Machine Learning has proved its ability to produce accurate models - but the deployment of these models outside the machine learning community has been hindered by the difficulties of interpreting these models.

We will comment on this in the discussion section.

Machine Learning has proved its ability to produce accurate models - but the deployment of these models outside the machine learning community has been hindered by the difficulties of interpreting these models.

Projection Pursuit is a classic exploratory technique for finding interesting projections of a dataset. We propose a method for recovering projections containing either Imbalanced Clusters or a Bernoulli-Rademacher distribution using a gradient-based technique to optimize the projection index. As sample complexity is a major limiting factor in Projection Pursuit, we analyze our algorithm's sample complexity within a Planted Vector setting where we can observe that Imbalanced Clusters can be recovered more easily than balanced ones. Additionally, we give a generalized result that works for a variety of data distributions and projection indices. We compare these results to computational lower bounds in the Low-Degree-Polynomial Framework. Finally, we experimentally evaluate our method's applicability to real-world data using FashionMNIST and the Human Activity Recognition Dataset, where our algorithm outperforms others when only a few samples are available.

After Rebuttal Thank you for the response and the clarifications. We still have some concerns about the complexity of the algorithm and its dependence on the iteration count -- in particular in cases where this may scale with the dimension d based on the specified termination criteria. This being said, it would be great to comment on this in the numerical results and include more timing studies to confirm this dependence. We also strongly suggest that the authors include the extensions listed in the improvement in the final revision. Before Rebuttal The authors present a novel algorithm with both theoretical analysis and empirical results. We have a few comments and suggestions for the work: In the introduction, we recommend that the authors also note alternative versions of finding OTM that are not based on solving a linear program (including non-discrete versions of OT based on solving ODEs or finding parametrized maps).

This paper studies the estimation of large-scale optimal transport maps (OTM), which is a well known challenging problem owing to the curse of dimensionality. Existing literature approximates the large-scale OTM by a series of one-dimensional OTM problems through iterative random projection. Such methods, however, suffer from slow or none convergence in practice due to the nature of randomly selected projection directions. Instead, we propose an estimation method of large-scale OTM by combining the idea of projection pursuit regression and sufficient dimension reduction. The proposed method, named projection pursuit Monge map (PPMM), adaptively selects the most informative'' projection direction in each iteration.

In the Naive Bayes classification model the class conditional densities are estimated as the products of their marginal densities along the cardinal basis directions. We study the problem of obtaining an alternative basis for this factorisation with the objective of enhancing the discriminatory power of the associated classification model. We formulate the problem as a projection pursuit to find the optimal linear projection on which to perform classification. Optimality is determined based on the multinomial likelihood within which probabilities are estimated using the Naive Bayes factorisation of the projected data. Projection pursuit offers the added benefits of dimension reduction and visualisation. We discuss an intuitive connection with class conditional independent components analysis, and show how this is realised visually in practical applications. The performance of the resulting classification models is investigated using a large collection of (162) publicly available benchmark data sets and in comparison with relevant alternatives. We find that the proposed approach substantially outperforms other popular probabilistic discriminant analysis models and is highly competitive with Support Vector Machines.

Two projection based feedforward network learning methods for model(cid:173) free regression problems are studied and compared in this paper: one is the popular back-propagation learning (BPL); the other is the projection pursuit learning (PPL). In terms of learning efficiency, both methods have comparable training speed when based on a Gauss(cid:173) Newton optimization algorithm while the PPL is more parsimonious. In terms of learning robustness toward noise outliers, the BPL is more sensi(cid:173) tive to the outliers.

High dimensional data modeling is difficult mainly because the so-called "curse of dimensionality". We propose a technique called "Gaussianiza(cid:173) tion" for high dimensional density estimation, which alleviates the curse of dimensionality by exploiting the independence structures in the data. Gaussianization is motivated from recent developments in the statistics literature: projection pursuit, independent component analysis and Gaus(cid:173) sian mixture models with semi-tied covariances. We propose an iter(cid:173) ative Gaussianization procedure which converges weakly: at each it(cid:173) eration, the data is first transformed to the least dependent coordinates and then each coordinate is marginally Gaussianized by univariate tech(cid:173) niques. Gaussianization offers density estimation sharper than traditional kernel methods and radial basis function methods.