>

mins2RL,z2RK

{�1(s) + �2(z)} subject to z = As<latexit sha1_base64="YNgXRDl6Wm0gSfTq/LX1MOE5VMA=">AAADHHichVHNahRBEK6Z+BPHn2ySg4KX0SWywbD0bAIJASGSi6CHZOMmgXQcpie9u23mj5nehWSYfQB9AA+BQAQP4sEn8KQHQTwq5BEkJ4ngxYM1PQNigrGG6ar6ur6vqygWeSKRhBxq+tC58xcuDl8yLl+5em2kMjq2moS92OUtN/TCeJ05CfdEwFtSSI+vRzF3fObxNba9mN+v9XmciDB4LHcivuk7nUC0hetIhOzKc+qLwE4p89MkoyIwqe/ILmNpM3vyaMrM8d2T+MOMerwtTZoaNOoK26oV9Mm7Km3UCtYkjUWnK2lmDAaKHWNRjz3lrhzIcJAhquru5e5+khl2pUrqRJl5OrDKoAqlLYWVd0BhC0JwoQc+cAhAYuyBAwl+G2ABgQixTUgRizES6p5DBgZye1jFscJBdBvPDmYbJRpgnmsmiu3iKx7+MTJNmCBfyGtyTD6SN+Qb+fVPrVRp5L3soGcFl0f2yLMbKz//y/LRS+j+YZ3BYFhddHZI3mJfR2fOJ6ENc2ougXNGCskndguN/u6L45X55kR6h7wkRzjrAaq+R9Wg/8N9tcybe5Avyzq5mtPBaqNuTdcbyzPVhblybcNwE25DDXczCwvwAJaghe9+165rpnZL39c/6J/0z0WprpWccfjL9K+/Af8+zEE=</latexit>

ADMMアルゴリズム:

sk+1 = arg mins2RL

⇢�1(s) +

1

2�||As� zk + vk||2

zk+1 = prox��2(Ask+1 + vk)

vk+1 = vk +Ask+1 � zk+1<latexit sha1_base64="Qv+0OkQVWPvzJwUcjAr60KwQE9A=">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</latexit>

Page 14: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

14SOAV最適化を解くアルゴリズム:交互方向乗数法 (ADMM)SOAV最適化の場合:

minx2RN

✓1

2||x� 1||1 +

1

2||x+ 1||1 + �||y �Hx||22

<latexit sha1_base64="cuezpABvh2mBKSdIv+Wj1p6STlg=">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</latexit>

minx,z1,z22RN

✓1

2||z1 � 1||1 +

1

2||z2 + 1||1 + �||y �Hx||22

subject to x = z1, x = z2<latexit sha1_base64="jzAznYqyZIhFLQm9X7V6yO0Cbno=">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</latexit>

�2(z) =1

2||z1 � 1||1 +

1

2||z2 + 1||1

<latexit sha1_base64="WUfrduHOXQMMdE+l2YhObwbgbuI=">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</latexit>

�1(x) = �||y �Hx||22<latexit sha1_base64="MEO1TNAQXVKFWKt1/AKSZVhHA2w=">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</latexit>

z = [zT1 zT

2 ]T

<latexit sha1_base64="tTHiddOPTWGYb8NTjVZEwk4ByAY=">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</latexit>

A = [I I]T<latexit sha1_base64="wFBGIHYmtrJAMAZY5ch773RAVCU=">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</latexit>

minx2RN ,z2R2N

{�1(x) + �2(z)} subject to z = Ax<latexit sha1_base64="0BGp0sF95cDVEdIAE6zKj/r5hSk=">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</latexit>

Page 15: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

15SOAV最適化の理論解析Box-SOAV最適化のシンボル誤り率特性ボックス制約付きSOAV最適化問題(Box-SOAV):

minx2[r1,rL]N

LX

l=1

ql||x� rl1||1 +1

2||y �Hx||22

!

<latexit sha1_base64="HSpvP7SZGdrmZv2V22nlF94LRv4=">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</latexit>

max�>0

min↵>0

(↵�

p�

2+

�2v�

p�

2↵� 1

2�2 � ↵�

2p�

+�p�

↵E

env ↵

�p

�f

✓X +

↵p�H

◆�)

<latexit sha1_base64="YH1q6Xh1BHq1lS39eM87CWqjZfw=">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</latexit>

||Q(x̂)� x||0/N<latexit sha1_base64="VPIAtaRwUx+B4yHTP6Tp4uh4VuM=">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</latexit>

シンボル誤り率:

1�LX

`=1

p` Pr

⇢Q✓prox ↵⇤

�⇤p�f

✓X +

↵⇤

p�H

◆◆= r` | X = r`

<latexit sha1_base64="Dz8urQIlWCbsjk9ZOavz82IzcSI=">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</latexit>

N,M ! 1, � = M/N<latexit sha1_base64="9WgqlHTQkfqWqxN1tXAVV/sbkAI=">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</latexit>

( )ただし, は以下の最適化問題の解↵⇤,�⇤

<latexit sha1_base64="7+DiQ19fbtCpitZvFYqBENaU5Ic=">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</latexit>

Convex Gaussian Mini-max Theoremを用いて導出

Page 16: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

16

2値(0,1)の場合のSER 3値(-1,0,1)の場合のSER

SOAV最適化の理論解析Box-SOAV最適化のシンボル誤り率特性

Page 17: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

17SOAV最適化の理論解析DAMPの理論解析(状態発展法による)

2値ベクトルの場合 離散値スパースベクトルの場合

ノイズレス, 値のときの理論限界: m >

✓1� 1

K

◆nK

<latexit sha1_base64="CilVUDGaTmJyb2W8rK0yelEibj8=">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</latexit>

R. Hayakawa and K. Hayashi, "Discreteness-Aware Approximate Message Passing for Discrete-Valued Vector Reconstruction," IEEE Trans. Signal Process., vol. 66, no. 24, pp. 6443-6457, Dec. 2018.

Page 18: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

18

SOAV最適化の繰り返し処理SOAV最適化:

R. Hayakawa and K. Hayashi, "Convex Optimization Based Signal Detection for Massive Overloaded MIMO Systems," IEEE Trans. Wireless Commun., Vol. 16, No. 11, pp. 7080-7091, Nov. 2017.

minx2RN

✓1

2||x� 1||1 +

1

2||x+ 1||1 + �||y �Hx||22

<latexit sha1_base64="cuezpABvh2mBKSdIv+Wj1p6STlg=">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</latexit>

W-SOAV最適化:

ベクトル毎の重みから,成分毎の重みへ

事前情報を重みとして取り込める→ ターボ信号処理との相性がよい

minx2RN

0

@NX

j=1

(w+j |xj � 1|+ w�

j |xj + 1|) + �||y �Hx||22

1

A

<latexit sha1_base64="3qbnJZLXZzW1XDh44vhPRWAjEWM=">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</latexit>

Page 19: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

19

W-SOAV最適化による過負荷MIMO信号検出

R. Hayakawa and K. Hayashi, "Convex Optimization Based Signal Detection for Massive Overloaded MIMO Systems," IEEE Trans. Wireless Commun., Vol. 16, No. 11, pp. 7080-7091, Nov. 2017.

W-SOAV単独の繰り返し W-SOAVとLDPC復号の繰り返し

n=100, m=64

Page 20: 凸最適化を用いた 過負荷MIMO信号検出 - IEICE The …...2 v過負荷MIMO信号検出のシナリオ v線形観測にもとづく連立方程式 vスパースベクトルの再構成(圧縮センシング)

20

SOAV最適化による過負荷MIMO-OFDM信号検出MIMO-OFDM受信信号モデル(プリコーディング無し) :2

64rf1...

rfM

3

75 =

2

64⇤1,1 · · · ⇤1,N...

...⇤M,1 · · · ⇤M,N

3

75

2

64s1...sN

3

75+

2

64vf1...

vfM

3

75

MIMO-OFDM受信信号モデル(プリコーディング有り):2

64rf1...

rfM

3

75 =

2

64⇤1,1X1 · · · ⇤1,NXN

......

⇤M,1X1 · · · ⇤M,NXN

3

75

2

64s1...sN

3

75+

2

64vf1...

vfM

3

75X1, . . . ,XN

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

プリコーディング行列: 共通のアダマール行列共通のDFT行列(SC-CP方式)

プリコーディング行列:

最適化による手法 (DFISTA)

システムサイズ小 大

プリコーディング

無 ☓ ○有 ○ ○

確率推論による手法(DAMP)

システムサイズ小 大

プリコーディング

無 ☓ ☓有 ☓ ◎

でOK

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IoT環境でのデータ収集過負荷MIMOとの違い→非アクティブ端末が存在

実部と虚部に分割して実数のアルゴリズムを用いると「実部が0(非アクティブ)なら,虚部も0」という事実が考慮されない

複素版のアルゴリズム

R. Hayakawa and K. Hayashi, “Reconstruction of Complex Discrete-Valued Vector via Convex Optimization with Sparse Regularizers,” IEEE Access, vol. 6, no. 1, pp. 66499-66512, Dec. 2018.

「OFDM信号のあるサブキャリアが0(非アクティブ)なら,他のサブキャリアも0」

グループスパース性を考慮したアルゴリズム

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SOAV最適化を用いたIoT環境でのデータ収集

IW: 繰り返し処理有り(T:繰り返し回数)DFISTA, BODAMP: 実数版アルゴリズム

SCSR: 複素版アルゴリズムDGS: グループスパース版アルゴリズム

アクティブ端末数: 10 アクティブ端末数: 70

変調方式:OFDM-QPSKプリコーディング:無し

送信端末数:N=80受信アンテナ数:M=60

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まとめv過負荷MIMO信号検出の問題設定v劣決定線形観測からの信号再構成v SOAV最適化v SOAV最適化のための凸最適化に基づくアルゴリズムv SOAV最適化の理論解析vW-SOAV最適化の繰り返し処理v SOAV最適化を用いた過負荷MIMO-OFDM信号検出v SOAV最適化を用いたIoT端末からのデータ収集