Probabilistic Principal Geodesic Analysis
Principal geodesic analysis (PGA) is a generalization of principal component analysis (PCA) for dimensionality reduction of data on a Riemannian manifold. Currently PGA is defined as a geometric fit to the data, rather than as a probabilistic model. Inspired by probabilistic PCA, we present a latent variable model for PGA that provides a probabilistic framework for factor analysis on manifolds. To compute maximum likelihood estimates of the parameters in our model, we develop a Monte Carlo Expectation Maximization algorithm, where the expectation is approximated by Hamiltonian Monte Carlo sampling of the latent variables. We demonstrate the ability of our method to recover the ground truth parameters in simulated sphere data, as well as its effectiveness in analyzing shape variability of a corpus callosum data set from human brain images.
The paper "Probabilistic Principal Geodesic Analysis" 
      introduces and discusses a probabilistic formulation of Principal 
      Geodesic Analysis. Principal Geodesic Analysis has been introduced 
      before as a generalization of standard PCA to non-linear manifolds, 
      i.e., to spaces with (potentially) other intrinsic geometries than 
      Euclidean spaces. The authors explicitly point out and discuss earlier 
      PGA results. It is argued that the contribution of the paper is a first 
      fully probabilistic formulation of PGA and accompanying solutions: In 
      analogy to probabilistic PCA, probabilistic PGA (PPGA) formulates the 
      problem as a probabilistic generative model. In this way, the solution 
      can be interpreted as maximum likelihood optimization. The authors 
      provide an approximate gradient optimization to find the optimal 
      parameters of the PPGA generative model. As a proof of concept, the 
      numerical results for (A) artificial sphere data and (B) for shape 
      estimation with corpus callosum data are provided. Evaluation: 
      The paper contributes to a very important research direction: 
      statistics for data within non-Euclidean spaces. A generative 
      formulation of PPCA within a non-Euclidean space is certainly 
      interesting and potentially of high relevance. I also agree with the 
      authors that formulations of latent variable models for non-Euclidean 
      spaces is an interesting future research subject. The 
      generative formulation of PPGA also makes sense and builds up on previous 
      work on PGA. The problems I have with the paper are (A) with the 
      formulation of the inference and learning scheme, and (B) with the 
      experiments. Formulation of inference and learning: The paper 
      first makes the impression of providing solutions for general PPGA. 
      While this may be the case for the probabilistic formulation of PPGA 
      generative model, the main contribution is rather the inference and 
      learning procedure. The latter is, however, only really provided for 
      symmetric spaces. This should be very clearly spelled out, in the 
      abstract and maybe even in the title. The inference and learning 
      procedure, and the experiments, do rely on symmetric spaces. In this 
      respect, please check formula (4). The text says the rhs is 
      proportional to the posterior probability but then the normalizer of 
      equation (2) seems to be missing. The normalizer would be relevant for 
      derivatives w.r.t. \mu and \tau it seems. The independence of \mu 
      already seems to be used for the gradient w.r.t. \mu but is only true 
      for symmetric spaces. The paper at least gives the impression that at 
      this point symmetric spaces are not assumed. For the \tau derivation, 
      symmetric spaces are explicitly assumed. A lot of more general 
      considerations like the locality of log map etc could presumably be 
      simplified if the paper focussed on symmetric spaces right from the 
      start. Experiments: The experiments should be clearly labelled 
      as proof of concept. They basically say that the algorithm seems to 
      work in principle. There is no comparison with other methods. It would 
      have been interesting to compare with non-probabilistic PGA. As the 
      methods are essentially equivalent for finding the principle geodesic 
      directions, results should be similar. PPGA can potentially provide 
      more information (other parameters, maybe a likelihood estimate) but 
      non of this is shown or discussed. Also potential local optima are not 
      discussed and other issues about the stability and complexity of the 
      algorithm. More generally: Work out and discuss in what 
      situations the approach really has a (potential) advantage over least 
      square formulations of PGA, or over projection methods such as azimuthal 
      equidistant projections. When does the generative formulation really 
      pay of? I think it can. Other points: I would cite 
      Roweis for PPGA alongside the other citations Optimization of the 
      PPCA likelihood is convex, can anything be said about PPGA 
      optimization (maybe for specific manifold properties). "requires 
      we compute", "evaulation", "coule", "the as points" After 
      author response: The author response has clarified major points. I 
      am also confident that the promised changes and discussions will 
      improve the manuscript. Especially an earlier mentioning of the 
      constraint (of the algorithm) to symmetric spaces will make this a clearer 
      paper. I suggested an explicit mentioning in the abstract because this 
      is an important point for the paper. The authors promise to point out 
      the constraint in the introduction, I still think it should be pointed 
      out in the abstract; why not? Symmetric spaces are still large and 
      interesting and potential later confusions can be removed. Eqn. 4 was 
      not complete and will be corrected - which will avoid related 
      confusions. In summary, yes, interesting paper, accept.