Data-Discriminants of Likelihood Equations

Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. The problem is to maximize the likelihood function with respect to given data on a statistical model. An algebraic approach to this problem is to solve a very structured parameterized polynomial system called likelihood equations. For general choices of data, the number of complex solutions to the likelihood equations is finite and called the ML-degree of the model. The only solutions to the likelihood equations that are statistically meaningful are the real/positive solutions. However, the number of real/positive solutions is not characterized by the ML-degree. We use discriminants to classify data according to the number of real/positive solutions of the likelihood equations. We call these discriminants data-discriminants (DD). We develop a probabilistic algorithm for computing DDs. Experimental results show that, for the benchmarks we have tried, the probabilistic algorithm is more efficient than the standard elimination algorithm. Based on the computational results, we discuss the real root classification problem for the 3 by 3 symmetric matrix~model.Comment: 2 table

Shape Generation using Spatially Partitioned Point Clouds

We propose a method to generate 3D shapes using point clouds. Given a point-cloud representation of a 3D shape, our method builds a kd-tree to spatially partition the points. This orders them consistently across all shapes, resulting in reasonably good correspondences across all shapes. We then use PCA analysis to derive a linear shape basis across the spatially partitioned points, and optimize the point ordering by iteratively minimizing the PCA reconstruction error. Even with the spatial sorting, the point clouds are inherently noisy and the resulting distribution over the shape coefficients can be highly multi-modal. We propose to use the expressive power of neural networks to learn a distribution over the shape coefficients in a generative-adversarial framework. Compared to 3D shape generative models trained on voxel-representations, our point-based method is considerably more light-weight and scalable, with little loss of quality. It also outperforms simpler linear factor models such as Probabilistic PCA, both qualitatively and quantitatively, on a number of categories from the ShapeNet dataset. Furthermore, our method can easily incorporate other point attributes such as normal and color information, an additional advantage over voxel-based representations.Comment: To appear at BMVC 201

Digitally interpreting traditional folk crafts

The cultural heritage preservation requires that objects persist throughout time to continue to communicate an intended meaning. The necessity of computer-based preservation and interpretation of traditional folk crafts is validated by the decreasing number of masters, fading technologies, and crafts losing economic ground. We present a long-term applied research project on the development of a mathematical basis, software tools, and technology for application of desktop or personal fabrication using compact, cheap, and environmentally friendly fabrication devices, including '3D printers', in traditional crafts. We illustrate the properties of this new modeling and fabrication system using several case studies involving the digital capture of traditional objects and craft patterns, which we also reuse in modern designs. The test application areas for the development are traditional crafts from different cultural backgrounds, namely Japanese lacquer ware and Norwegian carvings. Our project includes modeling existing artifacts, Web presentations of the models, automation of the models fabrication, and the experimental manufacturing of new designs and forms