CoChat
Sign Up
Cited by 55

The DOI matches its Crossref record; 2 of the 2 references it cites were found in a registry or an index.

The Partial Information Decomposition of Generative Neural Network Models

Tax, Tycho, Mediano, Pedro, Shanahan, Murray·2017·Entropy
DOI 10.3390/e19090474
Open this paper in CoChat — free account

Abstract

In this work we study the distributed representations learnt by generative neural network models. In particular, we investigate the properties of redundant and synergistic information that groups of hidden neurons contain about the target variable. To this end, we use an emerging branch of information theory called partial information decomposition (PID) and track the informational properties of the neurons through training. We find two differentiated phases during the training process: a first short phase in which the neurons learn redundant information about the target, and a second phase in which neurons start specialising and each of them learns unique information about the target. We also find that in smaller networks individual neurons learn more specific information about certain features of the input, suggesting that learning pressure can encourage disentangled representations.

Overview

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Integrity signals

Registry

Crossref · 10.3390/e19090474

Cites

2 of 2 references verified0 citation errors · 0 not checkable

Statistics

Figures

Figure 1 Graphical representation of the discriminative restricted Boltzmann machine (DRBM) and its components. Vectors x and y correspond to the training input and label, respectively, h is the activation of the hidden neurons, and U and W are the weight matrices to be learned. (Adapted from [23]).
Figure 1 Graphical representation of the discriminative restricted Boltzmann machine (DRBM) and its components. Vectors x and y correspond to the training input and label, respectively, h is the activation of the hidden neurons, and U and W are the weight matrices to be learned. (Adapted from [23]).CC-BY · From the PDF
Figure 2 Partial information (PI) diagram for two source variables and a target. The outer ellipse corresponds to the mutual information (MI) between both sources and the target, I ( Y ; { Z 1 , Z 2 } ) , and both inner circles (highlighted in black) to the MI between each source and the target, I ( Y ; Zi ) . Coloured areas represent the PI terms described in the text.
Figure 2 Partial information (PI) diagram for two source variables and a target. The outer ellipse corresponds to the mutual information (MI) between both sources and the target, I ( Y ; { Z 1 , Z 2 } ) , and both inner circles (highlighted in black) to the MI between each source and the target, I ( Y ; Zi ) . Coloured areas represent the PI terms described in the text.CC-BY · From the PDF
Figure 3 Classification error and mutual information between real and predicted labels, I ( ˆ Y , Y ) , calculated through training. Note: X -axis in the rightmost plot is reversed for illustration pusposes, so that training time goes from left to right.
Figure 3 Classification error and mutual information between real and predicted labels, I ( ˆ Y , Y ) , calculated through training. Note: X -axis in the rightmost plot is reversed for illustration pusposes, so that training time goes from left to right.CC-BY · From the PDF
Figure 4 Single-neuron entropy and mutual information follow non-trivial patterns during training. ( a ) Entropy quickly rises up to close to its maximum value of 1 bit. ( b ) Inter-neuron correlation as measured by pairwise MI peaks midway through training. ( c ) Histograms of single-neuron MI (SNMI) split midway through training, implying that some neurons actually lose information. Average SNMI is shown in black projected on the frame box.
Figure 4 Single-neuron entropy and mutual information follow non-trivial patterns during training. ( a ) Entropy quickly rises up to close to its maximum value of 1 bit. ( b ) Inter-neuron correlation as measured by pairwise MI peaks midway through training. ( c ) Histograms of single-neuron MI (SNMI) split midway through training, implying that some neurons actually lose information. Average SNMI is shown in black projected on the frame box.CC-BY · From the PDF
Figure 5 PI terms ( a ) and PI terms normalised by joint mutual information ( b ). Mutual information (MI) in black, redundancy ( R ) in blue, synergy ( S ) in green, and unique information ( U ) in red. MI increases consistently during training, but the PI terms reveal a transition between a redundancy-dominated phase and a unique-information-dominated phase.
Figure 5 PI terms ( a ) and PI terms normalised by joint mutual information ( b ). Mutual information (MI) in black, redundancy ( R ) in blue, synergy ( S ) in green, and unique information ( U ) in red. MI increases consistently during training, but the PI terms reveal a transition between a redundancy-dominated phase and a unique-information-dominated phase.CC-BY · From the PDF
Figure 6 Partial information decomposition (PID) analysis of larger groups of neurons in networks of different sizes. ( a ) Single-neuron MI is consistently smaller in bigger networks, indicating that, although the network as a whole is a better classifier, each individual neuron has a less-efficient encoding; ( b ) Normalised PIK synergy, with network size increasing from left to right and K from top to bottom. Network with maximum synergy for each PIK highlighted with a vertical dashed line. The PI group size with the highest synergy becomes larger in larger networks, indicating that in bigger networks one needs to consider larger groups to capture strong cooperative interactions.
Figure 6 Partial information decomposition (PID) analysis of larger groups of neurons in networks of different sizes. ( a ) Single-neuron MI is consistently smaller in bigger networks, indicating that, although the network as a whole is a better classifier, each individual neuron has a less-efficient encoding; ( b ) Normalised PIK synergy, with network size increasing from left to right and K from top to bottom. Network with maximum synergy for each PIK highlighted with a vertical dashed line. The PI group size with the highest synergy becomes larger in larger networks, indicating that in bigger networks one needs to consider larger groups to capture strong cooperative interactions.CC-BY · From the PDF

Verification

Claims the paper makes

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

The self-check needs the paper’s full text.

Citations

Cites

Show all 2 references

Research works printed but not linked: 35

  1. Srivastava, 2014, Dropout: A Simple Way to Prevent Neural Networks from Overfitting, J. Mach. Learn. Res., 15, 1929
  2. Gal, Y., and Ghahramani, Z. (2015). Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning. arXiv.
  3. Bengio, Y., Courville, A., and Vincent, P. (2012). Representation Learning: A Review and New Perspectives. arXiv.
  4. Higgins, I., Matthey, L., Glorot, X., Pal, A., Uria, B., Blundell, C., Mohamed, S., and Lerchner, A. (2016). Early Visual Concept Learning with Unsupervised Deep Learning. arXiv.
  5. Mathieu, M., Zhao, J., Sprechmann, P., Ramesh, A., and LeCun, Y. (2016). Disentangling Factors of Variation in Deep Representations Using Adversarial Training. arXiv.
  6. Siddharth, N., Paige, B., Van de Meent, J.W., Desmaison, A., Wood, F., Goodman, N.D., Kohli, P., and Torr, P.H.S. (2017). Learning Disentangled Representations with Semi-Supervised Deep Generative Models. arXiv.
  7. Garnelo, M., Arulkumaran, K., and Shanahan, M. (2016). Towards Deep Symbolic Reinforcement Learning. arXiv.
  8. Williams, P.L., and Beer, R.D. (2010). Nonnegative Decomposition of Multivariate Information. arXiv.
  9. Rieke, F., Bialek, W., Warland, D., and de Ruyter van Steveninck, R. (1997). Spikes: Exploring the Neural Code, MIT Press.
  10. Le, Q.V., Ranzato, M., Monga, R., Devin, M., Chen, K., Corrado, G.S., Dean, J., and Ng, A.Y. (2011). Building High-Level Features Using Large Scale Unsupervised Learning. arXiv.
  11. Zeiler, M.D., and Fergus, R. (2014, January 6–12). Visualizing and Understanding Convolutional Networks. Proceedings of the European Conference on Computer Vision, Zurich, Switzerland. · DOI · Paper page
  12. Choromanska, A., Henaff, M., Mathieu, M., Arous, G.B., and LeCun, Y. (2014). The Loss Surfaces of Multilayer Networks. arXiv.
  13. Kawaguchi, K. (2016). Deep Learning Without Poor Local Minima. arXiv.
  14. Sørngård, B. (2014). Information Theory for Analyzing Neural Networks. [Master’s Thesis, Norwegian University of Science and Technology].
  15. Schwartz-Ziv, R., and Tishby, N. (2017). Opening the Black Box of Deep Neural Networks via Information. arXiv.
  16. Achille, A., and Soatto, S. (2017). On the Emergence of Invariance and Disentangling in Deep Representations. arXiv. · DOI
  17. Tishby, N., and Zaslavsky, N. (2015). Deep Learning and the Information Bottleneck Principle. arXiv. · DOI
  18. Berglund, 2015, Measuring the Usefulness of Hidden Units in Boltzmann Machines with Mutual Information, Neural Netw., 64, 12, 10.1016/j.neunet.2014.09.004 · DOI
  19. Balduzzi, D., Frean, M., Leary, L., Lewis, J., Ma, K.W.D., and McWilliams, B. (2017). The Shattered Gradients Problem: If Resnets are the Answer, Then What is the Question?. arXiv.
  20. Smolensky, P. (1986). Information Processing in Dynamical Systems: Foundations of Harmony Theory, MIT Press. Technical Report, DTIC Document.
  21. Larochelle, H., and Bengio, Y. (2008, January 5–9). Classification Using Discriminative Restricted Boltzmann Machines. Proceedings of the 25th International Conference on Machine Learning, Helsinki, Finland. · DOI
  22. Hinton, 2006, A Fast Learning Algorithm for Deep Belief Nets, Neural Comput., 18, 1527, 10.1162/neco.2006.18.7.1527 · DOI · Paper page
  23. Tieleman, T. (2008, January 5–9). Training Restricted Boltzmann Machines Using Approximations to the Likelihood Gradient. Proceedings of the 25th International Conference on Machine Learning, Helsinki, Finland. · DOI
  24. Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, Wiley. · DOI · Paper page
  25. DeWeese, 1999, How to Measure the Information Gained from one Symbol, Netw. Comput. Neural Syst., 12, 325, 10.1088/0954-898X_10_4_303 · DOI
  26. Ince, R.A.A. (2017). Measuring Multivariate Redundant Information with Pointwise Common Change in Surprisal. Entropy, 19. · DOI
  27. Griffith, 2015, Quantifying Redundant Information in Predicting a Target Random Variable, Entropy, 17, 4644, 10.3390/e17074644 · DOI
  28. Harder, M., Salge, C., and Polani, D. (2013). Bivariate Measure of Redundant Information. Phys. Rev. E, 87. · DOI
  29. Gilbert, T., Kirkilionis, M., and Nicolis, G. Shared Information—New Insights and Problems in Decomposing Information in Complex Systems. Proceedings of the European Conference on Complex Systems 2012.
  30. Williams, P.L. (2011). Information Dynamics: Its Theory and Application to EmbodiedCognitive Systems. [Ph.D. Thesis, Indiana University].
  31. Lizier, J.T. (2010). The Local Information Dynamics of Distributed Computation in Complex Systems, Springer.
  32. Timme, 2014, Synergy, Redundancy, and Multivariate Information Measures: An Experimentalist’s Perspective, J. Comput. Neurosci., 36, 119, 10.1007/s10827-013-0458-4 · DOI
  33. Bertschinger, 2014, Quantifying Unique Information, Entropy, 16, 2161, 10.3390/e16042161 · DOI · Paper page
  34. Ay, 2015, Geometry and Expressive Power of Conditional Restricted Boltzmann Machines, J. Mach. Learn. Res., 16, 2405
  35. Kingma, D.P., and Welling, M. (2013). Auto-Encoding Variational Bayes. arXiv.

Cited by

Sorted by citations

Metadata

Authors
Tax, T., Mediano, P., & Shanahan, M.
Affiliations
  • Corti, Nørrebrogade 45E 2, 2200 Copenhagen N, Denmark
  • Department of Computing, Imperial College London, London SW7 2RH, UK
ORCID
Venue
Entropy
Publisher
MDPI AG
Published
2017-09-06
Volume
19
Issue
9
Pages
474
ISSN
1099-4300
DOI
10.3390/e19090474
Cited by
55
References
2
Language
en
Licence
creativecommons.org/licenses/by/4.0/ (vor)

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review