The spin-glass cornucopia

Transcription

1 The spin-glass cornucopia Marc Mézard! Ecole normale supérieure - PSL Research University and CNRS, Université Paris Sud LPT-ENS, January 2015

2 40 years of research

3 n 40 years of research 70 s:anomalies in the magnetic response of some obscure and useless alloys

4 n n 40 years of research 70 s:anomalies in the magnetic response of some obscure and useless alloys 1975: Edwards and Anderson define a model, an order parameter, a new phase of matter

5 n n n 40 years of research 70 s:anomalies in the magnetic response of some obscure and useless alloys 1975: Edwards and Anderson define a model, an order parameter, a new phase of matter : Theory develops along two lines: mean field theory (Sherrington-Kirkpatrick) and «real space» droplets

6 n n n n 40 years of research 70 s:anomalies in the magnetic response of some obscure and useless alloys 1975: Edwards and Anderson define a model, an order parameter, a new phase of matter : Theory develops along two lines: mean field theory (Sherrington-Kirkpatrick) and «real space» droplets : Solution of SK model by G. Parisi with replicas. Physical understanding (ultrametricity), cavity method (MPV) (MP)

7 Where do we stand? Real spin glasses: still an open problem A very sophisticated and powerful corpus of conceptual and methodological approaches has been developed, and used in many different fields Portrait of Ottavio Strada, Tintoretto, Venice 1567 Rijk s Museum Amsterdam

8 The cornucopia Statistical physics (glasses, disordered polymers ) Computer Science problems: satisfiability, coloring) Information Theory (error correction, inference Signal acquisition and processing - big data Significant information Gene expression networks Neural networks Interacting agents on a market

9 Why? Beginning of statistical physics and condensed matter theory : homogeneous systems. All atoms are alike. Easy (so to say)

10 Why? Beginning of statistical physics and condensed matter theory : homogeneous systems. All atoms are alike. Easy (so to say) Gradually : add dirt! Defects, dislocations, pinning sites, mixtures, walls, etc.

11 Why? Beginning of statistical physics and condensed matter theory : homogeneous systems. All atoms are alike. Easy (so to say) Gradually : add dirt! Defects, dislocations, pinning sites, mixtures, walls, etc. Last forty years : handle strongly disordered systems. Each atom has a different environment. Even the nature of the basic phases is hard to understand. Heterogeneous «agents»

12 Why? Beginning of statistical physics and condensed matter theory : homogeneous systems. All atoms are alike. Easy (so to say) Gradually : add dirt! Defects, dislocations, pinning sites, mixtures, walls, etc. Last forty years : handle strongly disordered systems. Each atom has a different environment. Even the nature of the basic phases is hard to understand. Heterogeneous «agents»

13 n n n Table of content Spin glasses dynamics, mean field description Information theory: phase transitions and glass phases in General setup: link to hardness) n Bird s eye view : compressed sensing

14 Magnets s i {±1} J ij j E = J ij s i s j i ij Equilibrium: P (s 1,...,s N )= 1 Z e E/T Ferromagnet: J ij = J>0 At low T: spins align, P concentrates on

15 Magnets s i {±1} J ij j E = J ij s i s j i ij Equilibrium: P (s 1,...,s N )= 1 Z e E/T Ferromagnet: J ij = J>0 Phases : hs i i = M «representative agent»

16 Magnets s i {±1} J ij j E = J ij s i s j i ij Equilibrium: P (s 1,...,s N )= 1 Z e E/T Ferromagnet: J ij = J>0 Phases : hs i i = M «representative agent» 0 1 M = X j J ij s j A ' tanh (zjm)

17 Magnets s i {±1} J ij j E = J ij s i s j i ij Equilibrium: P (s 1,...,s N )= 1 Z e E/T Ferromagnet: Phases : hs i i = M «representative agent» 0 J ij = J>0 1 M T Two states M = X j J ij s j A ' tanh (zjm)

18 CuMn i J ij j Spin glasses s i {±1} E = J ij s i s j ij P (s 1,...,s N )= 1 Z e E/T Spin glass: J ij sign depends on ij frustration What is the behavior

19 Spin Glasses Linear response to a small magnetic field: i J ij j χ (a.u.) T=12 K New dynamics Memory T=10 K t t 1 3 T=12 K t t t E. Vincent et al, SPEC time (min)

20 Mean-field lessons Energy Energy Magnetization 1 N σ i s i Configurations Ising Spin glass 1- Glass «phase» : Many pure states, unrelated by symmetry, organized in a hierarchical «ultrametric» structure 2- Many metastable states, unrelated by symmetry 3- «True» ground state : fragile!

21 Take-home Glass phases Proliferation of states (exponentially many) Hierarchical structures Various time scales : aging, modification of FDT Technically : powerful methods - Replicas - Dynamical field theories - Cavity method self-consistent equations for m i, magnetization of spin i in state and statistical analysis

22 An example from Information Theory Claude Shannon ( )

23 An example from Information Theory Claude Shannon ( ) Talk: S ~ Log N 2 or: S ~ N

24 An example from Information Theory More recent convergence: Error correcting codes, pool testing, compressed sensing Claude Shannon ( ) Talk: S ~ Log N 2 or: S ~ N

25 An example from Information Theory More recent convergence: Error correcting codes, pool testing, compressed sensing The new frontier : infinitely complex information Claude Shannon ( ) Talk: S ~ Log N 2 or: S ~ N

26 Information transmission and error correction

27

28 Principle of error correction : redundancy L = Transmission Encoder Decoder Channel a x r a Original Encoded Received Estimate of the message message message original message N M bits N bits N bits L = N M bits Encoding = add redundancy. Rate L/N

29 Principle of error correction : redundancy L = Transmission Encoder Decoder Channel a x r a Original Encoded Received Estimate of the message message message original message N M bits N bits N bits L = N M bits Encoding = add redundancy. Rate L/N e.g. repetition rate =1/3

30 Principle of error correction : redundancy L = Transmission Encoder Decoder Channel a x r a Original Encoded Received Estimate of the message message message original message N M bits N bits N bits L = N M bits Encoding = add redundancy. Rate L/N e.g. repetition rate =1/ Code Channel Majority decoding error probability p 3 +3p 2 (1 p) 3p 2

31 Principle of error correction : redundancy L = Transmission Encoder Decoder Channel a x r a Original Encoded Received Estimate of the message message message original message N M bits N bits N bits L = N M bits Encoding = add redundancy. Rate L/N Shannon s theorem: for a given noise level p, one can build a coder/decoder which transmits with r<r c (p)

32 Principle of error correction : redundancy L = Transmission Encoder Decoder Channel a x r a Original Encoded Received Estimate of the message message message original message N M bits N bits N bits L = N M bits Encoding = add redundancy. Rate L/N Shannon s theorem: for a given noise level p, one can build a coder/decoder which transmits with Two ingredients: - «Thermodynamic limit» N,L!1 - Ensemble of Random Codes ( Random Energy Model of spin glasses) r<r c (p)

33 Unit hypercube in dimensions N Shannon code ensemble codewords (random) sent codeword

34 Unit hypercube in dimensions N Shannon code ensemble codewords (random) sent codeword received word

35 Probability of perfect decoding: Phase transitions in decoding Decoding = find closest codeword 1 0 p = noise Shannon geometric phase transition = REM phase transition

36 Unit hypercube in dimensions N Shannon code ensemble codewords (random) sent codeword received word Perfect codes in principle, but impossible to decode in practice (structureless time ) O(e N )

37 Efficient codes : parity checks Shannon s code is useless (time O(e N ) ) Add redundancy, with structure allowing to decode a b c x 1 + x 4 + x 5 + x 7 = 0 (mod 2) x 2 + x 4 + x 6 + x 7 = 0 (mod 2) x 3 + x 5 + x 6 + x 7 = 0 (mod 2) codewords among words

38 Error decoding with parity checks variables, N 1 M = (1 R)N Random construction with K L equations variables per equation equations per variable N = 20,M = 10,R=1/2,L=3,K =6

39 Error decoding with parity checks variables, N 1 M = (1 R)N Random construction with K L equations variables per equation equations per variable

40 Error decoding with parity checks variables, N 1 M = (1 R)N Random construction with K L equations variables per equation equations per variable Flipped bits

41 Error decoding with parity checks variables, N 1 M = (1 R)N Random construction with K L equations variables per equation equations per variable violated parity constraints Flipped bits

42 Decoding =iteration of mean-field «cavity» equations. «Message passing» algorithm - «Belief propagation» Error decoding with parity checks variables, N 1 M = (1 R)N Random construction with K L equations variables per equation equations per variable violated parity constraints Flipped bits

43 Unit hypercube in dimensions N Error correction: decoding codewords sent codeword

44 Unit hypercube in dimensions N Error correction: decoding codewords sent codeword received word

45 Unit hypercube in dimensions N Error correction: decoding codewords sent codeword received word Seek a codeword at distance from received word pn

46 Unit hypercube in dimensions N Error correction: decoding codewords sent codeword received word metastable states Seek a codeword at distance from received word pn

47 Phase transitions in decoding Probability of perfect decoding: 1 0 p = noise Algo Max Prob Shannon

48 Phase transitions in decoding Probability of perfect decoding: 1 0 p = noise Algo Max Prob Shannon

49 Phase transitions in decoding Probability of perfect decoding: 1 Metastables states=traps 0 p = noise Algo Max Prob Shannon Glass phase

50 Phase transitions in decoding Probability of perfect decoding: 1 0 p = noise Algo Max Prob Shannon

51 Phase transitions in decoding Probability of perfect decoding: 1 0 Ex: 3-6 code on BSC p = noise Algo Max Prob Shannon p d =.084 p c =.101 p S =.110

52 Parity check code in easy phase, time O(N) 1 Phase transitions in decoding Parity-check code, in its glass phase, time O(e N ) «Ideal» code, time O(e N ) 0 p = noise Algo Max Prob Shannon

53 Parity check code in easy phase, time O(N) 1 Phase transitions in decoding Parity-check code, in its glass phase, time O(e N ) «Ideal» code, time O(e N ) 0 Ex: 3-6 code on BSC p = noise Algo Max Prob Shannon p d =.084 p c =.101 p S =.110

54 Example Repetition code k =3 s encoder t channel Coding Transmission f = 10% r decoder Decoding ŝ p =0.1 D. MacKay

55 Low density parity check code with rate 1/3 Transmission Decoding p =0.1 (1 f) 0 0 f 1 1 (1 f) 0 1 (1 f) f (1 f) 0 1 Errorless

56 Glasses and algorithms Phase transitions and glass phases are ubiquitous, in nature and in algorithms trying to solve constraint satisfaction problems with many variables and constraints. Phase transitions relate to the equilibrium system, reached after infinitely long (i.e. exponential) time Dynamic glass transitions relate to the practical performance of (polynomial time) algorithms Deep links with computer science

57 P versus NP SAT LDPC NP complete 3 Colouring TSP (d) 3SAT 3-d spin glass Hamiltonian cycle LDPC SAT TSP 3SAT 3-d spin glass (d) 3 Colouring Hamiltonian cycle NP NP = P = NP complete 2SAT Eulerian circuit P Assignment 2 colouring 2-d spin glass 2SAT Eulerian circuit Assignment 2 colouring 2-d spin glass Conjectured Possible P= NP P= NP

58 Links between NP-hardness and glass transition? Worst-case vs typical sample. Statistical physics = behavior of a typical large-size sample drawn from an ensemble - Importance of phase transitions - Importance of metastable states - Use of physics ideas to come up with new types of algorithms. Mean-field cavity message passing algorithms are among the most powerful algorithms for a large class of inference and constraint satisfaction problems

59 Example : Phase diagram of satisfiability and coloring SAT (E = 0 ) UNSAT (E >0) state Many states E=0 E>0 Many states E>0 α α = d c α =M/N Structural (geometrical) phase transition. Impacts all known algorithms. Survey propagation = best known algo

60 Significant data and knowledge extraction Images with pixels : Instantaneous patterns of firing neurons :

61 Significant data and knowledge extraction Images with pixels : Instantaneous patterns of firing neurons : Youtube : «How to draw George W Busch in 15 seconds»:

62 Significant data, and knowledge extraction Extract significant information from high-dimensional data : For the scientist, the citizen, the neural network, the computer scientist (machine learning)! NB: JPEG compression : from to

63 Compressed sensing Possible applications:! - Rapid Magnetic Resonance Imaging - Tomography - Electronic microscopy - Fluorescence microscopy - Astronomy - Generalized linear regression - Group Testing - Infer regulatory interactions among many genes using only a limited number of experimental conditions -...

64 Compressed sensing Possible applications:! - Rapid Magnetic Resonance Imaging - Tomography - Electronic microscopy - Fluorescence microscopy - Astronomy - Generalized linear regression - Group Testing - Infer regulatory interactions among many genes using only a limited number of experimental conditions How does the brain work?

65 The simplest problem: getting a signal from some measurement= linear transforms Consider a system of linear measurements Measurements y = 0 y 1.. y M 1 C A y = Fs Signal F = M N matrix s = 0 s 1... s N 1 C A Random F : «random projections» (incoherent with signal) s Pb: Find when M<N and is sparse s

66 Compressed sensing as an optimization problem: the norm approach L 1 Find a N - component vector x such that the M equations y = Fx are satisfied and x is minimal Hopefully: x = s x 0 : number of non-zero components x p = X i x i p Ideally, use x 0. In practice, use x 1 NB: «noisy version» : y = Fx+

67 Phase diagram - phase transitions Gaussian random matrix = M/N Number of measurements per variable Donoho 2006, Donoho Tanner 2005 Kabashima, Wadayama and Tanaka, JSTAT 2009 = R/N Fraction of nonzero variables Find a N - component vector x such that the M equations y = Fx are satisfied and x is minimal

68 Phase diagram - phase transitions Gaussian random matrix Number of measurements per variable = M/N Possible with L 1?? Donoho 2006, Donoho Tanner 2005 Kabashima, Wadayama and Tanaka, JSTAT 2009 Reconstruction impossible = R/N Fraction of nonzero variables Find a N - component vector x such that the M equations y = Fx are satisfied and x is minimal

69 Alternative approach, able to reach the optimal rate = Krzakala Sausset Mézard Sun Zdeborova Probabilistic approach = inference Message passing reconstruction of the signal Careful design of the measurement matrix NB: each of these three ingredients is crucial

70 Step 1: Probabilistic approach to compressed sensing Probability P (x) that the signal is x: 1) x must be compatible with the measurements: 1I) A priori measure on xfavours sparsity Gauss-Bernoulli prior: with probability : with probability 1 x i =0 X i F µi x i = y µ : drawn from Gaussian distribution Theorem: with this measure, the original signal most probable (not obvious!) x = s is the

71 Step 2: Sampling from the constrained measure P (x) = NY [(1 ) (x i )+ (x i )] i=1 PY µ=1 y µ! X F µi x i «Native configuration»= stored signal more probable than other configurations. Efficient sampling? Not so easy.!!!!!! i x i = s i Gaussian is infinitely

72 Step 2: Sampling from the constrained measure P (x) = NY [(1 ) (x i )+ (x i )] i=1 PY µ=1 y µ! X F µi x i «Native configuration»= stored signal more probable than other configurations. Efficient sampling? Not so easy.!!!!!! i x i = s i Gaussian is infinitely

73 Step 2: Sampling from the constrained measure P (x) = NY [(1 ) (x i )+ (x i )] i=1 PY µ=1 y µ! X F µi x i «Native configuration»= stored signal x i = s i is infinitely more probable than other configurations. Efficient sampling? Not so easy.!! constraints! Each constraint involves! all the variables: «long-range»! weak interactions (e.g. Curie! Weiss model for magnets). Mean field is exact* i Gaussian

74 Step 2: Sampling from the constrained measure P (x) = NY [(1 ) (x i )+ (x i )] i=1 PY µ=1 y µ! X F µi x i «Native configuration»= stored signal x i = s i is infinitely more probable than other configurations. Efficient sampling? Not so easy.!! «Mean field»: constraints! Each constraint involves belief propagation! all the variables: «long-range» (spin glass mean-field! weak interactions (e.g. Curie equations, TAP)! Weiss model for magnets). Mean field is exact* i Gaussian

75 Belief propagation = mean-field like equations µ i Local order parameters: a i!µ = hx i i µ v i!µ = hx 2 i i µ (hx i i µ ) 2 h.i µ where denotes the mean, in absence of constraint µ («cavity»-type measure)

76 Belief propagation = mean-field like equations µ i Local order parameters: a i!µ = hx i i µ v i!µ = hx 2 i i µ (hx i i µ ) 2 h.i µ where denotes the mean, in absence of constraint µ («cavity»-type measure)

77 Belief propagation = mean-field like equations µ i Local order parameters: a i!µ = hx i i µ v i!µ = hx 2 i i µ (hx i i µ ) 2 h.i µ where denotes the mean, in absence of constraint µ («cavity»-type measure) Closed self-consistent equations relating these order parameters («BP», «TAP», «G-AMP»,...)

78 Belief propagation = mean-field like equations µ i Local order parameters: a i!µ = hx i i µ v i!µ = hx 2 i i µ (hx i i µ ) 2 h.i µ where denotes the mean, in absence of constraint µ («cavity»-type measure) Closed self-consistent equations relating these order parameters («BP», «TAP», «G-AMP»,...) Four «messages» sent along each edge i µ ( numbers ) can be simplified to parameters 4NM O(N)

79 Performance of the probabilistic approach + message passing + parameter learning Simulations Analytic study of the large method, cavity method) N limit (replica

80 Analytic study: cavity equations, density evolution, replicas, state evolution Replica method allows to compute the «free entropy» (D) = lim N!1 1 N log P (D) where P (D) is the probability that reconstructed x is at distance from original signal s. D D = 1 N X (x i s i ) 2 Cavity method shows that the order parameters of the BP iteration flow according to the gradient of the replica free entropy («density evolution» eqns) analytic control of the BP equations NB rigorous: Bayati Montanari, Lelarge Montanari i

81 Φ(D) Free entropy log P (D) α = 0.62 α = 0.6 α = 0.58 α = =.4 Number of iterations BP convergence time α=ρ α rbp α L1 rbp Seeded BEP - L=10 Seeded BEP - L= D distance to native state α Dynamic glass transition When is too small, BP is trapped in a glass phase

82 Performance of BP : phase diagram L 1 1 BP α α L1 (ρ) α rbp (ρ) S-BEP α = ρ ρ

83 Step 3: design the measurement matrix in order to get around the glass transition Getting around the glass trap: design the matrix F so that one nucleates the naive state (crystal nucleation idea,...borrowed from error correcting codes!) Felström-Zigangirov, Kudekar Richardson Urbanke, Hassani Macris Urbanke,...! «Seeded BP»: A seed to nucleate the crystal

84 A construction inspired by the spatially coupled matrices developed in coding theory cf: Urbanke et al. Mixed mean-field and one-dimensional system: 1) Create many mean-field sub-systems subsystem 1 subsystem 2 subsystem 3 subsystem 4 subsystem 5

85 Mixed mean-field and one-dimensional system: 2) Add a first neighbor coupling subsystem 1 subsystem 2 subsystem 3 subsystem 4 subsystem 5

86 Mixed mean-field and one-dimensional system: subsystem 1 3) Choose parameters such that the first system is in the region of the phase diagram where there is no metastability subsystem 2 subsystem 3 subsystem 4 subsystem 5 large value of α Seed low values of α On average, α is still low!

87 Mixed mean-field and one-dimensional system: 4) The solution will appear in the first sub-system (with large α), and then propagate in the system subsystem 1 subsystem 2 subsystem 3 subsystem 4 subsystem 5 Seed

88 Numerical study Mean square error t=1 t=4 t=10 t=20 t=50 t=100 t=150 t=200 t=250 t= Block index L = 20 =.4 N = J 1 = 20 J 2 =.2 1 =1 =.5

89 Numerical study Mean square error t=1 t=4 t=10 t=20 t=50 t=100 t=150 t=200 t=250 t=300 t = 10 decoding of first block Block index L = 20 =.4 N = J 1 = 20 J 2 =.2 1 =1 =.5

90 Numerical study Mean square error t = 10 decoding of first block t=1 t=4 t=10 t=20 t=50 t=100 t=150 t=200 t=250 t= Block index t = 100 decoding of blocks 1 to 9 L = 20 =.4 N = J 1 = 20 J 2 =.2 1 =1 =.5

91 Seeded-BP : a threshold-reaching strategy L 1 1 BP seeded BP α α L1 (ρ) α rbp (ρ) S-BEP α = ρ ρ L 1 Theory: seeded-bp threshold at = when L!1 phase transition line moves up when using seeding F

92 Optimal performance on artificial signals (sparse but with independent components).! More realistic signals?

93 L 1 BEP Compressed sensing : measure «random projections» and reconstruct the signal α = ρ! Methods Linear relaxation Candès-Tao, Donoho, Mean field algorithm S-BEP α = 0.6 α = 0.5 α = 0.4 α = 0.3 α = 0.2 Mean field + special strategy to avoid glass Krzakala et al. glass transition Original image = compression factor optimal compression (information-theory)

94 Bird s eye view Failure of the «representative agent» mean-field idea Impossible to describe spin glasses (or structural glasses) with a single effective spin : misses all the richness Go beyond effective medium for e.g. epidemic propagation (SIR) or gene expression : beyond the law of mass action «Whom or why does the effective agent represent?» (A. Kirman) ; Minority games Substituted by statistical approaches to the heterogeneous behaviors of individual agents

95 Bird s eye view «Representative agent» mean-field idea is substituted by statistical approaches to the heterogeneous behaviors of individual agents + simulations and algorithms More complicated («ordered phases», equilibration times,etc.), but much much richer. This «curse of dimensionality, poses several challenges to both technical and conceptual progress in neuroscience» ( disciplines (big data) Infinitely complex???

96 The Replica Method, The Cavity Method, and all the conceptual, mathematical and physical challenges that appeared in the study of spin glasses have been a «source from which new knowledge and application have flowed profusely their invention 35 years ago Portrait of Ottavio Strada, Tintoretto, Venice 1567 Rijk s Museum Amsterdam * D. Sherrington

97 The Replica Method, The Cavity Method, and all the conceptual, mathematical and physical challenges that appeared in the study of spin glasses have been a «source from which new knowledge and application have flowed profusely their invention 35 years ago Two amusing remarks: Portrait of Ottavio Strada, Tintoretto, Venice 1567 Rijk s Museum Amsterdam * D. Sherrington

98 The Replica Method, The Cavity Method, and all the conceptual, mathematical and physical challenges that appeared in the study of spin glasses have been a «source from which new knowledge and application have flowed profusely their invention 35 years ago Two amusing remarks: Physical spin glasses still far from well understood Portrait of Ottavio Strada, Tintoretto, Venice 1567 Rijk s Museum Amsterdam * D. Sherrington

99 The Replica Method, The Cavity Method, and all the conceptual, mathematical and physical challenges that appeared in the study of spin glasses have been a «source from which new knowledge and application have flowed profusely their invention 35 years ago Two amusing remarks: Physical spin glasses still far from well understood Useless spin glasses have found applications in fields that were totally unexpected Portrait of Ottavio Strada, Tintoretto, Venice 1567 Rijk s Museum Amsterdam * D. Sherrington

100

101 0 y 1 0 F 1 0 s 1 Structured measurement matrix. Variances of the matrix elements F µi = 1 J 2 J 1 1 J 2 = J 1 J B C J 1 1 J 2 C J 1 1 J 2 J 1 1 J 2 A 0 J 1 1 J 2 J 1 1 : unit coupling : coupling J 1 : coupling J 2 : no coupling (null elements) C A independent random Gaussian variables, zero mean and variance Jb(µ)b(i) /N

102 0 y 1 0 F 1 0 s 1 Block 1 has a large value of M such that the solution arise in this block and then propagate sin the whole system! 1 J 2 J 1 1 J 2 = J 1 J B C J 1 1 J 2 C J 1 1 J 2 J 1 1 J 2 A 0 J 1 1 J 2 J 1 1 : unit coupling : coupling J 1 : coupling J 2 : no coupling (null elements) C A L =8 N i = N/L M i = i N/L 1 > BP j = 0 < BP j 2 = 1 L ( 1 +(L 1) 0 )

103 An example: tomography of binary mixtures

104 An example: tomography of binary mixtures L measurements for each angle Synchrotron light

105 An example: tomography of binary mixtures L measurements for each angle Synchrotron light L angles: L 2 measurements L 2 pixels

106 An example: tomography of binary mixtures L angles: L 2 measurements L 2 pixels size 1 size 1/L If the size of domains is reconstruct with L 2 pixel: possible to measurements

107 An example: tomography of binary mixtures size 1 size 1/L If the size of domains is reconstruct with L 2 pixel: possible to measurements

108 An example: tomography of binary mixtures This picture, digitalized on grid, can be reconstructed fom measurements with 16 angles size 1 size 1/L Gouillart et al., arxiv If the size of domains is reconstruct with L 2 pixel: possible to measurements