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© 2015 Nexidia, Inc.
Dale Smith, Ph.D.
Nexidia, Inc.
Tensor Decompositions and Machine
Learning
What is a Tensor?
• We’re moving from linear algebra to multilinear algebra…
• We know about vectors and matrices (linear ...
What is a Tensor Decomposition?
• Think of a hypercube in your data warehouse – can you do a decomposition
into lower-rank...
Tensor Decompositions in Machine Learning
• Feature discovery via hierarchical structures in high-dimensional data sets.
•...
Tensor Decompositions in Machine Learning
• Anima Anandkumar, et. al.: Fast Detection of Overlapping Communities via
Onlin...
Tensor Decomposition Software
MATLAB C++ R Python
Tensor
Toolbox
TensorDecomposition4Topic
Modeling
rTensor TensorToolbox
...
A Few References
• Survey papers with excellent visualizations of tensor decompositions
– Tensor Decompositions for Signal...
Recent Events
• Blog Posts and Podcasts
– Anima Anandkumar, Tensor Methods for Large-Scale Machine Learning, Strata+Hadoop...
Takeaways
• Do you have…
– Time series of matrices or cubes?
– Multiple aspects: time, location, type?
– Time-evolving gra...
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Dale Smith, Data Scientist, Nexidia at MLconf ATL - 9/18/15

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Tensor Decompositions and Machine Learning: We know about vectors and matrices (linear transformations) from Linear Algebra. But tensors are not so familiar. Think of a hypercube in your data warehouse – can you do a tensor decomposition into lower-rank objects that reveal hidden features or hierarchies?

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Dale Smith, Data Scientist, Nexidia at MLconf ATL - 9/18/15

  1. 1. © 2015 Nexidia, Inc. Dale Smith, Ph.D. Nexidia, Inc. Tensor Decompositions and Machine Learning
  2. 2. What is a Tensor? • We’re moving from linear algebra to multilinear algebra… • We know about vectors and matrices (linear transformations) from Linear Algebra. But tensors are not so familiar. A Rubik's Cube can help with visualization…and so can What's a Tensor? via YouTube! • Think of a hypercube in your data warehouse. Page 2© 2015 Nexidia, Inc.
  3. 3. What is a Tensor Decomposition? • Think of a hypercube in your data warehouse – can you do a decomposition into lower-rank objects that reveal hidden features or hierarchies? • A rank-2 tensor decomposition – the Singular Value Decomposition – computes A = UDVT. Principal Components Analysis approximates the original matrix by reducing the rank. • For rank higher than 2 there are two types of decompositions: a sum of rank-1 tensors and a decomposition into orthonormal subspaces. Oh, and there is a multilinear PCA as well… Page 3© 2015 Nexidia, Inc.
  4. 4. Tensor Decompositions in Machine Learning • Feature discovery via hierarchical structures in high-dimensional data sets. • Recommender systems – similarity measures between users and items. • Time-evolving network expressed as a 3-tensor. • Topic modeling to extract hidden topics in a document corpus. • Multilinear subspace learning – a technique of dimensionality reduction. • Tensor methods are fast, scalable, and in many cases are embarrassingly parallel. Page 4© 2015 Nexidia, Inc.
  5. 5. Tensor Decompositions in Machine Learning • Anima Anandkumar, et. al.: Fast Detection of Overlapping Communities via Online Tensor Methods, JMLR 2014 and Tensor Methods for Large-Scale Machine Learning, Strata+Hadoop, February 2015. • Computational issues with estimating latent variable models, e.g. topic models. – Local optima. – Slow convergence. – Markov Chain Monte Carlo can fail. • Tensor decomposition approaches guarantee finding the global optimum. • Observed speedups of 100x over the Expectation-Maximization and other estimation methods using data from Facebook and Yelp restaurant reviews. Page 5© 2015 Nexidia, Inc.
  6. 6. Tensor Decomposition Software MATLAB C++ R Python Tensor Toolbox TensorDecomposition4Topic Modeling rTensor TensorToolbox Tensorlab TH++ PyTensor scikit-tensor Page 6© 2015 Nexidia, Inc.
  7. 7. A Few References • Survey papers with excellent visualizations of tensor decompositions – Tensor Decompositions for Signal Processing Applications, Cichocki, De Lathauwer, et. al., IEEE Signal Processing Magazine, March 2015 (arXiv.org version). – Tensor Decompositions and Applications, Tamara G. Kolda and Brett W. Bader, SIAM Review, 2009 (Sandia pdf report). • 2009 Workshop Report: Future Directions in Tensor-Based Computation and Modeling. • Tutorials – Tutorial on Spectral and Tensor Methods for Guaranteed Learning. – Mining Large Time-evolving Data using Matrix and Tensor Tools. – Mining and Forecasting of Big Time-Series Data. Page 7© 2015 Nexidia, Inc.
  8. 8. Recent Events • Blog Posts and Podcasts – Anima Anandkumar, Tensor Methods for Large-Scale Machine Learning, Strata+Hadoop, February 2015. – Ben Lorica, Let’s build open source tensor libraries for data science, O’Reilly Radar, March 17, 2015. – The tensor renaissance in data science and associated podcast, O’Reilly Radar, May 7, 2015. • Tensor Decompositions in Smart Patient Monitoring, SIAM News, September 2015. • SIAM Conference on Applied Linear Algebra, October 26th – 30th, Atlanta. – Sessions on Tuesday the 27th. – Keynote on Friday the 30th. • Workshop on Tensor Decompositions and Applications, January 2016. Page 8© 2015 Nexidia, Inc.
  9. 9. Takeaways • Do you have… – Time series of matrices or cubes? – Multiple aspects: time, location, type? – Time-evolving graphs? – Primary source data with multiple metadata elements? • …then you may want to consider tensor decomposition methods to estimate models. Page 9© 2015 Nexidia, Inc.

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