Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation

By André Uschmajew. 2012

222 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Mechanics, Computational Theory and Mathematics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “…robustness of gradient-based and Newton-like methods [2,72,74,140,223,224], modifying and improving ALS (alternating least squares) [41, 57, 58, 90, 225], studying the convergence of ALS [51, 194, 253] and reducing the cost of the unfolding operations required during the approximation [222,269].”
Learning Hidden Features for Contextual Bandits (2016, background): “As a result, based on the proved convergence property of alternating least square in [23], the estimation of θ u and va is q-linearly convergent to the optimum (θv∗ u ,v∗ a).”
Low Complexity Damped Gauss-Newton Algorithms for CANDECOMP/PARAFAC (2012, methodology): “The Alternating Least Squares (ALS) algorithm [2–4, 10, 33] sequentially updates A using the update rule given by A = Y(n) ( ⊙ k,n A ) ( Γ )† , (n = 1, 2, .”
Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis (2014, background): “The convergence of ALS is not yet completely understood — it is quasi-linear close to the stationary point [55], while it becomes rather slow for ill-conditioned cases; for more detail we refer to [56], [57].”
Optimization problems in contracted tensor networks (2011, background): “For quadratic functionals F and the canonical tensor format UCP , the situation J = F ◦ UCP is considered in [20].”
LOCAL CONVERGENCE OF ALTERNATING SCHEMES FOR OPTIMIZATION OF CONVEX PROBLEMS IN THE TT FORMAT (2011, methodology): “Our proof is an adaptation of the concept followed in the recent paper [36], where one of the authors showed local convergence of PARAFAC-ALS: in the neighborhood of a solution, the ALS algorithm is identi ed as a perturbation of the linear block Gauss-Seidel method applied to the Hessian at the sol”
The geometry of algorithms using hierarchical tensors (2013, methodology): “In a completely different context, [35] proves convergence of the popular alternating least-squares (ALS) algorithm applied to CP tensors by considering orbits of equivalent tensor representations in a geometric way.”
Zur Theorie der Niedrigrangapproximation in Tensorprodukten von Hilberträumen (2013, background): “Die Darstellung ist alles andere als erschöpfend, sondern orientiert sich lose an den im Literaturverzeichnis gelisteten Arbeiten des Autors.”
Tensor Decomposition for Signal Processing and Machine Learning (2016, background): “In cases where ALS converges, it does so at a local linear rate (under some non-degeneracy condition), which makes it (locally) slower than some derivativebased algorithms [87], [88], see further.”
Factorization Bandits for Interactive Recommendation (2017, methodology): “In Lemma 1, 1 and 2 are the precision parameters for ALS, and q1 and q2 can be explicitly estimated as described in (Uschmajew 2012).”
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, unknown): “In recent papers by Uschmajew [41, 35] the local convergence of ALS is proven for the canonical and tensor train decompositions.”
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format (2013, methodology): “For rank-one tensors, somewhat better statements have been obtained in [42] (cf.”