Felipe Tobar

Jou-Hui Ho, Felipe Tobar

Standard online change point detection (CPD) methods tend to have large false discovery rates as their detections are sensitive to outliers. To overcome this drawback, we propose Greedy Online Change Point Detection (GOCPD), a computationally appealing method which finds change points by maximizing the probability of the data coming from the (temporal) concatenation of two independent models. We show that, for time series with a single change point, this objective is unimodal and thus CPD can be accelerated via ternary search with logarithmic complexity. We demonstrate the effectiveness of GOCPD on synthetic data and validate our findings on real-world univariate and multivariate settings.

Felipe Tobar, Elsa Cazelles, Taco de Wolff

We present a computationally-efficient strategy to initialise the hyperparameters of a Gaussian process (GP) avoiding the computation of the likelihood function. Our strategy can be used as a pretraining stage to find initial conditions for maximum-likelihood (ML) training, or as a standalone method to compute hyperparameters values to be plugged in directly into the GP model. Motivated by the fact that training a GP via ML is equivalent (on average) to minimising the KL-divergence between the true and learnt model, we set to explore different metrics/divergences among GPs that are computationally inexpensive and provide hyperparameter values that are close to those found via ML. In practice, we identify the GP hyperparameters by projecting the empirical covariance or (Fourier) power spectrum onto a parametric family, thus proposing and studying various measures of discrepancy operating on the temporal and frequency domains. Our contribution extends the variogram method developed by the geostatistics literature and, accordingly, it is referred to as the generalised variogram method (GVM). In addition to the theoretical presentation of GVM, we provide experimental validation in terms of accuracy, consistency with ML and computational complexity for different kernels using synthetic and real-world data.

Elsa Cazelles, Felipe Tobar

We propose a novel training objective for GPs constructed using lower-dimensional linear projections of the data, referred to as \emph{projected likelihood} (PL). We provide a closed-form expression for the information loss related to the PL and empirically show that it can be reduced with random projections on the unit sphere. We show the superiority of the PL, in terms of accuracy and computational efficiency, over the exact GP training and the variational free energy approach to sparse GPs over different optimisers, kernels and datasets of moderately large sizes.

Taco de Wolff, Alejandro Cuevas, Felipe Tobar

We present MOGPTK, a Python package for multi-channel data modelling using Gaussian processes (GP). The aim of this toolkit is to make multi-output GP (MOGP) models accessible to researchers, data scientists, and practitioners alike. MOGPTK uses a Python front-end, relies on the GPflow suite and is built on a TensorFlow back-end, thus enabling GPU-accelerated training. The toolkit facilitates implementing the entire pipeline of GP modelling, including data loading, parameter initialization, model learning, parameter interpretation, up to data imputation and extrapolation. MOGPTK implements the main multi-output covariance kernels from literature, as well as spectral-based parameter initialization strategies. The source code, tutorials and examples in the form of Jupyter notebooks, together with the API documentation, can be found at http://github.com/GAMES-UChile/mogptk

Felipe Tobar

Spectral estimation (SE) aims to identify how the energy of a signal (e.g., a time series) is distributed across different frequencies. This can become particularly challenging when only partial and noisy observations of the signal are available, where current methods fail to handle uncertainty appropriately. In this context, we propose a joint probabilistic model for signals, observations and spectra, where SE is addressed as an exact inference problem. Assuming a Gaussian process prior over the signal, we apply Bayes' rule to find the analytic posterior distribution of the spectrum given a set of observations. Besides its expressiveness and natural account of spectral uncertainty, the proposed model also provides a functional-form representation of the power spectral density, which can be optimised efficiently. Comparison with previous approaches, in particular against Lomb-Scargle, is addressed theoretically and also experimentally in three different scenarios. Code and demo available at https://github.com/GAMES-UChile/BayesianSpectralEstimation.

Christopher P. Ley, Felipe Tobar

We introduce the asynchronous graph generator (AGG), a novel graph attention network for imputation and prediction of multi-channel time series. Free from recurrent components or assumptions about temporal/spatial regularity, AGG encodes measurements, timestamps and channel-specific features directly in the nodes via learnable embeddings. Through an attention mechanism, these embeddings allow for discovering expressive relationships among the variables of interest in the form of a homogeneous graph. Once trained, AGG performs imputation by \emph{conditional attention generation}, i.e., by creating a new node conditioned on given timestamps and channel specification. The proposed AGG is compared to related methods in the literature and its performance is analysed from a data augmentation perspective. Our experiments reveal that AGG achieved state-of-the-art results in time series imputation, classification and prediction for the benchmark datasets \emph{Beijing Air Quality}, \emph{PhysioNet ICU 2012} and \emph{UCI localisation}, outperforming other recent attention-based networks.

Camilo Carvajal Reyes, Joaquín Fontbona, Felipe Tobar

Score-based generative models (SBM), also known as diffusion models, are the de facto state of the art for image synthesis. Despite their unparalleled performance, SBMs have recently been in the spotlight for being tricked into creating not-safe-for-work (NSFW) content, such as violent images and non-consensual nudity. Current approaches that prevent unsafe generation are based on the models' own knowledge, and the majority of them require fine-tuning. This article explores the use of external sources for ensuring safe outputs in SBMs. Our safe-for-work (SFW) sampler implements a Conditional Trajectory Correction step that guides the samples away from undesired regions in the ambient space using multimodal models as the source of conditioning. Furthermore, using Contrastive Language Image Pre-training (CLIP), our method admits user-defined NSFW classes, which can vary in different settings. Our experiments on the text-to-image SBM Stable Diffusion validate that the proposed SFW sampler effectively reduces the generation of explicit content while being competitive with other fine-tuning-based approaches, as assessed via independent NSFW detectors. Moreover, we evaluate the impact of the SFW sampler on image quality and show that the proposed correction scheme comes at a minor cost with negligible effect on samples not needing correction. Our study confirms the suitability of the SFW sampler towards aligned SBM models and the potential of using model-agnostic conditioning for the prevention of unwanted images.

Emily C. Ehrhardt, Felipe Tobar

We propose a novel approach to computationally efficient GP training based on the observation that square-exponential (SE) covariance matrices contain several off-diagonal entries extremely close to zero. We construct a principled procedure to eliminate those entries to produce a \emph{banded}-matrix approximation to the original covariance, whose inverse and determinant can be computed at a reduced computational cost, thus contributing to an efficient approximation to the likelihood function. We provide a theoretical analysis of the proposed method to preserve the structure of the original covariance in the 1D setting with SE kernel, and validate its computational efficiency against the variational free energy approach to sparse GPs.

Camilo Carvajal Reyes, Felipe Tobar

Flow-based models have proven successful for time-series generation, particularly when defined in lower-dimensional latent spaces that enable efficient sampling. However, how to design latent representations with desirable equivariance properties for time-series generative modelling remains underexplored. In this work, we propose a latent flow-matching framework in which equivariance is explicitly encouraged through a simple regularisation of a pre-trained autoencoder. Specifically, we introduce an equivariance loss that enforces consistency between transformed signals and their reconstructions, and use it to fine-tune latent spaces with respect to basic time-series transformations such as translation and amplitude scaling. We show that these equivariance-regularised latent spaces improve generation quality while preserving the computational advantages of latent flow models. Experiments on multiple real-world datasets demonstrate that our approach consistently outperforms existing diffusion-based baselines in standard time-series generation metrics, while achieving orders-of-magnitude faster sampling. These results highlight the practical benefits of incorporating geometric inductive biases into latent generative models for time series.

Augusto Tagle, Javier Ruiz-del-Solar, Felipe Tobar

Offline reinforcement learning (RL) recovers the optimal policy $π$ given historical observations of an agent. In practice, $π$ is modeled as a weighted version of the agent's behavior policy $μ$, using a weight function $w$ working as a critic of the agent's behavior. Though recent approaches to offline RL based on diffusion models have exhibited promising results, the computation of the required scores is challenging due to their dependence on the unknown $w$. In this work, we alleviate this issue by constructing a diffusion over both the actions and the weights. With the proposed setting, the required scores are directly obtained from the diffusion model without learning extra networks. Our main conceptual contribution is a novel guidance method, where guidance (which is a function of $w$) comes from the same diffusion model, therefore, our proposal is termed Self-Weighted Guidance (SWG). We show that SWG generates samples from the desired distribution on toy examples and performs on par with state-of-the-art methods on D4RL's challenging environments, while maintaining a streamlined training pipeline. We further validate SWG through ablation studies on weight formulations and scalability.

Felipe Tobar, Arnaud Robert, Jorge F. Silva

Let us consider the deconvolution problem, that is, to recover a latent source $x(\cdot)$ from the observations $\mathbf{y} = [y_1,\ldots,y_N]$ of a convolution process $y = x\star h + η$, where $η$ is an additive noise, the observations in $\mathbf{y}$ might have missing parts with respect to $y$, and the filter $h$ could be unknown. We propose a novel strategy to address this task when $x$ is a continuous-time signal: we adopt a Gaussian process (GP) prior on the source $x$, which allows for closed-form Bayesian nonparametric deconvolution. We first analyse the direct model to establish the conditions under which the model is well defined. Then, we turn to the inverse problem, where we study i) some necessary conditions under which Bayesian deconvolution is feasible, and ii) to which extent the filter $h$ can be learnt from data or approximated for the blind deconvolution case. The proposed approach, termed Gaussian process deconvolution (GPDC) is compared to other deconvolution methods conceptually, via illustrative examples, and using real-world datasets.

Matías Altamirano, Felipe Tobar

Kernel design for Multi-output Gaussian Processes (MOGP) has received increased attention recently. In particular, the Multi-Output Spectral Mixture kernel (MOSM) arXiv:1709.01298 approach has been praised as a general model in the sense that it extends other approaches such as Linear Model of Corregionalization, Intrinsic Corregionalization Model and Cross-Spectral Mixture. MOSM relies on Cramér's theorem to parametrise the power spectral densities (PSD) as a Gaussian mixture, thus, having a structural restriction: by assuming the existence of a PSD, the method is only suited for multi-output stationary applications. We develop a nonstationary extension of MOSM by proposing the family of harmonizable kernels for MOGPs, a class of kernels that contains both stationary and a vast majority of non-stationary processes. A main contribution of the proposed harmonizable kernels is that they automatically identify a possible nonstationary behaviour meaning that practitioners do not need to choose between stationary or non-stationary kernels. The proposed method is first validated on synthetic data with the purpose of illustrating the key properties of our approach, and then compared to existing MOGP methods on two real-world settings from finance and electroencephalography.

Jorge F. Silva, Felipe Tobar, Mario Vicuña et al.

Information-theoretic measures have been widely adopted in the design of features for learning and decision problems. Inspired by this, we look at the relationship between i) a weak form of information loss in the Shannon sense and ii) the operation loss in the minimum probability of error (MPE) sense when considering a family of lossy continuous representations (features) of a continuous observation. We present several results that shed light on this interplay. Our first result offers a lower bound on a weak form of information loss as a function of its respective operation loss when adopting a discrete lossy representation (quantization) instead of the original raw observation. From this, our main result shows that a specific form of vanishing information loss (a weak notion of asymptotic informational sufficiency) implies a vanishing MPE loss (or asymptotic operational sufficiency) when considering a general family of lossy continuous representations. Our theoretical findings support the observation that the selection of feature representations that attempt to capture informational sufficiency is appropriate for learning, but this selection is a rather conservative design principle if the intended goal is achieving MPE in classification. Supporting this last point, and under some structural conditions, we show that it is possible to adopt an alternative notion of informational sufficiency (strictly weaker than pure sufficiency in the mutual information sense) to achieve operational sufficiency in learning.

Bryan Sagredo, Sonia Español-Jiménez, Felipe Tobar

We present a framework for detecting blue whale vocalisations from acoustic submarine recordings. The proposed methodology comprises three stages: i) a preprocessing step where the audio recordings are conditioned through normalisation, filtering, and denoising; ii) a label-propagation mechanism to ensure the consistency of the annotations of the whale vocalisations, and iii) a convolutional neural network that receives audio samples. Based on 34 real-world submarine recordings (28 for training and 6 for testing) we obtained promising performance indicators including an Accuracy of 85.4\% and a Recall of 93.5\%. Furthermore, even for the cases where our detector did not match the ground-truth labels, a visual inspection validates the ability of our approach to detect possible parts of whale calls unlabelled as such due to not being complete calls.

Diego León, Felipe Tobar

In real-world settings, speech signals are almost always affected by reverberation produced by the working environment; these corrupted signals need to be \emph{dereverberated} prior to performing, e.g., speech recognition, speech-to-text conversion, compression, or general audio enhancement. In this paper, we propose a supervised dereverberation technique using \emph{U-nets with skip connections}, which are fully-convolutional encoder-decoder networks with layers arranged in the form of an "U" and connections that "skip" some layers. Building on this architecture, we address speech dereverberation through the lens of Late Reverberation Suppression (LS). Via experiments on synthetic and real-world data with different noise levels and reverberation settings, we show that our proposed method termed "LS U-net" improves quality, intelligibility and other performance metrics compared to the original U-net method and it is on par with the state-of-the-art GAN-based approaches.

Elsa Cazelles, Felipe Tobar, Joaquín Fontbona

We introduce weak barycenters of a family of probability distributions, based on the recently developed notion of optimal weak transport of mass by Gozlanet al. (2017) and Backhoff-Veraguas et al. (2020). We provide a theoretical analysis of this object and discuss its interpretation in the light of convex ordering between probability measures. In particular, we show that, rather than averaging the input distributions in a geometric way (as the Wasserstein barycenter based on classic optimal transport does) weak barycenters extract common geometric information shared by all the input distributions, encoded as a latent random variable that underlies all of them. We also provide an iterative algorithm to compute a weak barycenter for a finite family of input distributions, and a stochastic algorithm that computes them for arbitrary populations of laws. The latter approach is particularly well suited for the streaming setting, i.e., when distributions are observed sequentially. The notion of weak barycenter and our approaches to compute it are illustrated on synthetic examples, validated on 2D real-world data and compared to standard Wasserstein barycenters.

Felipe Tobar, Lerko Araya-Hernández, Pablo Huijse et al.

In a number of data-driven applications such as detection of arrhythmia, interferometry or audio compression, observations are acquired indistinctly in the time or frequency domains: temporal observations allow us to study the spectral content of signals (e.g., audio), while frequency-domain observations are used to reconstruct temporal/spatial data (e.g., MRI). Classical approaches for spectral analysis rely either on i) a discretisation of the time and frequency domains, where the fast Fourier transform stands out as the \textit{de facto} off-the-shelf resource, or ii) stringent parametric models with closed-form spectra. However, the general literature fails to cater for missing observations and noise-corrupted data. Our aim is to address the lack of a principled treatment of data acquired indistinctly in the temporal and frequency domains in a way that is robust to missing or noisy observations, and that at the same time models uncertainty effectively. To achieve this aim, we first define a joint probabilistic model for the temporal and spectral representations of signals, to then perform a Bayesian model update in the light of observations, thus jointly reconstructing the complete (latent) time and frequency representations. The proposed model is analysed from a classical spectral analysis perspective, and its implementation is illustrated through intuitive examples. Lastly, we show that the proposed model is able to perform joint time and frequency reconstruction of real-world audio, healthcare and astronomy signals, while successfully dealing with missing data and handling uncertainty (noise) naturally against both classical and modern approaches for spectral estimation.

Taco de Wolff, Alejandro Cuevas, Felipe Tobar

In Financial Signal Processing, multiple time series such as financial indicators, stock prices and exchange rates are strongly coupled due to their dependence on the latent state of the market and therefore they are required to be jointly analysed. We focus on learning the relationships among financial time series by modelling them through a multi-output Gaussian process (MOGP) with expressive covariance functions. Learning these market dependencies among financial series is crucial for the imputation and prediction of financial observations. The proposed model is validated experimentally on two real-world financial datasets for which their correlations across channels are analysed. We compare our model against other MOGPs and the independent Gaussian process on real financial data.

Elsa Cazelles, Arnaud Robert, Felipe Tobar

We propose the Wasserstein-Fourier (WF) distance to measure the (dis)similarity between time series by quantifying the displacement of their energy across frequencies. The WF distance operates by calculating the Wasserstein distance between the (normalised) power spectral densities (NPSD) of time series. Yet this rationale has been considered in the past, we fill a gap in the open literature providing a formal introduction of this distance, together with its main properties from the joint perspective of Fourier analysis and optimal transport. As the main aim of this work is to validate WF as a general-purpose metric for time series, we illustrate its applicability on three broad contexts. First, we rely on WF to implement a PCA-like dimensionality reduction for NPSDs which allows for meaningful visualisation and pattern recognition applications. Second, we show that the geometry induced by WF on the space of NPSDs admits a geodesic interpolant between time series, thus enabling data augmentation on the spectral domain, by averaging the dynamic content of two signals. Third, we implement WF for time series classification using parametric/non-parametric classifiers and compare it to other classical metrics. Supported on theoretical results, as well as synthetic illustrations and experiments on real-world data, this work establishes WF as a meaningful and capable resource pertinent to general distance-based applications of time series.

Felipe Tobar

We propose a novel class of Gaussian processes (GPs) whose spectra have compact support, meaning that their sample trajectories are almost-surely band limited. As a complement to the growing literature on spectral design of covariance kernels, the core of our proposal is to model power spectral densities through a rectangular function, which results in a kernel based on the sinc function with straightforward extensions to non-centred (around zero frequency) and frequency-varying cases. In addition to its use in regression, the relationship between the sinc kernel and the classic theory is illuminated, in particular, the Shannon-Nyquist theorem is interpreted as posterior reconstruction under the proposed kernel. Additionally, we show that the sinc kernel is instrumental in two fundamental signal processing applications: first, in stereo amplitude modulation, where the non-centred sinc kernel arises naturally. Second, for band-pass filtering, where the proposed kernel allows for a Bayesian treatment that is robust to observation noise and missing data. The developed theory is complemented with illustrative graphic examples and validated experimentally using real-world data.

Gonzalo Rios, Felipe Tobar

The Gaussian process (GP) is a nonparametric prior distribution over functions indexed by time, space, or other high-dimensional index set. The GP is a flexible model yet its limitation is given by its very nature: it can only model Gaussian marginal distributions. To model non-Gaussian data, a GP can be warped by a nonlinear transformation (or warping) as performed by warped GPs (WGPs) and more computationally-demanding alternatives such as Bayesian WGPs and deep GPs. However, the WGP requires a numerical approximation of the inverse warping for prediction, which increases the computational complexity in practice. To sidestep this issue, we construct a novel class of warpings consisting of compositions of multiple elementary functions, for which the inverse is known explicitly. We then propose the compositionally-warped GP (CWGP), a non-Gaussian generative model whose expressiveness follows from its deep compositional architecture, and its computational efficiency is guaranteed by the analytical inverse warping. Experimental validation using synthetic and real-world datasets confirms that the proposed CWGP is robust to the choice of warpings and provides more accurate point predictions, better trained models and shorter computation times than WGP.

Cristobal Valenzuela, Felipe Tobar

We propose a Bayesian nonparametric method for low-pass filtering that can naturally handle unevenly-sampled and noise-corrupted observations. The proposed model is constructed as a latent-factor model for time series, where the latent factors are Gaussian processes with non-overlapping spectra. With this construction, the low-pass version of the time series can be identified as the low-frequency latent component, and therefore it can be found by means of Bayesian inference. We show that the model admits exact training and can be implemented with minimal numerical approximations. Finally, the proposed model is validated against standard linear filters on synthetic and real-world time series.

Julio Backhoff-Veraguas, Joaquin Fontbona, Gonzalo Rios et al.

We introduce and study a novel model-selection strategy for Bayesian learning, based on optimal transport, along with its associated predictive posterior law: the Wasserstein population barycenter of the posterior law over models. We first show how this estimator, termed Bayesian Wasserstein barycenter (BWB), arises naturally in a general, parameter-free Bayesian model-selection framework, when the considered Bayesian risk is the Wasserstein distance. Examples are given, illustrating how the BWB extends some classic parametric and non-parametric selection strategies. Furthermore, we also provide explicit conditions granting the existence and statistical consistency of the BWB, and discuss some of its general and specific properties, providing insights into its advantages compared to usual choices, such as the model average estimator. Finally, we illustrate how this estimator can be computed using the stochastic gradient descent (SGD) algorithm in Wasserstein space introduced in a companion paper arXiv:2201.04232v2 [math.OC], and provide a numerical example for experimental validation of the proposed method.

Gonzalo Rios, Felipe Tobar

Gaussian processes (GPs) are Bayesian nonparametric generative models that provide interpretability of hyperparameters, admit closed-form expressions for training and inference, and are able to accurately represent uncertainty. To model general non-Gaussian data with complex correlation structure, GPs can be paired with an expressive covariance kernel and then fed into a nonlinear transformation (or warping). However, overparametrising the kernel and the warping is known to, respectively, hinder gradient-based training and make the predictions computationally expensive. We remedy this issue by (i) training the model using derivative-free global-optimisation techniques so as to find meaningful maxima of the model likelihood, and (ii) proposing a warping function based on the celebrated Box-Cox transformation that requires minimal numerical approximations---unlike existing warped GP models. We validate the proposed approach by first showing that predictions can be computed analytically, and then on a learning, reconstruction and forecasting experiment using real-world datasets.

Gabriel Parra, Felipe Tobar

Early approaches to multiple-output Gaussian processes (MOGPs) relied on linear combinations of independent, latent, single-output Gaussian processes (GPs). This resulted in cross-covariance functions with limited parametric interpretation, thus conflicting with the ability of single-output GPs to understand lengthscales, frequencies and magnitudes to name a few. On the contrary, current approaches to MOGP are able to better interpret the relationship between different channels by directly modelling the cross-covariances as a spectral mixture kernel with a phase shift. We extend this rationale and propose a parametric family of complex-valued cross-spectral densities and then build on Cramér's Theorem (the multivariate version of Bochner's Theorem) to provide a principled approach to design multivariate covariance functions. The so-constructed kernels are able to model delays among channels in addition to phase differences and are thus more expressive than previous methods, while also providing full parametric interpretation of the relationship across channels. The proposed method is first validated on synthetic data and then compared to existing MOGP methods on two real-world examples.

Felipe Tobar, Gonzalo Rios, Tomás Valdivia et al.

In sensing applications, sensors cannot always measure the latent quantity of interest at the required resolution, sometimes they can only acquire a blurred version of it due the sensor's transfer function. To recover latent signals when only noisy mixed measurements of the signal are available, we propose the Gaussian process mixture of measurements (GPMM), which models the latent signal as a Gaussian process (GP) and allows us to perform Bayesian inference on such signal conditional to a set of noisy mixture of measurements. We describe how to train GPMM, that is, to find the hyperparameters of the GP and the mixing weights, and how to perform inference on the latent signal under GPMM; additionally, we identify the solution to the underdetermined linear system resulting from a sensing application as a particular case of GPMM. The proposed model is validated in the recovery of three signals: a smooth synthetic signal, a real-world heart-rate time series and a step function, where GPMM outperformed the standard GP in terms of estimation error, uncertainty representation and recovery of the spectral content of the latent signal.

Felipe Tobar

Kernel adaptive filters, a class of adaptive nonlinear time-series models, are known by their ability to learn expressive autoregressive patterns from sequential data. However, for trivial monotonic signals, they struggle to perform accurate predictions and at the same time keep computational complexity within desired boundaries. This is because new observations are incorporated to the dictionary when they are far from what the algorithm has seen in the past. We propose a novel approach to kernel adaptive filtering that compares new observations against dictionary samples in terms of their unit-norm (normalised) versions, meaning that new observations that look like previous samples but have a different magnitude are not added to the dictionary. We achieve this by proposing the unit-norm Gaussian kernel and define a sparsification criterion for this novel kernel. This new methodology is validated on two real-world datasets against standard KAF in terms of the normalised mean square error and the dictionary size.

Iván Castro, Cristóbal Silva, Felipe Tobar

We present a probabilistic framework for both (i) determining the initial settings of kernel adaptive filters (KAFs) and (ii) constructing fully-adaptive KAFs whereby in addition to weights and dictionaries, kernel parameters are learnt sequentially. This is achieved by formulating the estimator as a probabilistic model and defining dedicated prior distributions over the kernel parameters, weights and dictionary, enforcing desired properties such as sparsity. The model can then be trained using a subset of data to initialise standard KAFs or updated sequentially each time a new observation becomes available. Due to the nonlinear/non-Gaussian properties of the model, learning and inference is achieved using gradient-based maximum-a-posteriori optimisation and Markov chain Monte Carlo methods, and can be confidently used to compute predictions. The proposed framework was validated on nonlinear time series of both synthetic and real-world nature, where it outperformed standard KAFs in terms of mean square error and the sparsity of the learnt dictionaries.