Raymundo Navarrete

Raymundo Navarrete, Divakar Viswanath

This article considers the numerical inversion of the power series $p(x)=1+b_{1}x+b_{2}x^{2}+\cdots$ to compute the inverse series $q(x)$ satisfying $p(x)q(x)=1$. Numerical inversion is a special case of triangular back-substitution, which has been known for its beguiling numerical stability since the classic work of Wilkinson (1961). We prove the numerical stability of inversion of power series and obtain bounds on numerical error. A range of examples show these bounds to be quite good. When $p(x)$ is a polynomial and $x=a$ is a root with $p(a)=0$, we show that root deflation via the simple division $p(x)/(x-a)$ can trigger instabilities relevant to polynomial root finding and computation of finite-difference weights. When $p(x)$ is a polynomial, the accuracy of the computed inverse $q(x)$ is connected to the pseudozeros of $p(x)$.

Jacob Miller, Vahan Huroyan, Raymundo Navarrete et al.

When visualizing a high-dimensional dataset, dimension reduction techniques are commonly employed which provide a single 2-dimensional view of the data. We describe ENS-t-SNE: an algorithm for Embedding Neighborhoods Simultaneously that generalizes the t-Stochastic Neighborhood Embedding approach. By using different viewpoints in ENS-t-SNE's 3D embedding, one can visualize different types of clusters within the same high-dimensional dataset. This enables the viewer to see and keep track of the different types of clusters, which is harder to do when providing multiple 2D embeddings, where corresponding points cannot be easily identified. We illustrate the utility of ENS-t-SNE with real-world applications and provide an extensive quantitative evaluation with datasets of different types and sizes.

Md Iqbal Hossain, Vahan Huroyan, Stephen Kobourov et al.

We describe MPSE: a Multi-Perspective Simultaneous Embedding method for visualizing high-dimensional data, based on multiple pairwise distances between the data points. Specifically, MPSE computes positions for the points in 3D and provides different views into the data by means of 2D projections (planes) that preserve each of the given distance matrices. We consider two versions of the problem: fixed projections and variable projections. MPSE with fixed projections takes as input a set of pairwise distance matrices defined on the data points, along with the same number of projections and embeds the points in 3D so that the pairwise distances are preserved in the given projections. MPSE with variable projections takes as input a set of pairwise distance matrices and embeds the points in 3D while also computing the appropriate projections that preserve the pairwise distances. The proposed approach can be useful in multiple scenarios: from creating simultaneous embedding of multiple graphs on the same set of vertices, to reconstructing a 3D object from multiple 2D snapshots, to analyzing data from multiple points of view. We provide a functional prototype of MPSE that is based on an adaptive and stochastic generalization of multi-dimensional scaling to multiple distances and multiple variable projections. We provide an extensive quantitative evaluation with datasets of different sizes and using different number of projections, as well as several examples that illustrate the quality of the resulting solutions.

Raymundo Navarrete, Divakar Viswanath

Prediction of dynamical time series with additive noise using support vector machines or kernel based regression has been proved to be consistent for certain classes of discrete dynamical systems. Consistency implies that these methods are effective at computing the expected value of a point at a future time given the present coordinates. However, the present coordinates themselves are noisy, and therefore, these methods are not necessarily effective at removing noise. In this article, we consider denoising and prediction as separate problems for flows, as opposed to discrete time dynamical systems, and show that the use of smooth splines is more effective at removing noise. Combination of smooth splines and kernel based regression yields predictors that are more accurate on benchmarks typically by a factor of 2 or more. We prove that kernel based regression in combination with smooth splines converges to the exact predictor for time series extracted from any compact invariant set of any sufficiently smooth flow. As a consequence of convergence, one can find examples where the combination of kernel based regression with smooth splines is superior by even a factor of $100$. The predictors that we compute operate on delay coordinate data and not the full state vector, which is typically not observable.