[01098] Elucidating theoretical biology and deep learning by algebraic statistics and topology
Session Time & Room : 1E (Aug.21, 17:40-19:20) @E504
Type : Proposal of Minisymposium
Abstract : Nonlinear algebra and topology are gaining popularity as a tool for studying theoretical biology including phylogenetics and mathematical neuroscience. Applying these modern mathematical fields can lead to a breakthrough in the important fields. However, there can be rather a limited access to the practical resources for the sophisticated mathematical methods. Thus, it is important to introduce the modern algebraic and topological methods and exchange their hands-on skills in person. In this minisymposium, each speaker will talk about the combinations of modern mathematical methods with statistical machine learning and their applications.
[04921] Judging unlearnability from structures of deep neural networks for low dimensional inputs
Format : Talk at Waseda University
Author(s) :
Keiji Miura (Kwansei Gakuin University)
Abstract : Zhang, Naitzat and Lim (2019) showed that a feedforward ReLU neural network is equivalent to a tropical rational map. Here we visualize the shapes of deep neural network functions by using the tropical algebra and judge its unlearnability of complicated boundaries. Especially, the limitation can be naturally interpreted by the tropical factorization of polynomials for the cases of one-dimensional input.
[04805] Hit and Run Sampling from the Space of Phylogenetic Trees
Format : Talk at Waseda University
Author(s) :
David Barnhill (Naval Postgraduate School)
Ruriko Yoshida (Naval Postgraduate School)
Keiji Miura (Kwansei Gakuin University)
Abstract : In this presentation we introduce a Markov Chain Monte Carlo (MCMC) Hit and Run (HAR) uniform sampler over a tropically convex space of ultrametrics. This is particularly important because by sampling from the space of ultrametrics, we are sampling from the space of phylogenetic trees, or tree space. This has wide ranging implications to statistical inference relating to drawing inference about the tree space. Specifically, we show how this HAR sampler can be employed to sample over the space of ultrametrics in order to non-parametrically estimate the phylogenetic tree distribution using what we call tropical density estimator (TDE) with the tropical metric. We compare the results of the TDE using the tropical metric against often used density estimation methods using the Billera-Holmes-Vogtman metric to show that TDE is more accurate and computationally less expensive.
[04960] Approximate Computation of Vanishing Ideals
Format : Talk at Waseda University
Author(s) :
Hiroshi Kera (Chiba University)
Abstract : The vanishing ideal of points is the set of all polynomials that vanish over the points. The approximate computation of generators has been developed at the intersection of computer algebra and machine learning in the last decade. Computer-algebraic algorithms have a rich theoretical background, whereas machine learning-oriented algorithms are designed for applications such as classification at the cost of some theoretical properties. This talk reviews the development of approximate computation of vanishing ideals.
