A Semi-Algebraic Framework for Approximate CP Tensor Decompositions via Simultaneous Matrix Diagonalizations


Jacobs Hall, Room 2512, Jacobs School of Engineering, 9500 Gilman Dr, La Jolla, San Diego, California 92093

Sponsored By:
Professor Piya Pal

Dr Martin Haardt
Ilmenau University of Technology
Communications Research Laboratory
Dr Martin Haardt


The Canonical Polyadic (CP) decomposition of R-way arrays is a powerful tool in multi-linear algebra. Algorithms to compute an approximate CP decomposition from noisy observations are often based on Alternating Least Squares (ALS) which may require a large number of iterations to converge. To avoid this drawback we investigate semi-algebraic approaches that algebraically reformulate the CP decomposition into a set of simultaneous matrix diagonalization (SMD) problems. To this end, we propose a SEmi-algebraic framework that allows the computation of approximate CP decompositions via SImultaneous Matrix Diagonalizations (SECSI). In contrast to previous Simultaneous Matrix Diagonalization (SMD)-based approaches, we use the tensor structure to construct not only one but the full set of possible SMDs. This concept facilitates a distributed implementation on a parallel high-performance architecture. Solving all SMDs, we obtain multiple estimates of the factor matrices and present strategies to choose the best estimate in a subsequent step. This SECSI framework retains the option to choose the number of SMDs to solve and to adopt various strategies for the selection of the final solution out of the multiple estimates. A best matching scheme based on an exhaustive search as well as heuristic selection schemes with a reduced computational complexity are devised to  flexibly adapt to specific applications. Several example algorithms with different accuracy- complexity trade-off points are compared to state-of-the-art algorithms. We obtain more reliable estimates and a reduced computational complexity. For tensors with R > 3 dimensions, it is beneficial to combine the SECSI framework with the concept of generalized unfoldings (SECSI-GU) in order to enhance their identifiability. These generalized unfoldings are known from the “Semi-Algebraic Tensor Decomposition” (SALT) algorithm. The resulting SECSI-GU framework offers a large number of degrees of freedom  to flexibly adapt the performance-complexity trade-off. As we show in numerical simulations, it outperforms SECSI and SALT for tensors with R > 3 dimensions. Several combined signal processing applications such as the joint processing of EEG and MEG data can benefit from coupled tensor decompositions, for instance, the coupled CP decomposition. It jointly decomposes tensors that have at least one factor matrix in common. Therefore, we also present an extension of the SECSI framework to the efficient calculation  of coupled CP decompositions and show its advantages compared to the traditional solution via coupled ALS. It also provides a better accuracy than the original SECSI framework in challenging scenarios.

Speaker Bio:
Martin Haardt has been a Full Professor in the Department of Electrical Engineering and Information Technology and Head of the Communications Research Laboratory at Ilmenau University of Technology, Germany, since 2001. He has also served as an Honorary Visiting Professor in the Department of Electronics at the University of York, United Kingdom, since 2012. After studying electrical engineering at the Ruhr-University Bochum, Germany, and at Purdue University, USA, he received his Diplom-Ingenieur (M.S.) degree from the Ruhr-University Bochum in 1991 and his Doktor-Ingenieur (Ph.D.) degree from Munich University of Technology in 1996. In 1997 he joint Siemens Mobile Networks in Munich, Germany, where he was responsible for strategic research for third generation mobile radio systems. From 1998 to 2001 he was the Director for International Projects and University Cooperations in the mobile infrastructure business of Siemens in Munich, where his work focused on mobile communications beyond the third generation. During his time at Siemens, he also taught in the international Master of Science in Communications Engineering program at Munich University of Technology. Martin Haardt has received the 2009 Best Paper Award from the IEEE Signal Processing Society, the Vodafone (formerly Mannesmann Mobilfunk) Innovations-Award for outstanding research in mobile communications, the ITG best paper award from the Association of Electrical Engineering, Electronics, and Information Technology (VDE), and the Rohde & Schwarz Outstanding Dissertation Award. In the fall of 2006 and the fall of 2007 he was a visiting professor at the University of Nice in Sophia-Antipolis, France, and at the University of York, UK, respectively. His research interests include wireless communications, array signal processing, high-resolution parameter estimation, as well as numerical linear and multi- linear algebra. Prof. Haardt has served as an Associate Editor for the IEEE Transactions on Signal Processing (2002-2006 and 2011-2015), the IEEE Signal Processing Letters (2006-2010), the Research Letters in Signal Processing (2007-2009), the Hindawi Journal of Electrical and Computer Engineering (since 2009), the EURASIP Signal Processing Journal (2011-2015), and as a guest editor for the EURASIP Journal on Wireless Communications and Networking. Since 2011, he has been an elected member of the Sensor Array and Multichannel (SAM) technical committee of the IEEE Signal Processing Society (since 2011), where he currently serves as the Vice Chair (since 2015). Moreover, he has served as the technical co-chair of PIMRC 2005 in Berlin, Germany, the European Wireless 2015 in Barcelona, Spain, as well as ISWCS 2010 in York, UK, and as the general co-chair of ISWCS 2013 in Ilmenau, Germany, CAMSAP 2013 in Saint Martin, French Antilles, WSA 2015 in Ilmenau, SAM 2016 in Rio de Janeiro, Brazil, and CAMSAP 2017 in Curacao, Dutch Antilles.

Bethany Carson
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