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OILMM

Implementation of the Orthogonal Instantaneous Linear Mixing Model

Citation:

@inproceedings{Bruinsma:2020:Scalable_Exact_Inference_in_Multi-Output,
    title = {Scalable Exact Inference in Multi-Output {Gaussian} Processes},
    year = {2020},
    author = {Wessel P. Bruinsma and Eric Perim and Will Tebbutt and J. Scott Hosking and Arno Solin and Richard E. Turner},
    booktitle = {Proceedings of 37th International Conference on Machine Learning},
    series = {Proceedings of Machine Learning Research},
    publisher = {PMLR},
    volume = {119},
    eprint = {https://arxiv.org/abs/1911.06287},
}

Contents:

• Requirements and Installation
• TLDR
• Basic Usage
  • Examples of Latent Process Models
    • Smooth Processes
    • Smooth Processes with a Rational Quadratic Kernel
    • Weakly Periodic Processes
    • Bayesian Linear Regression
• Advanced Usage
  • Use the OILMM Within Your Model
  • Kronecker-Structured Mixing Matrix
• Reproduce Experiments From the Paper

Requirements and Installation

See the instructions here. Then simply

pip install oilmm

TLDR

import numpy as np
from stheno import EQ, GP

# Use TensorFlow as the backend for the OILMM.
import tensorflow as tf
from oilmm.tensorflow import OILMM


def build_latent_processes(ps):
    # Return models for latent processes, which are noise-contaminated GPs.
    return [
        (
            p.variance.positive(1) * GP(EQ().stretch(p.length_scale.positive(1))),
            p.noise.positive(1e-2),
        )
        for p, _ in zip(ps, range(3))
    ]

# Construct model.  
prior = OILMM(tf.float32, build_latent_processes, num_outputs=6)

# Create some sample data.
x = np.linspace(0, 10, 100)
y = prior.sample(x)  # Sample from the prior.

# Fit the model to the data.
prior.fit(x, y, trace=True, jit=True)
prior.vs.print()  # Print all learned parameters.

# Make predictions.
posterior = prior.condition(x, y)  # Construct posterior model.
mean, var = posterior.predict(x)  # Predict with the posterior model.
lower = mean - 1.96 * np.sqrt(var)
upper = mean + 1.96 * np.sqrt(var)

Minimisation of "negative_log_marginal_likelihood":
    Iteration 1/1000:
        Time elapsed: 0.9 s
        Time left:  855.4 s
        Objective value: -0.1574
    Iteration 105/1000:
        Time elapsed: 1.0 s
        Time left:  15.5 s
        Objective value: -0.5402
    Done!
Termination message:
    CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH
latent_processes.processes[0].variance: 1.829
latent_processes.processes[0].length_scale: 1.078
latent_processes.processes[0].noise: 9.979e-03
latent_processes.processes[1].variance: 1.276
latent_processes.processes[1].length_scale: 0.9262
latent_processes.processes[1].noise: 0.03924
latent_processes.processes[2].variance: 1.497
latent_processes.processes[2].length_scale: 1.092
latent_processes.processes[2].noise: 0.04833
mixing_matrix.u:
    (6x3 array of data type float32)
    [[ 0.543 -0.237 -0.111]
     [ 0.578 -0.185 -0.357]
     [-0.204 -0.094 -0.567]
     [-0.554 -0.413 -0.081]
     [-0.12   0.571 -0.66 ]
     [-0.089 -0.636 -0.31 ]]
noise:      0.02245

Basic Usage

Examples of Latent Process Models

Smooth Processes

from stheno import GP, EQ

def build_latent_processes(ps):
    return [
        (
            p.variance.positive(1) * GP(EQ().stretch(p.length_scale.positive(1))),
            p.noise.positive(1e-2),
        )
        for p, _ in zip(ps, range(3))
    ]

Smooth Processes With A Rational Quadratic Kernel

from stheno import GP, RQ

def build_latent_processes(ps):
    return [
        (
            p.variance.positive(1)
            * GP(RQ(p.alpha.positive(1e-2)).stretch(p.length_scale.positive(1))),
            p.noise.positive(1e-2),
        )
        for p, _ in zip(ps, range(3))
    ]

Weakly Periodic Processes

from stheno import GP, EQ

def build_latent_processes(ps):
    return [
        (
            p.variance.positive(1)
            * GP(
                # Periodic component:
                EQ()
                .stretch(p.periodic.length_scale.positive(0.7))
                .periodic(p.periodic.period.positive(24))
                # Make the periodic component slowly change over time:
                * EQ().stretch(p.periodic.decay.positive(72))
            ),
            p.noise.positive(1e-2),
        )
        for p, _ in zip(ps, range(3))
    ]

Bayesian Linear Regression

from stheno import GP, Linear

num_features = 10


def build_latent_processes(ps):
    return [
        (
            GP(Linear().stretch(p.length_scales.positive(1, shape=(num_features,)))),
            p.noise.positive(1e-2),
        )
        for p, _ in zip(ps, range(3))
    ]

Advanced Usage

Use the OILMM Within Your Model

Kronecker-Structured Mixing Matrix

from matrix import Kronecker

p_left, m_left = 10, 3  # Shape of left factor in Kronecker product
p_right, m_right = 5, 2  # Shape of right factor in Kronecker product


def build_mixing_matrix(ps, p, m):
    return Kronecker(
        ps.left.orthogonal(shape=(p_left, m_left)),
        ps.right.orthogonal(shape=(p_right, m_right)),
    )


prior = OILMM(
    dtype,
    latent_processes=build_latent_processes,
    mixing_matrix=build_mixing_matrix,
    num_outputs=p_left * p_right
)

Reproduce Experiments From the Paper

TODO: Install requirements.

Scripts to rerun individual experiments from the paper can be found in the experiments folder. A shell script is provided to rerun all experiments from the paper at once:

sh run_experiments.sh

The results can then be found in the generated _experiments folder.