Multiview LL-LVM

This repository contains MATLAB code that implements the Multiview Locally Linear Latent Variable Model (MLL-LVM). MLL-LVM is a fully Bayesian model for manifold learning solved with Variational Inference. Observations are assumed to consist of multiple views or modes, i.e. different types of observations. In this sense the model is 'multiview'. Each view corresponds to a different mode of observation independent of the other views. All views, albeit conditionally independent, share a single set of embedding coordinates.

After learning the model with Variational Inference, one view can be inferred based on knowledge of the other views.

Graphical Model

The training set is assumed to contain N observations, each consisting of V views. Model parameters are either latent/probabilistic or deterministic/non-probabilistic.

• Latent variables
  • C : Local manifold tangents. A prior constrains tangents of neighbouring data to be close to one another.
  • x : Embedding coordinates. A prior constrains embeddings of neighbouring data to be close to one another.
  • y : Observations.
• Deterministic parameters
  • G : Data neighbourhood graph structure.
  • γ : Controls the prior on C.
  • α : Controls the prior on x.

Testing the model

In order to run the tests, change into the folder code/scripts/.

Learning the Swiss roll

The first two tests are run on the swiss-roll dataset. Run the swiss-roll scripts with:

• swissroll_demo Tests the single-view LL-LVM code base on Swiss roll (see comments on LL-LVM at the end of this readme).
• swissroll_demo_multi_splitintotwoviews Tests on Swiss roll data after splitting it into V=2 views: The dimension for the one view is 2, and for the other view it is 1. The resulting embedding coordinates are similar to the single-view case.

Estimation of missing views

For this test we shall assume a set of 3d points sampled off a helix, plus noise:

Info about the points is given as two views. View 1 comprises of one of the point variates, leaving the other two variates for View 2.

Given a number of points for which only view 2 is known, we aim to estimate view 1. The related test script can be run with:

• estimation_demo

Paper

The related paper has been presented in the 3^rd workshop on Bayesian and Graphical Models in Biomedical Imaging (BAMBI), held in conjunction with the 19^th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI).

The paper won the 'best paper award' of the workshop.

Please cite this work as:

@InProceedings{Sfikas16Bayesian,
  author       = "G. Sfikas and C. Nikou",
  title        = "Bayesian Multiview Manifold Learning applied to Hippocampus Shape and Clinical Score Data",
  booktitle    = "Proceedings of the $3^{rd}$ International Workshop on Bayesian and Graphical Models in Biomedical Imaging, held in conjunction with MICCAI'16",
  year         = "2016"
}

You can download the paper at http://www.cs.uoi.gr/~sfikas/16Sfikas_MLL-LVM.pdf .

Single-view model (LL-LVM)

The current model extends on the single-view locally-linear latent variable model LL-LVM presented in NIPS 2015. The Matlab code for this work can be found here.

In a nutshell, the current MLL-LVM extends on LL-LVM in these aspects:

• Data points can be seen as sets of views. Likewise, LL-LVM can be considered a special case of MLL-LVM for number of views V=1.
• Missing views can be inferred given observed views with a mechanism that derives from the model.