em-transfer-learning 1.0.0

Supervised linear transfer learning based on labelled Gaussian mixture models and expectation maximization in scikit-learn-compatible form.

Linear Supervised Transfer Learning

Copyright (C) 2019 - Benjamin Paassen
Machine Learning Research Group
Center of Excellence Cognitive Interaction Technology (CITEC)
Bielefeld University

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program; if not, see http://www.gnu.org/licenses/.

Introduction

This Python3 library provides several algorithms to learn a linear mapping from an $m$-dimensional source space to an $n$-dimensional target space, such that a classification model trained in the source space becomes applicable in the target space. The source space model is assumed to be a labelled mixture of Gaussians. Note that this library assumes that the relation between the source and target space is (approximately) linear and will necessarily fail if the relationship is highly nonlinear. Further note that this library requires a few labelled target space data points to work (typically, even ~10 data points are enough). However, not all classes are required, since the learned linear transformation generalizes across classes.

If you intend to use this library in academic work, please cite our paper.

Installation

This package is available on pypi as em_transfer_learning. You can install it via

pip install --user em_transfer_learning

QuickStart Guide

For a quick start we recommend to take a look at the demo in the notebook demo.ipynb. In this file we demonstrate how to perform transfer learning on example data. For the actual transfer learning, we recommend to initialize one of the following models, depending on your source space model:

1. em_transfer_learning.transfer_learning.LGMM_transfer_model : If you have a full labelled Gaussian mixture model.
2. em_transfer_learning.transfer_learning.SLGMM_transfer_model : If you have a labelled Gaussian mixture model with shared precision matrices.
3. em_transfer_learning.transfer_learning.Local_LVQ_transfer_model : If you have a learning vector quantization model with individual metric learning matrices.
4. em_transfer_learning.transfer_learning.LVQ_transfer_model : If you have a learning vector quantization model with shared metric learning matrix or no metric learning at all.

Note that models 2 and 4 are much faster to train compared to models 1 and 3 (refer to the next section for more information on that).

All these models follow the scikit-learn convention, i.e. you need to call the fit function with target space data first and then the predict function to map new target space data to the source space according to the learned mapping.

Background

The basic idea of our transfer learning approach is to maximize the likelihood of target space data according to the source space data distribution after the learned transfer function $h$ has been applied. More precisely, assume we have a data set $(\vec x_1, y_1), \ldots, (\vec x_m, y_m)$ of target data points $\vec x_j \in \mathbb{R}^n$ and their labels $y_j \in \{1, \ldots, L\}$. Then, we wish to maximize the joint probability

\max_h \prod_{j=1}^m p\Big(h(\vec x_j), y_j\Big)

To make this optimization problem feasible, we introduce two assumptions: First, that $p(x, y)$ can be modelled by a labelled Gaussian mixture model (lGMM) and, second, that $h$ can be approximated by a linear function. In more detail, that means the following.

Labelled Gaussian Mixture Models

A labelled Gaussian mixture model assumes that data is generated by a mixture of $K$ Gaussians, each of which has a prior $P(k)$, a data generating Gaussian density $p(\vec x|k)$, and a label generating distribution $P(y|k)$. Using these distributions, we can derive the joint probability density $p(\vec x, y)$ as follows.

p(\vec x, y) = \sum_{k=1}^K p(\vec x, y, k) = \sum_{k=1}^K p(\vec x, y|k) \cdot P(k)

Our model assumes that $\vec x$ and $y$ are conditionally independent given the component index $k$, such that we can re-write:

p(\vec x, y) = \sum_{k=1}^K p(\vec x|k) \cdot P(y|k) \cdot P(k)

Note that $p(\vec x|k)$ is a multivariate Gaussian probability density with parameters for the mean $\vec \mu_k$ and the precision matrix $\Lambda_k$. Also note that this model is a proper generalization over standard Gaussian mixture models and that many of the GMM properties translate directly to lGMMs. More precisely, we obtain a standard GMM by setting the label distribution $P(y|k)$ to a uniform distribution and leaving it unchanged during training. Alternatively, we also obtain a standard GMM by assigning the same label to all data points.

Also note that lGMMs generalize over learning vector quantization models if we apply a scaling trick to the precision matrices (for more details on this, refer to our paper).

Expectation Maximization transfer learning

Our assumption that the transfer function $h$ is approximately linear implies that $h$ can be re-written as $h(\vec x) \approx H \cdot \vec x$ for some matrix $H$. Thus, our transfer learning problem becomes:

\max_H \prod_{j=1}^m \sum_{k=1}^K p(H \cdot \vec x|k) \cdot P(y|k) \cdot P(k)

Due to the product of sums, a direct optimization of this expression is infeasible. However, we can apply an expectation maximization scheme. In particular, we initialize $H$ with the identity matrix (padded with zeros wherever necessary) and then iteratively perform the following two steps:

1. Expectation: We compute the posterior $p(k|H \cdot \vec x_j, y_j)$ for the current transfer matrix $H$, all data points $j$ and all Gaussian components $k$, yielding a matrix $\Gamma \in \mathbb{R}^{K \times m}$ with entries $\gamma_{k,j} = p(k|H \cdot \vec x_j, y_j)$. The full expression for the posterior is given in our paper.

2. Maximization: We maximize the expected log likelihood under fixed posterior, i.e.

   \max_H \sum_{j=1}^m \sum_{k=1}^K \gamma_{k, j} \cdot \log\big[p(H \cdot \vec x_j, y_j| k)\big]
   

   This optimization problem can be shown to be convex und thus lends itself for optimization techniques like l-BFGS. Even better, if the precision matrix $\Lambda_k$ is shared across all Gaussians $k$, the problem has a closed-form solution, namely

   H = W \cdot \Gamma \cdot X^T \cdot (X \cdot X^T + \lambda \cdot I)^{-1}
   

   where $W = (\vec \mu_1, \ldots, \vec \mu_K)$, $X = (\vec x_1, \ldots, \vec x_m)$, $\lambda$ is a (small) regularization constant, and $I$ is the identity matrix. Due to this closed form solution, the SLGMM_transfer_model and the LVQ_model are much faster to train compared to the LGMM_transfer_model and the Local_LVQ_transfer_model.

For more detailed background, please refer to our paper.

Contents

This library contains the following files.

• demo.ipynb : A demo script illustrating how to use this library.
• LICENSE : A copy of the GPLv3 license.
• em_transfer_learning/lgmm.py : A file to train labelled Gaussian mixture models with or without shared precision matrices.
• em_transfer_learning/transfer_learning.py : The actual transfer learning models.
• lgmm_test.py : A set of unit tests for lgmm.py.
• README.md : This file.
• transfer_learning_test.py : A set of unit tests for transfer_learning.py.

Licensing

This library is licensed under the GNU General Public License Version 3.

Dependencies

This library depends on NumPy for matrix operations, on scikit-learn for the base interfaces and on SciPy for optimization.

Literature

• Paassen, B., Schulz, A., Hahne, J., and Hammer, B (2018). Expectation maximization transfer learning and its application for bionic hand prostheses. Neurocomputing, 298, 122-133. doi:10.1016/j.neucom.2017.11.072. Link