Scalable tensor factorization
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Scalable Tensor Factorization

Ext-RESCAL is a memory efficient implementation of RESCAL, a state-of-the-art algorithm for DEDICOM-like tensor factorization. Ext-RESCAL is written in Python and leverages the SciPy Sparse module.

Current Version



  • 3-D sparse tensor factorization [1]
  • Joint 3-D sparse tensor and 2-D sparse matrix factorization (extended version) [2-3]
  • Handy input format
  • Support of float values as tensor values
  • The implementation provably scales well to the domains with millions of nodes on the affordable hardware

[1] M. Nickel, V. Tresp, H. Kriegel. A Three-way Model for Collective Learning on Multi-relational Data // Proceedings of the 28th International Conference on Machine Learning (ICML'2011). - 2011.

[2] M. Nickel, V. Tresp, H. Kriegel. Factorizing YAGO: Scalable Machine Learning for Linked Data // Proceedings of the 21st international conference on World Wide Web (WWW'2012). - 2012.

[3] Nickel, Maximilian. Tensor factorization for relational learning. Diss. München, Ludwig-Maximilians-Universität, Diss., 2013, 2013.

Expected Applications

  • Link Prediction
  • Collaborative Filtering
  • Entity Search


  • Python 2.7+
  • Numpy 1.6+
  • SciPy 0.12+

Usage Examples

1) Let's imagine we have the following semantic graph:


Each tensor slice represents an adjacency matrix of the corresponding predicate (member-of, genre, cites). We run the RESCAL algorithm to decompose a 3-D tensor with 2 latent components and zero regularization on the test data:

python --latent 2 --lmbda 0 --input tiny-example --outputentities entity.embeddings.csv --outputfactors latent.factors.csv --log rescal.log

The test data set represents a tiny entity graph of 3 adjacency matrices (frontal tensor slices) in the row-column representation. See the directory tiny-example. Ext-RESCAL will output the latent factors for the entities into the file entity.embeddings.csv.

2) Then, we assume that there is an entity-term matrix:


Then, we run the extended version of RESCAL algorithm to decompose a 3-D tensor and 2-D matrix with 2 latent components and regularizer equal to 0.001 on the test data (entity graph and entity-term matrix):

python --latent 2 --lmbda 0.001 --input tiny-mixed-example --outputentities entity.embeddings.csv --outputterms term.embeddings.csv --outputfactors latent.factors.csv --log extrescal.log

If we plot the resulting embeddings, we would get the following picture, which reveals the similarity of entities and words in the same latent space:


In case of float values in the entity-term matrix (e.g. TF-IDF weighted vectors), one may use script for calculation, providing the file ext-matrix-elements in the input directory, which contains the values.

Development and Contribution

Ext-RESCAL has been developed by Nikita Zhiltsov. This project is a fork of the original code base provided by Maximilian Nickel (see his latest implementation available under GPL license). Ext-RESCAL may contain some bugs, so, if you find any of them, feel free to contribute the patches via pull requests into the develop branch. If you want to contribute, but have no idea how, please ask on Ext-RESCAL Google Group.


Feel free to ask any questions on Ext-RESCAL Google Group.

Release Notes

0.7 (October 8, 2014):

  • Grealy improve the memory consumption for all scripts after refactoring to using csr_matrix
  • Fix the eigenvalue initialization
  • Improve (speed up) initialization of A by summation

0.6 (March 21, 2014):

  • Make the extended algorigthm output fixed (by replacing random initialization)
  • Add handling of float values in the extended task
  • Add the util for matrix pseudoinversion
  • Switch to Apache License 2.0

0.5 (March 29, 2013):

  • Greatly improve the convergence speed via initialization of starting matrices with eigenvectors

0.4 (March 14, 2013):

  • Add efficient computation of the exact objective value via trick with trace

0.3 (March 12, 2013):

  • Fix random sampling for the basic task
  • Add output of latent factors

0.2 (February 26, 2013):

  • Add an opportunity to approximate the objective function via random sampling
  • Bug fixes
  • Change the default settings

0.1 (January 31, 2013):

  • The basic implementation of both the algorithms


The original algorithms are an intellectual property of the authors of the cited papers.


The author is not responsible for implications from the use of this software.


Licensed under the Apache License, Version 2.0:


If you have used this software in your research work, please cite the following paper:

Zhiltsov, N., Agichtein, E. Improving Entity Search over Linked Data by Modeling Latent Semantics. Proceedings of the International Conference on Information and Knowledge Management (CIKM 2013). ACM, 2013.