arXiv subjects

Raymundo Navarrete

Publications and source records attributed to Raymundo Navarrete.

6 recordsLinked to original sources

arXiv2024

ENS-t-SNE: Embedding Neighborhoods Simultaneously t-SNE

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.

cs.LG↗

arXiv2020

Multi-Perspective, Simultaneous Embedding

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.

cs.DS↗

arXiv2018

Prediction of Dynamical time Series Using Kernel Based Regression and Smooth Splines

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.

stat.ML↗

arXiv2018

Delay Embedding of Periodic Orbits Using a Fixed Observation Function

Delay coordinates are a widely used technique to pass from observations of a dynamical system to a representation of the dynamical system as an embedding in Euclidean space. Current proofs show that delay coordinates of a given dynamical system result in embeddings generically over a space of observations (Sauer, Yorke, Casdagli, J. Stat. Phys., vol. 65 (1991), p. 579-616). Motivated by applications of the embedding theory, we consider the situation where the observation function is fixed. For example, the observation function may simply be some fixed coordinate of the state vector. For a fixed observation function (any nonzero linear combination of coordinates) and for the special case of periodic solutions, we prove that delay coordinates result in an embedding generically over the space of flows in the $C^{r}$ topology with $r\geq2$.

math.DS↗

arXiv2018

Prevalence of Delay Embeddings with a Fixed Observation Function

Let $x_{j+1}=ϕ(x_{j})$, $x_{j}\in\mathbb{R}^{d}$, be a dynamical system with $ϕ$ being a diffeomorphism. Although the state vector $x_{j}$ is often unobservable, the dynamics can be recovered from the delay vector $\left(o(x_{1}),\ldots,o(x_{D})\right)$, where $o$ is the scalar-valued observation function and $D$ is the embedding dimension. The delay map is an embedding for generic $o$, and more strongly, the embedding property is prevalent. We consider the situation where the observation function is fixed at $o=π_{1}$, with $π_{1}$ being the projection to the first coordinate. However, we allow polynomial perturbations to be applied directly to the diffeomorphism $ϕ$, thus mimicking the way dynamical systems are parametrized. We prove that the delay map is an embedding with probability one with respect to the perturbations. Our proof introduces a new technique for proving prevalence using the concept of Lebesgue points.

math.DS↗

arXiv2014

Accuracy and stability of inversion of power series

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 $p(x)$.

math.NA↗