Approximate Message Passing
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1 Approximate Message Passing Mohsen Bayati, David Donoho, Adel Javanmard Iain Johnstone, Marc Lelarge, Arian Maleki, Andrea Montanari Stanford University December 8, 2012 Andrea Montanari (Stanford) AMP December 8, / 68
2 Statistical estimation y = f (; noise)! Unknown object y! Observations f ( ; noise)! Parametric model Problem: Estimate from observations y. Andrea Montanari (Stanford) AMP December 8, / 68
3 Statistical estimation y = f (; noise)! Unknown object y! Observations f ( ; noise)! Parametric model Problem: Estimate from observations y. Andrea Montanari (Stanford) AMP December 8, / 68
4 A broad convergence Statistics [Genomics,... ] Data mining [Collaborative ltering, Predictive analytics,... ] Signal processing [Compressive sampling,... ] Inverse problems [Medical imaging, Seismographic imaging,... ] +Data, + Computation, Exploit hidden structure Andrea Montanari (Stanford) AMP December 8, / 68
5 How should we think about these problems? Andrea Montanari (Stanford) AMP December 8, / 68
6 How should we think about these problems? Optimization? maximize Likelihood(jy) Complexity() `Separation principle' Modeler/statistician proposes convex cost function. Optimization expert proposes simple iterative algorithm. Run for 20 iterations and hope for the best. Andrea Montanari (Stanford) AMP December 8, / 68
7 How should we think about these problems? Beyond separation? y! ^ 1! ^ 2! ^ 3! : : : Constrained complexity per iteration Fixed number of iterations (say 20) What is minimum MSE achievable? Andrea Montanari (Stanford) AMP December 8, / 68
8 Outline A long example (algorithm + heuristics) A list of theorems/pointers Andrea Montanari (Stanford) AMP December 8, / 68
9 A long example Andrea Montanari (Stanford) AMP December 8, / 68
10 What type of example? Image processing (because they make nice gures) Compressed sensing (simple) WILL NOT MENTION SPARSITY! Andrea Montanari (Stanford) AMP December 8, / 68
11 What type of example? Image processing (because they make nice gures) Compressed sensing (simple) WILL NOT MENTION SPARSITY! Andrea Montanari (Stanford) AMP December 8, / 68
12 What type of example? Image processing (because they make nice gures) Compressed sensing (simple) WILL NOT MENTION SPARSITY! Andrea Montanari (Stanford) AMP December 8, / 68
13 An image = 2 C n Unknown object (n = : ) Andrea Montanari (Stanford) AMP December 8, / 68
14 Noiseless linear measurements y = A = A Want to reconstruct Andrea Montanari (Stanford) AMP December 8, / 68
15 Noiseless linear measurements y = A = A Want to reconstruct Andrea Montanari (Stanford) AMP December 8, / 68
16 Measurement structure A = e F R ef = subsampled Fourier matrix R = = random modulation! y 2 C m, m = 0:17 n Andrea Montanari (Stanford) AMP December 8, / 68
17 An approach popular in this community y = A + z ; z N(0; 2 I m m ) n 1 o p jy (jy) / exp ky 2 2 Ak2 2 p () my n 1 / exp y a A T 2 o ny a a=1 2 2 i=1 p i (i ) Andrea Montanari (Stanford) AMP December 8, / 68
18 An approach popular in this community y = A + z ; z N(0; 2 I m m ) n 1 o p jy (jy) / exp ky 2 2 Ak2 2 p () my n 1 / exp y a A T 2 o ny a a=1 2 2 i=1 p i (i ) Andrea Montanari (Stanford) AMP December 8, / 68
19 An approach popular in this community y = A + z ; z N(0; 2 I m m ) n 1 o p jy (jy) / exp ky 2 2 Ak2 2 p () my n 1 / exp y a A T 2 o ny a a=1 2 2 i=1 p i (i ) Andrea Montanari (Stanford) AMP December 8, / 68
20 Factor graph! 1 a m 1 i n p jy (jy) / my a=1 n 1 exp 2 2 y a A T 2 o ny a p i (i ) i=1 Use BP! Andrea Montanari (Stanford) AMP December 8, / 68
21 Many issues Anyone knows the prior distribution of natural images? Computation per iteration, memory (mn). Very loopy graph.... Let us try something simpler! Andrea Montanari (Stanford) AMP December 8, / 68
22 Many issues Anyone knows the prior distribution of natural images? Computation per iteration, memory (mn). Very loopy graph.... Let us try something simpler! Andrea Montanari (Stanford) AMP December 8, / 68
23 Constructing a rst estimate y = A Matched lter ^ 1 = 1 m Ay y Andrea Montanari (Stanford) AMP December 8, / 68
24 How good is this? E ^ 1 = (one line calculation) = Andrea Montanari (Stanford) AMP December 8, / 68
25 Check it out ^ 1 = A y y = = Does not look that good! Andrea Montanari (Stanford) AMP December 8, / 68
26 Check it out ^ 1 = A y y = = Does not look that good! Andrea Montanari (Stanford) AMP December 8, / 68
27 Idea = + `noise' Andrea Montanari (Stanford) AMP December 8, / 68
28 Idea = + Andrea Montanari (Stanford) AMP December 8, / 68
29 How big is the `noise'? Efk^ 1 k 2 2g = (two lines calculation) = 1 kk 2 2 Andrea Montanari (Stanford) AMP December 8, / 68
30 Matched lter blows up noise MSE 1 out = MSE in Andrea Montanari (Stanford) AMP December 8, / 68
31 Let's check MSE out MSE in Andrea Montanari (Stanford) AMP December 8, / 68
32 Denoising ^ 1 + z ; z i N(0; 1) Idea: Treat ^ 1 as eective observations in denoising Andrea Montanari (Stanford) AMP December 8, / 68
33 Denoising ^ 1 + z ; z i N(0; 1) Idea: Treat ^ 1 as eective observations in denoising Andrea Montanari (Stanford) AMP December 8, / 68
34 Denoising by nonlocal means by = + z ; ^i = P j W (i ; j )by j P j W (i ; j ) ; ( 1 if kpatch(i ; by) Patch(j ; by)k ; W (i ; j ) = 0 otherwise [Buades, Coll, Morel, 2005] ^ (by) Andrea Montanari (Stanford) AMP December 8, / 68
35 Denoising by nonlocal means by = + z ; ^i = P j W (i ; j )by j P j W (i ; j ) ; ( 1 if kpatch(i ; by) Patch(j ; by)k ; W (i ; j ) = 0 otherwise [Buades, Coll, Morel, 2005] ^ (by) Andrea Montanari (Stanford) AMP December 8, / 68
36 Denoising by nonlocal means by = + z ; ^i = P j W (i ; j )by j P j W (i ; j ) ; ( 1 if kpatch(i ; by) Patch(j ; by)k ; W (i ; j ) = 0 otherwise [Buades, Coll, Morel, 2005] ^ (by) Andrea Montanari (Stanford) AMP December 8, / 68
37 Patches i j Andrea Montanari (Stanford) AMP December 8, / 68
38 Will it work? ^ 2 = (^ 1 ) = (A y y) = Andrea Montanari (Stanford) AMP December 8, / 68
39 Let's try ^ 1 = A y y = ^2 = (A y y) = Better than garbage! Andrea Montanari (Stanford) AMP December 8, / 68
40 How much better? MSE out MSE in Andrea Montanari (Stanford) AMP December 8, / 68
41 How much better? ? MSE out c 2 p x c 1 x MSE in Andrea Montanari (Stanford) AMP December 8, / 68
42 Let us repeat the denoising experiment Andrea Montanari (Stanford) AMP December 8, / 68
43 Let us repeat the denoising experiment: y = + z y (y) = 1 = 0:5 = 0:25 = 0:12 Andrea Montanari (Stanford) AMP December 8, / 68
44 Quantitatively MSE out c 2 p x c 1 x MSE in Andrea Montanari (Stanford) AMP December 8, / 68
45 Quantitatively 0.1 MSE out c 2 p x c 1 x e MSE in Andrea Montanari (Stanford) AMP December 8, / 68
46 Approximate denoiser characterization MSE out = c p MSE in (enough for our purposes) (see also Maleki, Baraniuk, Narayan, 2012) (Arias-Castro, Willett, 2012) Andrea Montanari (Stanford) AMP December 8, / 68
47 What we achieved so far MSE out MSE in 0.1 MSE out MSE in Matched lter Denoiser Andrea Montanari (Stanford) AMP December 8, / 68
48 What we achieved so far MSE in MSE out 0.1 MSE out MSE in Matched lter Denoiser Andrea Montanari (Stanford) AMP December 8, / 68
49 What we achieved so far MSE new MSE old Andrea Montanari (Stanford) AMP December 8, / 68
50 What we achieved so far MSE new MSE old What about iterating? Andrea Montanari (Stanford) AMP December 8, / 68
51 What we achieved so far MSE new MSE old What about iterating? Andrea Montanari (Stanford) AMP December 8, / 68
52 How do we iterate? Will tell you later! Andrea Montanari (Stanford) AMP December 8, / 68
53 t = 1 ^ 1 = Andrea Montanari (Stanford) AMP December 8, / 68
54 t = 2 ^ 2 = Andrea Montanari (Stanford) AMP December 8, / 68
55 t = 3 ^ 3 = Andrea Montanari (Stanford) AMP December 8, / 68
56 t = MSE new Non obvious! MSE old Andrea Montanari (Stanford) AMP December 8, / 68
57 t = 4 ^ 4 = Andrea Montanari (Stanford) AMP December 8, / 68
58 t = MSE new MSE old Andrea Montanari (Stanford) AMP December 8, / 68
59 t = 5 ^ 5 = Andrea Montanari (Stanford) AMP December 8, / 68
60 t = MSE new MSE old Andrea Montanari (Stanford) AMP December 8, / 68
61 t = 6 ^ 6 = Andrea Montanari (Stanford) AMP December 8, / 68
62 t = MSE new MSE old Andrea Montanari (Stanford) AMP December 8, / 68
63 t = 7 ^ 7 = Andrea Montanari (Stanford) AMP December 8, / 68
64 t = MSE new MSE old Andrea Montanari (Stanford) AMP December 8, / 68
65 t = 8 ^ 8 = Andrea Montanari (Stanford) AMP December 8, / 68
66 t = MSE new MSE old Andrea Montanari (Stanford) AMP December 8, / 68
67 t = 9 ^ 9 = Andrea Montanari (Stanford) AMP December 8, / 68
68 t = MSE new MSE old Andrea Montanari (Stanford) AMP December 8, / 68
69 t = 0; 1; 2; 3; : : : ; 20 Andrea Montanari (Stanford) AMP December 8, / 68
70 t = 0; 1; 2; 3; : : : ; MSE new MSE old Andrea Montanari (Stanford) AMP December 8, / 68
71 How do we iterate? Andrea Montanari (Stanford) AMP December 8, / 68
72 Approximate Message Passing (AMP) ^ 2t = (^ 2t 1 ) ^ 2t+1 = ^ 2t + A y r t r t = y A^ 2t + b t r t 1 b t = 1 m div(^ 2t 1 ) (can be computed explicitly) [Thouless, Anderson, Palmer, 1977, Kabashima, 2003, Donoho, Maleki, Montanari, 2009, Donoho, Johnstone, Montanari, 2012] Andrea Montanari (Stanford) AMP December 8, / 68
73 Connection with Belief Propagation m i!j = m i + "i!j Linearize in "i!j Very dierent from naive mean eld! Andrea Montanari (Stanford) AMP December 8, / 68
74 Connection with Belief Propagation m i!j = m i + "i!j Linearize in "i!j Very dierent from naive mean eld! Andrea Montanari (Stanford) AMP December 8, / 68
75 Connection with Perturbation, Optimization, Statistics? Robustness wrt p X 2DistributionClass ( = minimax denoider in DistributionClass) Can rigorously track evolution over A random (What about A = A det + A rand?) Andrea Montanari (Stanford) AMP December 8, / 68
76 Connection with Perturbation, Optimization, Statistics? Robustness wrt p X 2DistributionClass ( = minimax denoider in DistributionClass) Can rigorously track evolution over A random (What about A = A det + A rand?) Andrea Montanari (Stanford) AMP December 8, / 68
77 A list of theorems/pointers Andrea Montanari (Stanford) AMP December 8, / 68
78 A list of theorems/pointers Connection with optimization [Bayati, Montanari, 2011] Minimax theory for sparse/block-sparse/tv [Donoho, Johnstone, Montanari, 2012] Bayesian reconstruction up to information dimension [Donoho, Javanmard, Montanari, 2012] Universality [Bayati, Lelarge, Montanari, 2012] Analysis of Generalized AMP [Javanmard, Montanari, 2012] Application to sparse PCA [In preparation, 2013] Andrea Montanari (Stanford) AMP December 8, / 68
79 Related work (Non-rigorous) replica method. [Tanaka 2002, Guo, Verdú 2005, Kabashima, Tanaka 2009, Rangan, Fletcher, Goyal 2009, Caire, Tulino, Shamai, Verdú ] Alternative argument for robust regression (e.g. min ky Ak 1 ) [Bean, Bickel, El Karoui, Lim, Yu 2012] Generalized linear models [Rangan 2011] Graphical model priors [Schniter et al ] Low-rank matrices [Rangan, Fletcher 2012] Andrea Montanari (Stanford) AMP December 8, / 68
80 Conclusion Andrea Montanari (Stanford) AMP December 8, / 68
81 Conclusion Can do message passing/bp without Bayesian assumptions! There is something between naive mean eld and BP Thanks! Andrea Montanari (Stanford) AMP December 8, / 68
82 Noise distribution? Frequency error Andrea Montanari (Stanford) AMP December 8, / 68
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