Project Proposal (PDF) - Oxford Brookes University

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FP7-ICT-2011-9 STREP proposal 18/01/12 v1 [Dynact] Based on the general ideas and methods developed by the SYSTeMS-IDSIA teams and described recently in [62,63], we will derive formulae for the probabilities of such sequences in imprecise HMMs that lend themselves to backwards recursion, and therefore to generalisations of the backward algorithms to precise HMMs, essentially linear in the length of the Markov chain. SubTask 1.2.2. - Computation of lower and upper likelihoods These are the lower and upper probabilities of the observation sequences. Backwards recursion calculations for these entities should also be possible, based on the seminal work in [62,63]. SubTask 1.2.3. - Most probable explanation The "most probable explanation" for precise HMMs consists on finding the state sequence (explanation) with the highest probability, conditional on the observation sequence. In an imprecise probabilities context, there are basically two ways of formulating this problem [94]. The "maximin" method is a pessimistic approach which seeks the state sequence with the highest lower probability, conditional on the observations. In the "maximality" method, one state sequence is considered to be a better explanation than another if it has a higher posterior probability in all the precise models that are consistent with a given imprecise model. Preliminary analyses [64] hint that the latter method is most likely to be able to be implemented efficiently. Task 1.3 - Imprecise EM for hidden Markov models Task 1.2 is about making inference on imprecise HMMs, however learnt. Yet, learning imprecise HMMs from video sequences is not trivial, as the number of operation to be performed for each iteration of the algorithm can grow exponentially with the input size. This task involves: SubTask 1.3.1 - Specialisation of the EM algorithms to the HMM topology The imprecise EM developed in Task 1.1 has to be first specialized to the special case of HMMs, based on the inference algorithms developed in Task 1.2. SubTask 1.3.2 - Implementation of recursive formulae for parameter updating Recursive formulae describing the revision of model parameters have to be determined. SubTask 1.3.3 - Convergence and initialisation: empirical study The convergence properties of the EM algorithm, specialised to the HMM topology, have to be assessed. Its robustness and the quality of the estimates with respect to parameter initialisation (a crucial issue even in the precise case) needs to be considered as well. Task 1.4 - Imprecise Dynamical Graphical Models HMMs are an example of probabilistic graphical models with a dynamic component, in a way the simplest example of dynamic Bayesian networks. A general theory for dynamic BNs is well established, but a similar comprehensive framework is still missing for credal networks, the imprecise-probabilistic generalisations of Bayesian networks. Contributing to the development of such a general theory is the final aim of this work package, paying special attention to higher-order HMM topologies able to provide a better model of the correlation between the frames of a video sequence depicting an action. We also intend to explore the possibility of modelling these relations by means of undirected graphs, providing a generalisation of random Markov fields (which are already widely used in computer vision) to the imprecise probabilistic framework. SubTask 1.4.1 - Dynamical (and object-oriented) credal networks A crucial step consist on investigating whether the ideas pioneered in [62,63] for inference in credal trees can be extended to more general types of probabilistic graphical models. Methods will then have to be devised for (recursively) constructing joint models from the local imprecise-probabilistic models attached to the nodes of such credal networks. SubTask 1.4.2 - Inference algorithms and complexity results for dynamic credal networks Based on those (recursive) methods for constructing joints, recursive or message-passing algorithms need to be developed for updating these networks based on new observations, computing lower and upper probabilities, likelihoods, and most probable explanation. SubTask 1.4.3 - Imprecise Markov random fields Markov random fields are graphical models based on undirected graphs particularly suited for modelling pixel-to-pixel correlation in image analysis [85]. The extension of our inference techniques to imprecise Markov random fields is potentially of enormous impact and will be pursued towards the end of WP1. WP2 – Classification of generative dynamical models Once represented video sequences as either precise or imprecise-probabilistic graphical models (for instance of the class of imprecise hidden Markov models), gesture and action recognition reduce to classifying such generative models. Different competing approaches to this issue can be foreseen. Task 2.1 – Dissimilarity measures for imprecise graphical models SubTask 2.1.1 - Modelling similarity between sets of probability distributions <strong>Proposal</strong> Part B: page [14] of [67]
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 <strong>Proposal</strong> Part B: page [15] of [67]
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Project Proposal (PDF) - Oxford Brookes University

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