keras-lmu 0.3.0

Keras implementation of Legendre Memory Units

KerasLMU: Recurrent neural networks using Legendre Memory Units

Paper

This is a Keras-based implementation of the Legendre Memory Unit (LMU). The LMU is a novel memory cell for recurrent neural networks that dynamically maintains information across long windows of time using relatively few resources. It has been shown to perform as well as standard LSTM or other RNN-based models in a variety of tasks, generally with fewer internal parameters (see this paper for more details). For the Permuted Sequential MNIST (psMNIST) task in particular, it has been demonstrated to outperform the current state-of-the-art results. See the note below for instructions on how to get access to this model.

The LMU is mathematically derived to orthogonalize its continuous-time history – doing so by solving d coupled ordinary differential equations (ODEs), whose phase space linearly maps onto sliding windows of time via the Legendre polynomials up to degree d − 1 (the example for d = 12 is shown below).

A single LMU cell expresses the following computational graph, which takes in an input signal, x, and couples a optimal linear memory, m, with a nonlinear hidden state, h. By default, this coupling is trained via backpropagation, while the dynamics of the memory remain fixed.

The discretized A and B matrices are initialized according to the LMU’s mathematical derivation with respect to some chosen window length, θ. Backpropagation can be used to learn this time-scale, or fine-tune A and B, if necessary.

Both the kernels, W, and the encoders, e, are learned. Intuitively, the kernels learn to compute nonlinear functions across the memory, while the encoders learn to project the relevant information into the memory (see paper for details).

Note

The paper branch in the lmu GitHub repository includes a pre-trained Keras/TensorFlow model, located at models/psMNIST-standard.hdf5, which obtains a psMNIST result of 97.15%. Note that the network is using fewer internal state-variables and neurons than there are pixels in the input sequence. To reproduce the results from this paper, run the notebooks in the experiments directory within the paper branch.

Nengo Examples

• LMUs in Nengo (with online learning)

• Spiking LMUs in Nengo Loihi (with online learning)

• LMUs in NengoDL (reproducing SotA on psMNIST)

Citation

@inproceedings{voelker2019lmu,
  title={Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks},
  author={Aaron R. Voelker and Ivana Kaji\'c and Chris Eliasmith},
  booktitle={Advances in Neural Information Processing Systems},
  pages={15544--15553},
  year={2019}
}

Release history

0.3.0 (November 6, 2020)

Changed

• Renamed module from lmu to keras_lmu (so it will now be imported via import keras_lmu), renamed package from lmu to keras-lmu (so it will now be installed via pip install keras-lmu), and changed any references to “NengoLMU” to “KerasLMU” (since this implementation is based in the Keras framework rather than Nengo). In the future the lmu namespace will be used as a meta-package to encapsulate LMU implementations in different frameworks. (#24)

0.2.0 (November 2, 2020)

Added

• Added documentation for package description, installation, usage, API, examples, and project information. (#20)

• Added LMU FFT cell variant and auto-switching LMU class. (#21)

• LMUs can now be used with any Keras RNN cell (e.g. LSTMs or GRUs) through the hidden_cell parameter. This can take an RNN cell (like tf.keras.layers.SimpleRNNCell or tf.keras.layers.LSTMCell) or a feedforward layer (like tf.keras.layers.Dense) or None (to create a memory-only LMU). The output of the LMU memory component will be fed to the hidden_cell. (#22)

• Added hidden_to_memory, memory_to_memory, and input_to_hidden parameters to LMUCell, which can be used to enable/disable connections between components of the LMU. They default to disabled. (#22)

• LMUs can now be used with multi-dimensional memory components. This is controlled through a new memory_d parameter of LMUCell. (#22)

• Added dropout parameter to LMUCell (which applies dropout to the input) and recurrent_dropout (which applies dropout to the memory_to_memory connection, if it is enabled). Note that dropout can be added in the hidden component through the hidden_cell object. (#22)

Changed

• Renamed lmu.lmu module to lmu.layers. (#22)

• Combined the *_encoders_initializer``parameters of ``LMUCell into a single kernel_initializer parameter. (#22)

• Combined the *_kernel_initializer parameters of LMUCell into a single recurrent_kernel_initializer parameter. (#22)

Removed

• Removed Legendre, InputScaled, LMUCellODE, and LMUCellGating classes. (#22)

• Removed the method, realizer, and factory arguments from LMUCell (they will take on the same default values as before, they just cannot be changed). (#22)

• Removed the trainable_* arguments from LMUCell. This functionality is largely redundant with the new functionality added for enabling/disabling internal LMU connections. These were primarily used previously for e.g. setting a connection to zero and then disabling learning, which can now be done more efficiently by disabling the connection entirely. (#22)

• Removed the units and hidden_activation parameters of LMUCell (these are now specified directly in the hidden_cell. (#22)

• Removed the dependency on nengolib. (#22)

• Dropped support for Python 3.5, which reached its end of life in September 2020. (#22)

0.1.0 (June 22, 2020)

Initial release of KerasLMU 0.1.0! Supports Python 3.5+.

The API is considered unstable; parts are likely to change in the future.

Thanks to all of the contributors for making this possible!