partially linear additive gaussian graphical models13-11-00)-13... · 2019-06-09 · partially...
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Partially Linear Additive Gaussian GraphicalModels
Presented by: Sinong GengPrinceton University
June 13th, 2019 @ ICML 2019
Authors
Sinong GengPrincetonUniversity
Minhao YanCornell
University
Mladen KolarThe University of
Chicago
Sanmi KoyejoUniversity of
Illinois
Poster: Partially Linear Additive Gaussian Graphical ModelsThu Jun 13th 06:15 – 09:00 PM @ Pacific Ballroom
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Brain Functional Connectivity Analysis
Estimation is distorted by physiological noise [Van Dijk et al.,2012, Goto et al., 2016].The noise sources are observable e.g. motion, breathing…
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Brain Functional Connectivity Analysis
Estimation is distorted by physiological noise [Van Dijk et al.,2012, Goto et al., 2016].
The noise sources are observable e.g. motion, breathing…
3
Brain Functional Connectivity Analysis
Estimation is distorted by physiological noise [Van Dijk et al.,2012, Goto et al., 2016].The noise sources are observable e.g. motion, breathing… 3
Model Formulation: Goals
→ A general formulation of the effects caused by the noise.
→ Stronger theoretical guarantees compared tp methodswith hidden variables.
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Model Formulation
→ Z denotes the observed fMRI data, and random variableG, the physiological noise.
→ Z | G = g follows a Gaussian graphical model [Yang et al.,2015] with a parameter matrix, denoted by Ω(g):
P(Z = z;Ω(g) | G = g) ∝ exp
p∑
j=1
Ωjj(g)zjp∑
j=1
p∑j′>j
Ωjj′(g)zjzj′ −1
2
p∑j
z2j
.
→ Parameter matrices are additive:
Ω(g) := Ω0 + R(g).
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Model Formulation: Ω(g)Goals:
• Identifiable parameters• A general formulation
Assumptions:• R(g) = 0 for any g satisfying |g| ≤ g∗.• R(g), and Ω(g) are smooth enough to be recoveredby kernel methods.
Existing assumptions:• R(g) = 0 [Van Dijk et al., 2012, Power et al., 2014].• E(R(g)) = 0 [Lee and Liu, 2015, Geng et al., 2018].
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Model Formulation: Ω(g)Goals:
• Identifiable parameters• A general formulation
Assumptions:• R(g) = 0 for any g satisfying |g| ≤ g∗.• R(g), and Ω(g) are smooth enough to be recoveredby kernel methods.
Existing assumptions:• R(g) = 0 [Van Dijk et al., 2012, Power et al., 2014].• E(R(g)) = 0 [Lee and Liu, 2015, Geng et al., 2018].
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Model Formulation: Ω(g)Goals:
• Identifiable parameters• A general formulation
Assumptions:• R(g) = 0 for any g satisfying |g| ≤ g∗.• R(g), and Ω(g) are smooth enough to be recoveredby kernel methods.
Existing assumptions:• R(g) = 0 [Van Dijk et al., 2012, Power et al., 2014].• E(R(g)) = 0 [Lee and Liu, 2015, Geng et al., 2018].
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Parameter EstimationLog Pseudo Likelihood:
•We summarize the varying effects as Mij := x⊤ij Ωi·j,where x⊤ij denotes the ith row vector of xj.•
ℓPL
(zi,gii∈[n] ;R(·),Ω0
)=
n∑i=1
p∑j=1
zij
(x⊤ij Ω0·j +Mij
)− 1
2z2ij
− 1
2
(x⊤ij Ω0·j +Mij
)2.
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Parameter Estimation• Pseudo-Profile Likelihood [Fan et al., 2005]• Suppose that Assumptions are satisfied. Then, for anyϵ > 0, with probability of at least 1− ϵ, there exists C4 > 0,so that Ω0 shares the same structure with the underlyingtrue parameter Ω∗
0, if for some constant C5 > 0,
C5
√logpn
≥ λ ≥ 4
αC4
√logpn
,
r := 4C2λ ≤ ∥Ω∗0S∥∞ ,
and n ≥(64C5C2
2C3/α)2 logp.
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Parameter Estimation
Sparsistency: The underlying structure can be recoveredwith a high probability.
√n Convergence: The smallest scale of the non-zero
component that the PPL method can distinguish fromzero converges to zero at a rate of
√n.
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Overall Performance
→ LR-GGM→ fMRI dataset with control
subjects and those withSchizophrenia.
→ Diagnosis using the re-covered structure by twodifferent methods. 0
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0.50 0.55 0.60 0.65 0.70 0.75
AUCs
Frequ
en
cy
MethodsPLA−GGMLR−GGM
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Thank you!
References I
J. Fan, T. Huang, et al. Profile likelihood inferences onsemiparametric varying-coefficient partially linearmodels. Bernoulli, 11(6):1031–1057, 2005.
S. Geng, M. Kolar, and O. Koyejo. Joint nonparametricprecision matrix estimation with confounding. arXivpreprint arXiv:1810.07147, 2018.
M. Goto, O. Abe, T. Miyati, H. Yamasue, T. Gomi, andT. Takeda. Head motion and correction methods inresting-state functional mri. Magnetic Resonance inMedical Sciences, 15(2):178–186, 2016.
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References II
W. Lee and Y. Liu. Joint estimation of multiple precisionmatrices with common structures. The Journal of MachineLearning Research, 16(1):1035–1062, 2015.
J. D. Power, A. Mitra, T. O. Laumann, A. Z. Snyder, B. L.Schlaggar, and S. E. Petersen. Methods to detect,characterize, and remove motion artifact in resting statefmri. Neuroimage, 84:320–341, 2014.
K. R. Van Dijk, M. R. Sabuncu, and R. L. Buckner. Theinfluence of head motion on intrinsic functionalconnectivity mri. Neuroimage, 59(1):431–438, 2012.
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References III
E. Yang, P. Ravikumar, G. I. Allen, and Z. Liu. Graphicalmodels via univariate exponential family distributions.Journal of Machine Learning Research, 16(1):3813–3847,2015.
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