Browsing by Subject "Random projection"

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Open Access
Büyük ölçekli doğrusal denklem sistemleri için hızlı ve gürbüz çözüm teknikleri
(IEEE, 2019-04) Özaslan, İbrahim K.; Pilancı, Mert; Arıkan, Orhan
Büyük ölçekli doğrusal sistemlerin veri matrisi, sütunlar arası yüksek ilintiye ve genellikle yüksek durum numaralarına sahiptir. Bilinmeyenlerin, ölçümlerden En Küçük Kareler (EKK) tekniğiyle üretilmesi, ölçüm gürültüsünün, sonucu kabul edilemez şekilde etkilemesine neden olmaktadır. Bu nedenle gürbüz çözüm tekniklerine ihtiyaç duyulmaktadır. Bu bildiride, yüksek durum numarasına sahip büyük ölçekli ölçüm matrislerinin yer aldığı doğrusal sistemlerin, Momentum - Yinelemeli Hessian Krokileme (Momentum - Iterative Hessian Sketch (M-IHS)) çözücüsü kullanılarak nasıl düzgelenebileceği incelenmiştir. Önerilen çözücü, tüm iterasyonlar için tek bir düzgeleme parametresi bulmak yerine, her bir iterasyon için düzgeleme parametresini başka bir parametre ayarı yapmadan otomatik olarak bulmakta ve daha sonra hızlı yaklaşım sağlayan momentum parametrelerini buna göre belirlemektedir. Yapılan analizde her ne kadar Genelleştirilmiş Çapraz Dogrulama (GCV) tekniği kullanılmış olsa da, M-IHS, bildiride açıklanan adımlar kullanılarak, herhangi bir risk tahmini ile düzgelenebilir.
Open Access
Fast and robust solution techniques for large scale linear least squares problems
(2020-07) Özaslan, İbrahim Kurban
Momentum Iterative Hessian Sketch (M-IHS) techniques, a group of solvers for large scale linear Least Squares (LS) problems, are proposed and analyzed in detail. Proposed M-IHS techniques are obtained by incorporating the Heavy Ball Acceleration into the Iterative Hessian Sketch algorithm and they provide signiﬁcant improvements over the randomized preconditioning techniques. By using approximate solvers along with the iterations, the proposed techniques are capable of avoiding all matrix decompositions and inversions, which is one of the main advantages over the alternative solvers such as the Blendenpik and the LSRN. Similar to the Chebyshev Semi-iterations, the M-IHS variants do not use any inner products and eliminate the corresponding synchronization steps in hierarchical or distributed memory systems, yet the M-IHS converges faster than the Chebyshev Semi-iteration based solvers. Lower bounds on the required sketch size for various randomized distributions are established through the error analyses of the M-IHS variants. Unlike the previously proposed approaches to produce a solution approximation, the proposed M-IHS techniques can use sketch sizes that are proportional to the statistical dimension which is always smaller than the rank of the coeﬃcient matrix. Additionally, hybrid schemes are introduced to estimate the unknown ℓ2-norm regularization parameter along with the iterations of the M-IHS techniques. Unlike conventional hybrid methods, the proposed Hybrid M-IHS techniques estimate the regularization parameter from the lower dimensional sub-problems that are constructed by random projections rather than the deterministic projections onto the Krylov Subspaces. Since the lower dimensional sub-problems that arise during the iterations of the Hybrid M-IHS variants are close approximations to the Newton sub-systems and the accuracy of their solutions increase exponentially, the parameters estimated from them rapidly converge to a proper regularization parameter for the full problem. In various numerical experiments conducted at several noise levels, the Hybrid M-IHS variants consistently estimated better regularization parameters and constructed solutions with less errors than the direct methods in far fewer iterations than the conventional hybrid methods. In large scale applications where the coeﬃcient matrix is distributed over a memory array, the proposed Hybrid M-IHS variants provide improved eﬃciency by minimizing the number of distributed matrix-vector multiplications with the coeﬃcient matrix.
Open Access
M-IHS: An accelerated randomized preconditioning method avoiding costly matrix decompositions
(Elsevier Inc., 2023-08-29) Ozaslan, I. K.; Pilanci, M.; Arıkan, Orhan
Momentum Iterative Hessian Sketch ([Formula presented]) techniques, a group of solvers for large scale regularized linear Least Squares (LS) problems, are proposed and analyzed in detail. The proposed [Formula presented] techniques are obtained by incorporating Polyak's heavy ball acceleration into the Iterative Hessian Sketch algorithm and they provide significant improvements over the randomized preconditioning techniques. By solving the linear systems arising in the sub-problems during the iterations approximately, the proposed techniques are capable of avoiding all matrix decompositions and inversions, which is one of the main advantages over the alternative solvers such as the Blendenpik and the LSRN. Similar to the Chebyshev semi-iterations, the [Formula presented] variants do not use any inner products and eliminate the corresponding synchronization steps in hierarchical or distributed memory systems, yet the [Formula presented] converges faster than the Chebyshev semi-iteration based solvers. Lower bounds on the required sketch size for various randomized distributions are established through the error analyses. Unlike the previously proposed approaches to produce a solution approximation, the proposed [Formula presented] techniques can use sketch sizes that are proportional to the statistical dimension which is always smaller than the rank of the coefficient matrix. © 2023 Elsevier Inc.