Project Proposal (PDF) - Oxford Brookes University

FP7-ICT-2011-9 STREP proposal
18/01/12 v1 [Dynact]
A first sensible option is to consider imprecise graphical models, and iHMMs in particular, as convex sets of
probability distributions themselves. A distance/dissimilarity between two imprecise models is then just a
special case of distance between two sets of distributions. We can then compute the latter as the vector of the
distances between all the possible pairs of distributions in the two sets, and take the minimum or the
maximum distance, or we can identify a single representative element in the credal set (e.g., the maximum
entropy element or the center of mass), to which distance measures for single distributions can be applied.
These ideas have already been formalised by OBU and IDSIA: yet the evaluation of such distances require to
define an optimisation problem whose efficient solution still needs to be developed.
SubTask 2.1.2 - Specialisation to the class of imprecise hidden Markov models (iHMMs)
The results achieved in the previous task should later be specialised to the sets of (joint) probability
distributions associated with an iHMM. The challenge, here, is to perform the distance evaluation efficiently,
by exploiting the algorithms developed in Task 1.2.
SubTask 2.1.3 - Supervised learning of similarity measures for precise generative models
A second option is to learn in a supervised way the “best” metric for a given training set of models [3, 4, 48].
A natural optimisation criterion consists on maximising the classification performance achieved by the learnt
metric, a problem which has elegant solutions in the case of linear mappings [50, 57]. An interesting tool is
provided by the formalism of pullback metrics: if the models belong to a Riemannian manifold M, any
diffeomorphism of itself (or “automorphism”) induces such a metric on M (see Figure 5). By designing a
suitable parameterized family of automorphisms we obtain a family of pullback metrics we can optimise on.
OBU already has relevant experience in this sense [13]. The second goal of Task 2.1 will be the formulation
of a general manifold learning framework for the classification of generative (precise) models, starting from
hidden Markov models, later to be extended to more powerful classes of dynamical models (such as, for
instance, semi-Markov [51], hierarchical or variable length [26] Markov models) able to describe complex
activities rather than elementary actions.
Figure 5. Once encoded each training video sequence as a graphical model (for instance a hidden Markov
model, left) we obtain a training set of such models in the appropriate model space M (right). Any
automorphism F on M induces a map of tangent vectors on M, which in turn induces a pullback metric.
SubTask 2.1.4 - Supervised learning of similarity measures for imprecise-probabilistic models
Most relevant to the current project is the extension of metric learning approaches to imprecise HMMs first,
and other classes of imprecise graphical models later. We will be able to use the results of SubTasks 2.1.1
and 2.1.2 to define a base metric/similarity for iHMMs, on which to apply our metric optimisation approach.
Task 2.2 - Classification of imprecise graphical models
Task 2.2 is designed to use a-apriori or learnt distance measures (developed in Task 2.1) to classify imprecise
models (2.2.1) and imprecise HMMs in particular (2.2.2), but also to pursue alternative classification
strategies based on kernel SVMs in an approach called “structured learning” (2.2.3), or adopt large scale
classification techniques based on randomised projections onto a reduced, measurement space (2.2.4).
SubTask 2.2.1 - Distance-based clustering and classification of imprecise models
Distance functions between convex sets of probabilities (2.1.1) can be used in distance-based classifiers to
classify arbitrary imprecise models representing credal sets. Such an approach can be extended to more
general classes of imprecise graphical models.
SubTask 2.2.2 - Efficient specialisation to the case of imprecise HMMs
The results of SubTask 2.1.2 can in the same way be used to deliver distance-based classifiers for imprecise
Markov models as a relevant special case, given the importance of HMMs in action recognition. A
computationally efficient specialisation of these techniques to the HMM topology is sought.
SubTask 2.2.3 - Kernel SVM classification of imprecise models: a structured learning approach
Kernel SVM classification is a powerful tool, which allows us to find optimal implicit linear classifiers in
sufficiently large feature spaces without having to compute feature maps explicitly, thanks to the so-called
Proposal Part B: page [15] of [67]