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Randomized Matrix-Free Trace and Log-Determinant Estimators Ilse Ipsen Joint with: Alen Alexanderian & Arvind Saibaba NorthCarolinaStateUniversity Raleigh,NC,USA ResearchsupportedbyDARPAXData Our Contribution to this Minisymposium Inverse problems: Bayesian OED Big data: Randomized estimators Given: Hermitian positive semi-definite matrix A Want: trace(A) and logdet(I+A) Our estimators • Matrix free, simple implementation • Much higher accuracy than Monte Carlo • Informative probabilistic bounds, even for small dimensions Inverse Problem: Diffusive Contaminant Transport Forwardproblem: Time-dependentadvection-diffusionequation Inverseproblem: Re-constructuncertaininitialconcentration frommeasurementsof35sensors,at3timepoints Prior-preconditionedFisherinformation H≡C1/2F∗Γ−1 FC1/2 prior noise prior SensitivityofOED: trace(H) BayesianD-optimaldesigncriterion: logdet(I+H) Our estimators • Fast and accurate for Bayesian inverse problem • Matrix free, simple implementation • Much higher accuracy than Monte Carlo • Informative probabilistic bounds, even for small dimensions Our Contribution to this Minisymposium Inverse problems: Bayesian OED Big data: Randomized estimators Given: Hermitian positive semi-definite matrix A Want: trace(A) and logdet(I+A) Our Contribution to this Minisymposium Inverse problems: Bayesian OED Big data: Randomized estimators Given: Hermitian positive semi-definite matrix A Want: trace(A) and logdet(I+A) Our estimators • Fast and accurate for Bayesian inverse problem • Matrix free, simple implementation • Much higher accuracy than Monte Carlo • Informative probabilistic bounds, even for small dimensions Existing Estimators Small Explicit Matrices: Direct Methods Trace = sum of diagonal elements (cid:88) trace(A) = a jj j Logdet (1) Convert to trace logdet(I+A) = trace(log(I+A)) (2) Cholesky factorization I+A = LL∗ (cid:89) logdet(I+A) = log|det(L)|2 = 2log l jj j Large Sparse or Implicit Matrices Trace: Monte Carlo methods N (cid:88) trace(A) ≈ 1 z∗Az N j j j=1 N independent random vectors z j Rademacher, standard Gaussian ⇒ unbiased estimator Hutchinson1989, Avron&Toledo2011 Roosta-Khorasani&Ascher2015, Lin2016 Logdet: Expansion of log   (cid:88) logdet(I+A) ≈ trace pj(A) j Barry&Pace1999, Pace&LeSage2004, Zhangetal. 2008 Chenetal. 2011, Anitescuetal. 2012,Boutsidisetal. 2015 Hanetal. 2015 Our Estimators Randomized Subspace Iteration Given: n×n matrix A with k dominant eigenvalues Compute low rank approximation T: (1) Pick random starting guess Ω with (cid:96) ≥ k columns (2) Subspace iteration Y = AqΩ (3) Orthonormalize: Thin QR Y = QR (4) Compute (cid:96)×(cid:96) matrix T = Q∗AQ Estimators: trace(T) ≈ trace(A) logdet(I+T) ≈ logdet(I+A)

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Big data: Randomized estimators. Given: Hermitian positive semi-definite matrix A. Want: trace(A) and log det(I + A). Our estimators. • Matrix free

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Big data: Randomized estimators. Given: Hermitian positive semi-definite matrix A. Want: trace(A) and log det(I + A). Our estimators. • Matrix free

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Publication Year:	2017
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Language:	English
File Size:	3.28
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