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# Supervised Spectral Latent Variable Models

*Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics*, PMLR 5:33-40, 2009.

#### Abstract

We present a probabilistic structured prediction method for learning input-output dependencies where correlations between outputs are modeled as low-dimensional manifolds constrained by both geometric, distance preserving output relations,and predictive power of inputs. Technically this reduces to learning a probabilistic, input conditional model, over latent (manifold) and output variables using an alternation scheme. In one round, we optimize the parameters of an input-driven manifold predictor using latent targets given by preimages (conditional expectations) of the current manifold-to-output model. In the next round, we use the distribution given by the manifold predictor in order to maximize the probability of the outputs with an additional, implicit distance preserving constraint on the manifold. The resulting Supervised Spectral Latent Variable Model (SSLVM) combines the properties of probabilistic geometric manifold learning (accommodates geometric constraints corresponding to any spectral embedding method including PCA, ISOMAP or Laplacian Eigenmaps), with the additional supervisory information to further constrain it for predictive tasks. We demonstrate the superiority of the method over baseline PPCA + regression frameworks and show its potential in difficult realworld computer vision benchmarks designed for the reconstruction of three-dimensional human poses from monocular image sequences.