lsmmdma 0.0.7

Scaling MMD-MA.

Large-Scale MMD-MA

Installation | Examples | Command line instructions | Output

The objective of MMD-MA is to match points coming from two different spaces in a lower dimensional space. To this end, two sets of points are projected, from two different spaces endowed with a positive definite kernel, to a shared Euclidean space of lower dimension low_dim. The mappings from high to low dimensional space are obtained using functions belonging to the respective RKHS. To obtain the mappings, we minimise a loss function that is composed of three terms:

• an MMD term between the low dimensional representations of the two views, which encourages them to have the same distribution. The RBF kernel is used to compute MMD.
• two non-collapsing penalty terms (corresponding to the pen_dual or pen_primal functions), one for each view. These terms ensure that the low dimensional representations are mutually orthogonal, preventing collapsing.
• two distortion penalties (corresponding to the dis_dual or dis_primal functions), one for each view. These terms encourage the low dimensional representation to obtain the same pairwise structure as the original views.

MMD-MA can be formulated using either the primal (when we use the linear kernel in the input spaces) or the dual problem. Each has advantages or disadvantages depending on the input data. For each view, when the number of features p is larger than the number of samples n p >> n, then the dual formulation is beneficial in terms of runtime and memory, while if n >> p, the primal formulation is favorable.

Additionally, in order to scale the computation of MMD to a large number of samples, we propose to use the library KeOps which lets you compute large kernel operations on GPUs without memory overflow.

Installation

To install the latest release of lsmmdma, use the following command:

$ pip install lsmmdma

To install the development version, use the following command instead:

$ pip install git+https://github.com/google-research/large-scale-mmdma

Alternatively, it can be installed from sources with the following command:

$ python setup.py install

In Google Colab, use the following command:

$ !pip install lsmmdma

The KeOps library might require to be installed separately in advance, according to the given instructions.

Command line instructions

The algorithm can be run with a command line using:

python3 -m lsmmmda.main [flags]

A set of flags is available to determine the IO, the model, the hyperparameters and the seed.

Input/Output It is possible to give as input either a path and filenames pointing to the user input or to choose to generate data with data_pipeline.py. In the case one wants to generate simulation data, the input flags are:

• --data: str, it can be either 'branch', 'triangle' or None (default). The simulated data is described in the pydoc of data_pipeline.generate_data. None means that simulated data is not used.
• --n: int (default None), number of samples for both views.
• --p: int (default None), number of features for both views.

For data given by the user, the input flags are:

• --input_dir: str (default None), input directory.
• --input_fv: str (default None), filename of the array (n1 x p1 or n1 x n1) that serves as first set of points.
• --input_sv: str (default None), filename of the array (n2 x p2 or n2 x n2) that serves as second set of points.
• --rd_vec: str (default None), filename of the vector that contains the indices of the samples from the first view that match (ground truth) the samples from the second view. This is only used at evaluation time. If --rd_vec is not used, we assume that the samples of both views follow the same ordering.

In both cases, two other flags are also available:

• --kernel: bool (default False), whether the input data given by the user is a kernel (n x n instead of n x p). This parameter can only be set to True when --mode is 'dual'.
• --output_dir: str (default None), output directory.

Model The flags allow you to choose between four algorithms, using either the 'primal' or 'dual' formulation, and using KeOps or not.

• --mode: str, either 'primal' or 'dual' (default).
• --keops: bool, either True (default) or False.

Seeds The seed for the training phase, and for generating the data when --data is not None, is fixed with the flag --seed (int, default value is 0). If one wishes to use multipe starts when training (X seeds and selection of the best one based on the value of the loss), it is possible to also define the number of seeds to try with: --ns (int, default value is 1).

Model hyperparameters Several hyperparameters ought to be fixed in advance:

• --d: int (default 5), dimension of the latent space.
• --init: str (default 'uniform,0.,0.1'), initialisation for the learned parameters. It can be sampled from a 'uniform', 'normal', 'xavier_uniform' or 'xavier_normal' distributions. The parameters of the initialisation functions are passed to the same flag separated by a coma. See initializers.py and train.py.
• --l1: float (default 1e-4), hyperparameter in front of both penalty terms.
• --l2: float (default 1e-4), hyperparameter in front of both distortion terms.
• --lr: float (default 1e-5), learning rate.
• --s: float (default 1.0), scale parameter of the RBF kernel in MMD.

Training and evaluation Several flags structure the training loop:

• --e: int (default 5001), number of epochs for the training process.
• --ne: int (default 100), regular interval at which the evaluation (call to metrics.SupervisedEvaluation) is done, every 'ne' epochs. 0 means that the results are never evaluated.
• --nr: int (default 100), regular interval at which the loss and components of the loss are recorded, every 'ne' epochs. 0 means that the loss and its components are never recorded.
• --pca: int (default 100), regular interval at which PCA is performed on the embeddings, every 'pca' epochs. 0 means that PCA is not used on the output.

Timing Timing the method is possible when the flag --time is set to True (default False).

We show now three examples of usage of the command line to run the algorithm.

1. To run the algorithm on simulated data from data_pipeline.py, a minimal set of commands is:

python3 -m lsmmdma.main --output_dir outdir \
--data branch --n 300 --p 400 \
--e 1001 --d 5 --keops False --m dual

2. To run the algorithm on simulated data from data_pipeline.py, one can also choose when to record the intermediate results, the hyperparameters and to allow for 5 multiple starts:

python3 -m lsmmdma.main --output_dir outdir \
--data branch --n 300 --p 400 \
--k 4 --ns 5 \
--e 1001 --d 5 --nr 100 --ne 100 --keops False --m dual --pca 100 \
--lr 1e-5 --l1 1e-4 --l2 1e-4 --s 1.0 --init 'uniform,0,0.1'

3. To run the algorithm on user input data, in the form n_sample x p_feature. --data should be '' (default) and --kernel should be False (default). The argument --keops can be True or False, --mode can be 'dual' or 'primal'.

python3 -m lsmmdma.main --input_dir datadir --output_dir outdir \
--input_fv my_data_1 --input_sv my_data_2 --kernel False \
--k 4 --ns 5 \
--e 1001 --d 5 --nr 100 --ne 100 --keops True --m dual --pca 100 \
--lr 1e-5 --l1 1e-4 --l2 1e-4 --s 1.0 --init 'uniform,0,0.1'

4. To run the algorithm on user kernel data, in the form n_sample x n_sample. --data should be '' (default) and --kernel should be True. The argument --keops can be True or False, --mode can only be 'dual'.

python3 -m lsmmdma.main --input_dir datadir --output_dir outdir \
--input_fv my_data_1 --input_sv my_data_2 --kernel True \
--k 4 --ns 5 \
--e 1001 --d 5 --nr 100 --ne 100 --keops True --m dual --pca 100 \
--lr 1e-5 --l1 1e-4 --l2 1e-4 --s 1.0 --init 'uniform,0,0.1'

Output

When one uses main.py, several files are saved in the output directory FLAGS.output_dir:

• [key:val].tsv: results of the model at the last epoch.
• [key:val]_tracking.json: loss and its components during training, evaluation metrics during training, seed, number of epochs.
• [key:val]_model.json: model and optimiser state dictionaries, loss, number of epochs, seed.
• [key:val]_pca.npy: 2D representation obtained with PCA on the embeddings during training.
• [key:val]_embeddings.npy: embeddings during training.
• generated_data_X: first view, second view and rd_vec generated by data_pipeline.generate_data.

[key:val] represent a list of key:value as determined by the flags and written in main.py.

Disclaimer

This is not an officially supported Google product.