Abstract

The performances of various estimators for wavefront sensing applications such as adaptive optics (AO) are compared. Analytical expressions for the bias and variance terms in the mean squared error (MSE) are derived for the minimum-norm maximum likelihood (MNML) and the maximum a posteriori (MAP) reconstructors. The MAP estimator is analytically demonstrated to yield an optimal trade-off that reduces the MSE, hence leading to a better Strehl ratio. The implications for AO applications are quantified thanks to simulations on 8-m- and 42-m-class telescopes. We show that the MAP estimator can achieve twice as low MSE as MNML methods do. Large AO systems can thus benefit from the high quality of MAP reconstruction in O(n) operations, thanks to the fast fractal iterative method (FrIM) algorithm (Thiébaut and Tallon, submitted to J. Opt. Soc. Am. A).

© 2009 Optical Society of America

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References

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2006 (3)

2005 (1)

2004 (1)

2003 (2)

2002 (4)

2000 (1)

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at paranal from gsm instrument and surface layer contribution,” Astrophys. J., Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

1994 (1)

1992 (2)

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Electr. Eng. 18, 451-466 (1992).
[CrossRef]

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209-224 (1992).
[CrossRef]

1983 (1)

1980 (1)

1978 (2)

1977 (2)

1976 (1)

R. J. Noll, “Dynamic atmospheric turbulence correction,” Proc. SPIE 75, 39-42 (1976).

1970 (1)

H. Sorenson, “Least-square estimation: from Gauss to Kalman,” IEEE Spectrum 7, 63-68 (1970).
[CrossRef]

Agabi, A.

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at paranal from gsm instrument and surface layer contribution,” Astrophys. J., Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Anderson, E.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users' Guide, 2nd ed. (SIAM, 1995).

Bai, Z.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users' Guide, 2nd ed. (SIAM, 1995).

Baudoz, P.

Beuzit, J.-L.

Bischof, C.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users' Guide, 2nd ed. (SIAM, 1995).

Borgnino, J.

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at paranal from gsm instrument and surface layer contribution,” Astrophys. J., Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Brase, J.

Charton, J.

Conan, J.-M.

Conan, R.

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at paranal from gsm instrument and surface layer contribution,” Astrophys. J., Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

R. Conan, “Modélisation des effets de l'échelle externe de cohérence spatiale du front d'onde pour l'observation à haute résolution angulaire en astronomie,” Ph.D. thesis (Université de Nice-Sophia Antipolis, 2000).

Dainty, J. C.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209-224 (1992).
[CrossRef]

Demmel, J.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users' Guide, 2nd ed. (SIAM, 1995).

Dohlen, K.

Dongarra, J.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users' Guide, 2nd ed. (SIAM, 1995).

Du Croz, J.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users' Guide, 2nd ed. (SIAM, 1995).

Ellerbroek, B.

Ellerbroek, B. L.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Fried, D. L.

Fusco, T.

Gavel, D.

Gavel, D. T.

Gilles, L.

Glindemann, A.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209-224 (1992).
[CrossRef]

Greenbaum, A.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users' Guide, 2nd ed. (SIAM, 1995).

Hammarling, S.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users' Guide, 2nd ed. (SIAM, 1995).

Herrmann, J.

Hudgin, R. H.

Kasper, M.

Kulcsar, C.

Lane, R. G.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209-224 (1992).
[CrossRef]

Le Louarn, M.

Le Roux, B.

Looze, D.

Macintosh, B.

Markey, J. K.

Martin, F.

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at paranal from gsm instrument and surface layer contribution,” Astrophys. J., Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

McKenney, A.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users' Guide, 2nd ed. (SIAM, 1995).

Michau, V.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323-336 (2006).
[CrossRef]

Mouillet, D.

Mugnier, L.

Nicolle, M.

Noll, R.

Noll, R. J.

R. J. Noll, “Dynamic atmospheric turbulence correction,” Proc. SPIE 75, 39-42 (1976).

Ostrouchov, S.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users' Guide, 2nd ed. (SIAM, 1995).

Pelat, D.

D. Pelat, Course titled “Bruits et signaux” (École Doctorale d'Ile de France, Département Astronomie-Astrophysique, Paris, 2005).

Petit, C.

Poyneer, L.

Poyneer, L. A.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Puget, P.

Raynaud, H.-F.

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Electr. Eng. 18, 451-466 (1992).
[CrossRef]

Rousset, G.

Sarazin, M.

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at paranal from gsm instrument and surface layer contribution,” Astrophys. J., Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Sauvage, J.-F.

Sorensen, D.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users' Guide, 2nd ed. (SIAM, 1995).

Sorenson, H.

H. Sorenson, “Least-square estimation: from Gauss to Kalman,” IEEE Spectrum 7, 63-68 (1970).
[CrossRef]

Tallon, M.

M. Le Louarn and M. Tallon, “Analysis of modes and behavior of a multiconjugate adaptive optics system,” J. Opt. Soc. Am. A 19, 912-925 (2002).
[CrossRef]

E. Thiébaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescopes,” submitted to J. Opt. Soc. Am. A.

Tarantola, A.

A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (SIAM, 2005).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, 1961).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Thiébaut, E.

E. Thiébaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescopes,” submitted to J. Opt. Soc. Am. A.

Thomas, S.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323-336 (2006).
[CrossRef]

Tokovinin, A.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323-336 (2006).
[CrossRef]

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at paranal from gsm instrument and surface layer contribution,” Astrophys. J., Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Trinquet, H.

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at paranal from gsm instrument and surface layer contribution,” Astrophys. J., Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Troy, M.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Wallner, E. P.

Wang, J. Y.

Ziad, A.

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at paranal from gsm instrument and surface layer contribution,” Astrophys. J., Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Appl. Opt. (1)

Astrophys. J., Suppl. Ser. (1)

F. Martin, R. Conan, A. Tokovinin, A. Ziad, H. Trinquet, J. Borgnino, A. Agabi, and M. Sarazin, “Optical parameters relevant for high angular resolution at paranal from gsm instrument and surface layer contribution,” Astrophys. J., Suppl. Ser. 144, 39-44 (2000).
[CrossRef]

Comput. Electr. Eng. (1)

M. C. Roggemann, “Optical performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Electr. Eng. 18, 451-466 (1992).
[CrossRef]

IEEE Spectrum (1)

H. Sorenson, “Least-square estimation: from Gauss to Kalman,” IEEE Spectrum 7, 63-68 (1970).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (6)

Mon. Not. R. Astron. Soc. (1)

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack-Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323-336 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (2)

R. J. Noll, “Dynamic atmospheric turbulence correction,” Proc. SPIE 75, 39-42 (1976).

L. A. Poyneer, “Advanced techniques for Fourier transform wavefront reconstruction,” Proc. SPIE 4839, 1023-1034 (2002).
[CrossRef]

Waves Random Media (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209-224 (1992).
[CrossRef]

Other (8)

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users' Guide, 2nd ed. (SIAM, 1995).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

E. Thiébaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescopes,” submitted to J. Opt. Soc. Am. A.

F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
[CrossRef]

D. Pelat, Course titled “Bruits et signaux” (École Doctorale d'Ile de France, Département Astronomie-Astrophysique, Paris, 2005).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, 1961).

R. Conan, “Modélisation des effets de l'échelle externe de cohérence spatiale du front d'onde pour l'observation à haute résolution angulaire en astronomie,” Ph.D. thesis (Université de Nice-Sophia Antipolis, 2000).

A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (SIAM, 2005).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Models of wavefront structure functions. Dotted line, Kolmogorov theoretical five-thirds law. Dashed line, saturated Kolmogorov model with minimal saturation set for a 42 - m telescope. Solid line, saturated Kolmogorov model with minimal saturation set for an 8 - m telescope. Dashed–dotted curve, von Kármán structure function for r 0 = 0.159 m and L 0 = 25 m .

Fig. 2
Fig. 2

Noise propagation on a square aperture for zonal MNML. Small black squares, noise propagation coefficient γ MNML averaged over 100 simulations of zonal MNML reconstruction of a null wavefront (dispersion given by error bars); solid line, law fitted on these values. Dashed line, Fried’s law for γ MNML over a square aperture. Large white diamonds, ratio ϵ MNML 2 σ e 2 estimated from averaging 50 simulations of zonal MNML reconstructions on Kolmogorov turbulence (dispersion given by corresponding error bars). The conditions of the simulations are r 0 = 0.5 d (where d is the sampling step size) and σ e = 0.1 rad. /sub-aperture. The number of wavefront samples are n = ( 2 q + 1 ) 2 = 5 2 , 9 2 , 17 2 , 33 2 , 65 2 , 129 2 , 257 2 , and 513 2 .

Fig. 3
Fig. 3

Comparison of γ MNML on square and circular apertures. Dashed line, Fried’s law for square aperture. Dashed–dotted line, Noll’s law for circular aperture. Dotted curve, Poyneer et al. [27] law for circular aperture with FTR, γ FTR on Fried’s geometry. Other symbols are average results of 100 simulations of zonal MNML reconstructions on null signal. Squares, for a square aperture. Circles, for a circular aperture. Number of subapertures across the pupil diameter: 16, 32, 64, 128, 256, and 512. Other simulation parameters are the same as for Fig. 2.

Fig. 4
Fig. 4

Comparison of noise propagation coefficient γ MNML and MSE normalized by noise variance for zonal MNML reconstructions on square and circular apertures. Dashed line, Fried’s law on square apertures. Solid line, our experimental law on circular apertures; see Eq. (57). White symbols, ϵ MNML 2 σ e 2 from our simulations. Black symbols, noise propagation coefficient γ MNML from our simulations. Squares, square aperture. Circles, circular aperture. System dimensions are the same as in Fig. 3. Other simulation parameters are the same as for Fig. 2.

Fig. 5
Fig. 5

Comparison of zonal and modal MSE normalized by the noise variance. Dashed line, Fried’s law for γ MNML . Large white symbols, ϵ MNML 2 σ e 2 , i.e., reconstruction by zonal MNML. Small black symbols, ϵ KL 2 σ e 2 , i.e., reconstruction by modal MNML. Symbols correspond to the average of 50 reconstructions of Kolmogorov turbulence over circular apertures. Noise level was set to σ e = 0.1 rad /subaperture. System dimensions are the same as in Fig. 3. For every system size, the turbulence conditions are: r 0 = 0.1 d , 0.5 d , d, and 2 d .

Fig. 6
Fig. 6

Comparison of KL-MNML and MAP performance on an 8 - m telescope. Dashed line, Fried’s law for γ MNML on square apertures. White symbols, ϵ KL 2 σ e 2 , i.e., reconstruction by KL-MNML. Black symbols, ϵ MAP 2 σ e 2 , i.e., reconstruction by FrIM MAP. Markers stand for average results over 200 simulations of von Kármán turbulent wavefronts on circular apertures. Iso- μ 0 are linked with solid lines. μ 0 varies along with noise variance values σ e 2 = 5 × 10 3 , 10 2 , 10 1 , 1, and 2 rad 2 per subaperture. L 0 = 24 m , r 0 = 25 cm , D d = 16 , 32, 40, and 64.

Fig. 7
Fig. 7

Comparison of KL-MNML and MAP performance on a 42 - meter telescope. Legend is the same as for Fig. 6. Changing parameters values are L 0 = 21 m , r 0 = 32 cm , D d = 64 , 84, 128, and 256.

Fig. 8
Fig. 8

Ratio between MSE obtained by the FrIM MAP method and MSE obtained by KL-MNML reconstruction for the same data as plotted in Figs. 6, 7. White symbols, 8 - m telescope. The same system dimensions are linked with dashed–dotted lines. Black symbols, 42 - m telescope. Iso-n are linked with curves.

Equations (97)

Equations on this page are rendered with MathJax. Learn more.

d = S ( w ) + e ,
w ( x , y ) j = 1 n w j f j ( x , y ) .
d = S w + e ,
C ̱ e = C e a e ,
w = K u ,
C w = a w K C u K T = a w K K T ,
w = a w K u
ϵ 2 = 1 n tr ( w w ̂ ) ( w w ̂ ) T w , e ,
w ̂ = R d .
ϵ 2 = b + v ,
b = tr ( ( I R S ) C ̱ w ( I R S ) T ) a w n ,
v = tr ( R C ̱ e R T ) a e n ,
ϵ 2 = 1 n tr P ( w w ̂ ) ( w w ̂ ) T P T w , e ,
b = tr ( P ( I R S ) C ̱ w ( I R S ) T P T ) a w n ,
v = tr ( P R C ̱ e R T P T ) a e n .
w ̂ ML = arg min w ̂ ( d S w ̂ ) T C ̱ e 1 ( d S w ̂ ) ,
Ω ML = { w R n S T W e S w = S T W e d }
w ̂ MNML = arg min w ̂ Ω ML w ̂ 2 = R MNML d
R MNML = ( S T W e S ) S T W e ,
w ̂ KL = arg min w ̂ Ω ML w ̂ KL = R KL d ,
R KL = K ( K T S T W e S K ) K T S T W e .
R MAP = arg min R ϵ 2 = ( S T W e S + μ 0 W w ) 1 S T W e ,
R MAP = Q U ( Σ T Σ + μ 0 I ) 1 Σ T V 1 T L 1 ,
R KL = Q U Σ V 1 T L 1 ,
R MNML = Q U Z Σ V 1 T L 1 ,
R = Q U G V 1 T L 1
G KL = Σ ,
G MAP = ( Σ T Σ + μ 0 I ) 1 Σ T ,
G MNML = Z Σ .
b = tr ( W ( I G Σ ) ( I G Σ ) T ) a w ,
v = tr ( W G G T ) a e ,
W = 1 n U T U ,
W = 1 n U T Q T P T P Q U .
b KL = a e i = p + 1 i = n W i , i μ 0 ,
v KL = a e i = 1 i = p W i , i σ i 2 ,
ϵ KL 2 = a e ( i = 1 i = p W i , i σ i 2 + i = p + 1 i = n W i , i μ 0 ) ,
b MAP = a e ( i = 1 i = p μ 0 W i , i ( μ 0 + σ i 2 ) 2 + i = p + 1 i = n W i , i μ 0 ) ,
v MAP = a e ( i = 1 i = p σ i 2 W i , i ( μ 0 + σ i 2 ) 2 ) ,
ϵ MAP 2 = a e ( i = 1 i = p W i , i μ 0 + σ i 2 + i = p + 1 i = n W i , i μ 0 ) .
b MNML = b KL + b 0
v MNML = tr ( P ( S T W e S ) P T ) a e n = tr ( S T W e S ) a e n ,
D w ( r ) = [ w ( r ) w ( r + r ) ] 2 r = 6.88 × ( r r 0 ) 5 3 ,
D w ( r ) = 2 σ w 2 2 C w ( r ) ,
C w ( r ) = α ( L 0 r 0 ) 5 3 ( 2 π r L 0 ) 5 6 K 5 6 ( 2 π r L 0 ) ,
α = [ 12 5 Γ ( 6 5 ) ] 5 6 Γ ( 11 6 ) π 8 3 0.0858 .
C w ( r ) = ( D r 0 ) 5 3 C ̱ w ( r ) ,
C ̱ w ( r ) = 0.0858 × ( L 0 D ) 5 3 ( 2 π r L 0 ) 5 6 K 5 6 ( 2 π r L 0 ) ,
C ̱ w ( r ) = 3.44 × [ ( 2 ( N 1 ) d D ) 5 3 ( r D ) 5 3 ] ,
w = ( D r 0 ) 5 6 K u
ϵ 2 σ e 2 = γ + η μ 0 ,
γ = def v σ e 2 ,
η = def ( D r 0 ) 5 3 b = μ 0 b σ e 2
γ MNML = 1 n tr ( S T S )
γ MNML 0.6558 + 0.16028 ln n ,
γ MNML 0.748 + 0.138 ln n .
γ MNML 0.0136 + 0.1592 ln n .
γ MNML 0.842 + 0.127 ln n .
γ FTR 0.1456 ( ln n ) 2 1.7922 ln n + 7.6175 .
W e = L T L 1
M = L 1 S
M K = V 1 Σ V 2 T ,
Σ = [ Σ 11 0 0 0 ] with Σ 11 = diag ( σ 1 , σ 2 , , σ p ) ,
K V 2 = Q U ,
U = [ U 11 U 12 0 U 22 ] ,
M = V 1 Σ U 1 Q T ,
K 1 = V 2 U 1 Q T .
S T W e = Q U T Σ T V 1 T L 1 ,
S T W e S = M T M = Q U T Σ T Σ U 1 Q T ,
W w = K T K 1 = Q U T U 1 Q T ,
K T S T W e S K = V 2 Σ T Σ V 2 T ,
( K T S T W e S K ) = V 2 ( Σ T Σ ) V 2 T ,
( Σ T Σ ) Σ T = Σ = ( Σ 11 1 0 0 0 )
Q U T Σ T Σ U 1 Q T w = Q U T Σ T V 1 T L 1 d .
Σ T Σ y = Σ T z with z = V 1 T L 1 d .
y 1 = Σ 11 1 z 1 .
w 2 2 = U y 2 2 = U 11 y 1 + U 12 y 2 2 2 + U 22 y 2 2 2 ,
( U 12 T U 12 + U 22 T U 22 ) y 2 = U 12 T U 11 y 1 .
y MNML = ( y 1 y 2 ) = Z Σ z with Z = ( I 11 Z 12 Z 21 Z 22 ) ,
Z 21 = ( U 12 T U 12 + U 22 T U 22 ) 1 U 12 T U 11 .
w MNML = Q U Z Σ V 1 T L 1 d ,
b KL = a w tr ( W ( I Σ Σ ) ( I Σ Σ ) T ) = a w tr ( W 22 ) ,
v KL = a e tr ( W Σ ( Σ ) T ) = a e tr ( W 11 Σ 11 2 ) ,
ϵ KL 2 = a e tr ( W 11 Σ 11 2 ) + a w tr ( W 22 ) ,
W = ( W 11 W 12 W 21 W 22 ) ,
I G MAP Σ = μ 0 ( Σ T Σ + μ 0 I ) 1 ,
b MAP = μ 0 a e tr ( W ( Σ T Σ + μ 0 I ) 2 ) = μ 0 a e tr ( W 11 ( Σ 11 2 + μ 0 I 11 ) 2 ) + a w tr ( W 22 ) ,
v MAP = a e tr ( W Σ T Σ ( Σ T Σ + μ 0 I ) 2 ) = a e tr ( W 11 Σ 11 2 ( Σ 11 2 + μ 0 I 11 ) 2 ) ,
ϵ MAP 2 = a e tr ( W ( Σ T Σ + μ 0 I ) 1 ) .
I G MNML Σ = I Z Σ Σ = ( 0 0 Z 21 I 22 ) ,
( I G MNML Σ ) ( I G MNML Σ ) T = ( 0 0 0 I 22 + Z 21 Z 21 T ) ,
Z 21 = U 22 1 U 22 T U 12 T ( I 11 + B 11 ) 1 U 11 ,
B 11 = U 12 U 22 1 U 22 T U 12 T .
U G MNML = ( Y 11 Σ 11 1 0 Y 21 Σ 11 1 0 )
Y 11 = U 11 + U 12 Z 21 = ( I 11 + B 11 ) 1 U 11 ,
Y 21 = U 22 Z 21 = U 22 T U 12 T ( I 11 + B 11 ) 1 U 11 .
v MNML = tr ( U G MNML G MNML T U T ) a e n = tr ( Y 11 T Y 11 Σ 11 2 ) a e n + tr ( Y 21 T Y 21 Σ 11 2 ) a e n = tr ( U 11 T U 11 Σ 11 2 ( I 11 + B 11 ) 1 ) a e n ,
v MNML = a e tr ( W 11 Σ 11 2 ( I 11 + B 11 ) 1 ) .

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