Machine Learning

, Volume 94, Issue 3, pp 303–351 | Cite as

Learning with tensors: a framework based on convex optimization and spectral regularization

  • Marco Signoretto
  • Quoc Tran Dinh
  • Lieven De Lathauwer
  • Johan A. K. Suykens


We present a framework based on convex optimization and spectral regularization to perform learning when feature observations are multidimensional arrays (tensors). We give a mathematical characterization of spectral penalties for tensors and analyze a unifying class of convex optimization problems for which we present a provably convergent and scalable template algorithm. We then specialize this class of problems to perform learning both in a transductive as well as in an inductive setting. In the transductive case one has an input data tensor with missing features and, possibly, a partially observed matrix of labels. The goal is to both infer the missing input features as well as predict the missing labels. For induction, the goal is to determine a model for each learning task to be used for out of sample prediction. Each training pair consists of a multidimensional array and a set of labels each of which corresponding to related but distinct tasks. In either case the proposed technique exploits precise low multilinear rank assumptions over unknown multidimensional arrays; regularization is based on composite spectral penalties and connects to the concept of Multilinear Singular Value Decomposition. As a by-product of using a tensor-based formalism, our approach allows one to tackle the multi-task case in a natural way. Empirical studies demonstrate the merits of the proposed methods.


Spectral regularization Matrix and tensor completion Tucker decomposition Multilinear rank Transductive and inductive learning Multi-task learning 



Research supported by: ERC AdG A-DATADRIVE-B, Research Council KUL: GOA/10/09 MaNet, CoE EF/05/006 Optimization in Engineering (OPTEC) en PFV/10/002 (OPTEC), CIF1 and STRT1/08/023, IOF-SCORES4CHEM, several PhD/postdoc and fellow grants; Flemish Government: FWO: PhD/postdoc grants, projects: G0226.06, G.0302.07, G.0320.08, G.0427.10N, G.0558.08, G.0557.08, G.0588.09; IWT: PhD Grants, Eureka-Flite+, SBO LeCoPro, SBO Climaqs, SBO POM, O&O-Dsquare. Belgian Federal Science Policy Office: IUAP P6/04 (DYSCO, Dynamical systems, control and optimization, 2007–2011); IBBT; EU: ERNSI; FP7-HD-MPC (INFSO-ICT-223854), COST intelliCIS, FP7-EMBOCON (ICT-248940), FP7-SADCO (MC ITN-264735).


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Copyright information

© The Author(s) 2013

Authors and Affiliations

  • Marco Signoretto
    • 1
  • Quoc Tran Dinh
    • 2
  • Lieven De Lathauwer
    • 3
  • Johan A. K. Suykens
    • 1
  1. 1.ESAT-SCD/SISTAKatholieke Universiteit LeuvenLeuvenBelgium
  2. 2.Laboratory for Information and Inference Systems (LIONS)Ecole Polytechnique Federale de Lausanne (EPFL)LausanneSwitzerland
  3. 3.Group Science, Engineering and TechnologyKatholieke Universiteit LeuvenKortrijkBelgium

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