Felipe Tobar
now a Research Fellow, University of Chile
Publications
Learning Stationary Time Series using Gaussian Process with Nonparametric Kernels
Felipe Tobar, Thang D. Bui, Richard E. Turner, Dec 2015. (In Advances in Neural Information Processing Systems 29). Montréal CANADA.
Abstract▼ URL
We introduce the Gaussian Process Convolution Model (GPCM), a two-stage nonparametric generative procedure to model stationary signals as the convolution between a continuous-time white-noise process and a continuous-time linear filter drawn from Gaussian process. The GPCM is a continuous-time nonparametricwindow moving average process and, conditionally, is itself a Gaussian process with a nonparametric kernel defined in a probabilistic fashion. The generative model can be equivalently considered in the frequency domain, where the power spectral density of the signal is specified using a Gaussian process. One of the main contributions of the paper is to develop a novel variational freeenergy approach based on inter-domain inducing variables that efficiently learns the continuous-time linear filter and infers the driving white-noise process. In turn, this scheme provides closed-form probabilistic estimates of the covariance kernel and the noise-free signal both in denoising and prediction scenarios. Additionally, the variational inference procedure provides closed-form expressions for the approximate posterior of the spectral density given the observed data, leading to new Bayesian nonparametric approaches to spectrum estimation. The proposed GPCM is validated using synthetic and real-world signals.
Unsupervised State-Space Modeling Using Reproducing Kernels
Felipe Tobar, Petar M. Djurić, Danilo P. Mandic, 2015. (IEEE Transactions on Signal Processing).
Abstract▼ URL
A novel framework for the design of state-space models (SSMs) is proposed whereby the state-transition function of the model is parametrized using reproducing kernels. The nature of SSMs requires learning a latent function that resides in the state space and for which input-output sample pairs are not available, thus prohibiting the use of gradient-based supervised kernel learning. To this end, we then propose to learn the mixing weights of the kernel estimate by sampling from their posterior density using Monte Carlo methods. We first introduce an offline version of the proposed algorithm, followed by an online version which performs inference on both the parameters and the hidden state through particle filtering. The accuracy of the estimation of the state-transition function is first validated on synthetic data. Next, we show that the proposed algorithm outperforms kernel adaptive filters in the prediction of real-world time series, while also providing probabilistic estimates, a key advantage over standard methods.
High-Dimensional Kernel Regression: A Guide for Practitioners
Felipe Tobar, Danilo P. Mandic, 2015. (In Trends in Digital Signal Processing: A Festschrift in Honour of A.G. Constantinides). Edited by Y. C. Lim, H. K. Kwan, W.-C. Siu. CRC Press.
Design of Positive-Definite Quaternion Kernels
Felipe Tobar, Danilo P. Mandic, 2015. (IEEE Signal Processing Letters).
Abstract▼ URL
Quaternion reproducing kernel Hilbert spaces (QRKHS) have been proposed recently and provide a high-dimensional feature space (alternative to the real-valued multikernel approach) for general kernel-learning applications. The current challenge within quaternion-kernel learning is the lack of general quaternion-valued kernels, which are necessary to exploit the full advantages of the QRKHS theory in real-world problems. This letter proposes a novel way to design quaternion-valued kernels, this is achieved by transforming three complex kernels into quaternion ones and then combining their real and imaginary parts. Building on this general construction, our emphasis is on a new quaternion kernel of polynomial features, which is assessed in the prediction of bodysensor networks applications.
Modelling of Complex Signals using Gaussian Processes
Felipe Tobar, Richard E. Turner, 2015. (In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)).
Abstract▼ URL
In complex-valued signal processing, estimation algorithms require complete knowledge (or accurate estimation) of the second order statistics, this makes Gaussian processes (GP) well suited for modelling complex signals, as they are designed in terms of covariance functions. Dealing with bivariate signals using GPs require four covariance matrices, or equivalently, two complex matrices. We propose a GP-based approach for modelling complex signals, whereby the second-order statistics are learnt through maximum likelihood; in particular, the complex GP approach allows for circularity coefficient estimation in a robust manner when the observed signal is corrupted by (circular) white noise. The proposed model is validated using climate signals, for both circular and noncircular cases. The results obtained open new possibilities for collaboration between the complex signal processing and Gaussian processes communities towards an appealing representation and statistical description of bivariate signals.