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The Joint Manifold Model for Semi-supervised Multi-valued RegressionRamanan Navaratnam 1 Andrew W. Fitzgibbon 2 Roberto Cipolla 11 University of CambridgeTrumpington St, Cambridge, UKhttp://mi.eng.cam.ac.uk/{ ˜ rn246, ˜ cipolla}2Microsoft Research, Cambridge7 JJ Thomson Ave, Cambridge, UKhttp://www.research.microsoft.com/ ˜ awfAbstractMany computer vision tasks may be expressed as theproblem of learning a mapping between image space anda parameter space. For example, in human body pose es-timation, recent research has directly modelled the map-ping from image features (z) to joint angles (θ). Fittingsuch models requires training data in the form of labelled(z,θ) pairs, from which are learned the conditional den-sities p(θ|z). Inference is then simple: given test imagefeatures z, the conditional p(θ|z) is immediately computed.However large amounts of training data are required to fitthe models, particularly in the case where the spaces arehigh dimensional.We show how the use of unlabelled data—samples fromthe marginal distributions p(z) and p(θ)—may be used toimprove fitting. This is valuable because it is often signif-icantly easier to obtain unlabelled than labelled samples.We use a Gaussian process latent variable model to learnthe mapping from a shared latent low-dimensional manifoldto the feature and parameter spaces. This extends existingapproaches to (a) use unlabelled data, and (b) representone-to-many mappings.Experiments on synthetic and real problems demonstratehow the use of unlabelled data improves over existing tech-niques. In our comparisons, we include existing approachesthat are explicitly semi-supervised as well as those whichimplicitly make use of unlabelled examples.1. IntroductionMany computer vision algorithms can be viewed as thedesign of a function which takes images as inputs and re-turns parameters of the imaged scene. In human body poseestimation, for example, the input to the function is a vec-tor of image features, and the desired output is a probabilitydensity over the pose parameters of the human in the image.Much recent research [1, 6, 7, 14, 16, 19, 20] has adoptedthis “vision as regression” paradigm: given training exam-ples comprising corresponding pairs of image features (z)and joint angles (θ), learn a function θ = f(z). This is anattractive paradigm because it promises fast and determinis-tic inference, loading most of the computational effort intothe learning or regression phase.This is a rather bare characterization of the paradigm,however, with a number of difficulties which are immedi-ately apparent. First, in most cases of interest, the mappingbetween the pose space and image space can be many-to-many, so that f must be a one-to-many mapping. This isresolved by learning instead the conditional density, so thatgiven an observed image with features z, one can obtain adistribution over the pose, namely p(θ|z).The second difficulty, which we address in this paper,is that the dimensionality of the feature and pose spaces istypically relatively high, meaning that learning p(θ|z) re-quires a considerable amount of labelled examples, i.e. cor-responding (θ,z) pairs. We show in this paper how to makeuse of unlabelled data to improve the estimate of the map-ping. Unlabelled data are sets of pose parameters withoutcorresponding images, as might be found in a motion cap-ture database; or image features obtained from generic im-ages of humans in motion.This significantly reduces the number of training exam-ples needed to learn complex mappings, meaning that ap-plications which would previously have required too muchlabelled training data to be feasible are now possible.1.1. BackgroundWe combine a number of recent research results in orderto achieve this. First we observe that most papers adoptingthe regression approach already do make use of unlabelleddata, even though they may not mention semi-supervisedlearning. This is because many papers begin by projectingraw image features and pose parameters into a lower di-mensional space before learning the mapping. Early workused principal components analysis [1], while more re-1