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Enhancing the PRIMME Eigensolver for Computing Accurately Singular Triplets of Large Matrices Lingfei Wu and Andreas Stathopoulos Department of Computer Science College of William and Mary April 10th, 2014 CopperMountain2014 1 / 24

Introduction: applications of SVD • Introduction Social network analysis: voting similarities among • The applications politicians • The problems • The methods • Convergence issue • Accuracy issue Related work primme svds: why choose the two stage strategy primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 2 / 24

Introduction: applications of SVD • Introduction Social network analysis: voting similarities among • The applications politicians • The problems • The methods • Image-processing: calculation of Eigenfaces in face • Convergence issue • Accuracy issue recognition Related work primme svds: why choose the two stage strategy primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 2 / 24

Introduction: applications of SVD • Introduction Social network analysis: voting similarities among • The applications politicians • The problems • The methods • Image-processing: calculation of Eigenfaces in face • Convergence issue • Accuracy issue recognition Related work • Textual database searching: Google, Yahoo, and Baidu primme svds: why choose the two stage strategy primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 2 / 24

Introduction: applications of SVD • Introduction Social network analysis: voting similarities among • The applications politicians • The problems • The methods • Image-processing: calculation of Eigenfaces in face • Convergence issue • Accuracy issue recognition Related work • Textual database searching: Google, Yahoo, and Baidu primme svds: why • choose the two stage Numerical linear algebra: least square fitting, rank, strategy low-rank approximation, computation of pseudospectrum primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 2 / 24

Introduction: applications of SVD • Introduction Social network analysis: voting similarities among • The applications politicians • The problems • The methods • Image-processing: calculation of Eigenfaces in face • Convergence issue • Accuracy issue recognition Related work • Textual database searching: Google, Yahoo, and Baidu primme svds: why • choose the two stage Numerical linear algebra: least square fitting, rank, strategy low-rank approximation, computation of pseudospectrum primme svds: how to develop the two stage • Variance reduction in Monte Carlo method strategy Evaluations Conclusions CopperMountain2014 2 / 24

Introduction: what is SVD ? Assume A ∈ ℜ m × n is a large, sparse matrix: Introduction • The applications • The problems • The methods A = U Σ V T • Convergence issue U T U = I, V T V = I, Σ = Diag • Accuracy issue Related work primme svds: why choose the two stage strategy primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 3 / 24

Introduction: what is SVD ? Assume A ∈ ℜ m × n is a large, sparse matrix: Introduction • The applications • The problems • The methods A = U Σ V T • Convergence issue U T U = I, V T V = I, Σ = Diag • Accuracy issue Related work primme svds: why Our Problem : find k smallest singular values and choose the two stage strategy corresponding left and right singular vectors of A primme svds: how to develop the two stage strategy Av i = σ i u i , σ 1 ≤ . . . ≤ σ k Evaluations Conclusions CopperMountain2014 3 / 24

Introduction: how to compute SVD ? • Introduction A Hermitian eigenvalue problem on • The applications • The problems • The methods • Convergence issue • Accuracy issue Related work primme svds: why choose the two stage strategy primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 4 / 24

Introduction: how to compute SVD ? • Introduction A Hermitian eigenvalue problem on • The applications Normal equations matrix C = A T A or C = AA T • The problems ◦ • The methods � � A T 0 • Convergence issue Augmented matrix B = ◦ • Accuracy issue A 0 Related work primme svds: why choose the two stage strategy primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 4 / 24

Introduction: how to compute SVD ? • Introduction A Hermitian eigenvalue problem on • The applications Normal equations matrix C = A T A or C = AA T • The problems ◦ • The methods � � A T 0 • Convergence issue Augmented matrix B = ◦ • Accuracy issue A 0 Related work primme svds: why • Lanczos bidiagonalization method (LBD) choose the two stage strategy primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 4 / 24

Introduction: how to compute SVD ? • Introduction A Hermitian eigenvalue problem on • The applications Normal equations matrix C = A T A or C = AA T • The problems ◦ • The methods � � A T 0 • Convergence issue Augmented matrix B = ◦ • Accuracy issue A 0 Related work primme svds: why • Lanczos bidiagonalization method (LBD) choose the two stage strategy A = PB d Q T primme svds: how to develop the two stage strategy Evaluations B d = X Σ Y T Conclusions Where U = PX and V = QY CopperMountain2014 4 / 24

Introduction: difference between methods • Introduction Convergence speed • The applications • The problems • The methods • Convergence issue • Accuracy issue Related work primme svds: why choose the two stage strategy primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 5 / 24

Introduction: difference between methods • Introduction Convergence speed • The applications ◦ Eigen fast for largest SVs • The problems • The methods methods on C ◦ slow for smallest SVs • Convergence issue • Accuracy issue Related work primme svds: why choose the two stage strategy primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 5 / 24

Introduction: difference between methods • Introduction Convergence speed • The applications ◦ Eigen fast for largest SVs • The problems • The methods methods on C ◦ slow for smallest SVs • Convergence issue • Accuracy issue Related work primme svds: why Eigen ◦ slower for largest SVs choose the two stage methods on B strategy ◦ extremely slow for smallest SVs primme svds: how to (interior eigenvalue problem) develop the two stage strategy Evaluations Conclusions CopperMountain2014 5 / 24

Introduction: difference between methods • Introduction Convergence speed • The applications ◦ Eigen fast for largest SVs • The problems • The methods methods on C ◦ slow for smallest SVs • Convergence issue • Accuracy issue Related work primme svds: why Eigen ◦ slower for largest SVs choose the two stage methods on B strategy ◦ extremely slow for smallest SVs primme svds: how to (interior eigenvalue problem) develop the two stage strategy Evaluations LBD on A ◦ fast for largest SVs Conclusions similar to C but exhibits irregular ◦ convergence for smallest SVs CopperMountain2014 5 / 24

Introduction: difference between methods • Introduction Accuracy • The applications • The problems • The methods • Convergence issue • Accuracy issue Related work primme svds: why choose the two stage strategy primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 6 / 24

Introduction: difference between methods • Introduction Accuracy • The applications ◦ Eigen can only achieve accuracy of • The problems • The methods methods on C O ( κ ( A ) � A � ǫ mach ) • Convergence issue • Accuracy issue Related work primme svds: why choose the two stage strategy primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 6 / 24

Introduction: difference between methods • Introduction Accuracy • The applications ◦ Eigen can only achieve accuracy of • The problems • The methods methods on C O ( κ ( A ) � A � ǫ mach ) • Convergence issue • Accuracy issue Related work ◦ primme svds: why Eigen can achieve accuracy of choose the two stage methods on B O ( � A � ǫ mach ) strategy primme svds: how to develop the two stage strategy Evaluations Conclusions CopperMountain2014 6 / 24

Introduction: difference between methods • Introduction Accuracy • The applications ◦ Eigen can only achieve accuracy of • The problems • The methods methods on C O ( κ ( A ) � A � ǫ mach ) • Convergence issue • Accuracy issue Related work ◦ primme svds: why Eigen can achieve accuracy of choose the two stage methods on B O ( � A � ǫ mach ) strategy primme svds: how to develop the two stage strategy LBD on A ◦ can achieve accuracy of Evaluations O ( � A � ǫ mach ) Conclusions CopperMountain2014 6 / 24