Target distribution:p(x,y)} (x,y)~卫 Question:What are these measures? ·[Z12] 88m8=&8m8.-89 ·[Z12] R(p)=flogrank+(P)] -P=[p(x,y)]xy
Target distribution: {𝑝 𝑥, 𝑦 } • Question: What are these measures? • [Z’12] RCorr(𝑝) = RComm(𝑝). −𝑅(𝑝) QCorr(𝑝) = QComm(𝑝), −𝑄(𝑝) • [Z’12] 𝑅(𝑝) = ⌈log 𝑟𝑎𝑛𝑘+(𝑃)⌉ – 𝑃 = 𝑝 𝑥, 𝑦 𝑥,𝑦 (𝑥, 𝑦) 𝑝
R(p)=logrank+(P) For (entry-wise)nonnegative matrices,we can define more variants. ·Nonnegative rank rank(M)=minfr:M=>=1Mi,rank(Mi)=13 rank+(M)=min{r:M =k=1Mk,rank(Mk)=1,Mk =0 Extensively-studied in linear algebra and engineering.Many connections to (T)CS
𝑅(𝑝) = ⌈log 𝑟𝑎𝑛𝑘+(𝑃)⌉ • For (entry-wise) nonnegative matrices, we can define more variants. • Nonnegative rank 𝑟𝑎𝑛𝑘 𝑀 = min 𝑟: 𝑀 = σ𝑖=1 𝑟 𝑀𝑖 , 𝑟𝑎𝑛𝑘 𝑀𝑖 = 1 𝑟𝑎𝑛𝑘+ 𝑀 = min 𝑟:𝑀 = σ𝑘=1 𝑟 𝑀𝑘 , 𝑟𝑎𝑛𝑘 𝑀𝑘 = 1,𝑀𝑘 ≥ 0 • Extensively-studied in linear algebra and engineering. Many connections to (T)CS