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 in 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 prevention of unwanted images.

Information-theoretic measures have been widely adopted for machine learning (ML) feature design. Inspired by this, we look at the relationship between information loss in the Shannon sense and the operation loss in the minimum probability of error (MPE) sense when considering a family of lossy representations. 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 encoder. When considering a general family of lossy continuous representations, we show that a form of vanishing information loss (a weak informational sufficiency (WIS)) implies a vanishing MPE loss. Our findings support the observation that selecting/designing representations that capture informational sufficiency is appropriate for learning. However, this selection is rather conservative if the intended goal is achieving MPE in classification. Supporting this, 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. Furthermore, our new WIS condition is used to demonstrate the expressive power of digital encoders and the capacity of two existing compression-based algorithms to achieve lossless prediction in ML.

Kernel design for Multi-output Gaussian Processes (MOGP) has received increased attention recently, in particular, the Multi-Output Spectral Mixture kernel (MOSM) 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 Cramer’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 processes. 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.

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) operational loss in the minimum probability of error (MPE) sense when considering a family of lossy continuous representations of an observation. 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 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 design principle is a rather conservative if the intended goal is achieving MPE in classification. On this last point, we discuss about studying weak forms of informational sufficiencies to achieve operational sufficiency in learning settings.

We introduce weak barycenters of a family of probability distributions, based on the recently developed notion of optimal weak transport of mass by Gozlan et 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.

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.

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.

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.

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.

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 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 is addressed theoretically, showing that the proposed method is an infinite-dimensional variant of the Lomb-Scargle approach, and also empirically through three experiments.

Timely detection and accurate description of extreme events, such as natural disasters and other crisis situations, are crucial for emergency management and mitigation. Extreme-event detection is challenging, since one has to rely upon reports from human observers appointed to specific geographical areas, or on an expensive and sophisticated infrastructure. In the case of earthquakes, geographically dense sensor networks are expensive to deploy and maintain. Therefore, only some regions-or even countries-are able to acquire useful information about the effects of earthquakes in their own territory. An inexpensive and viable alternative to this problem is to detect extreme real-world events through people's reactions in online social networks. In particular, Twitter has gained popularity within the scientific community for providing access to real-time “citizen sensor” activity. Nevertheless, the massive amount of messages in the Twitter stream, along with the noise it contains, underpin a number of difficulties when it comes to Twitter-based event detection. We contribute to address these challenges by proposing an online method for detecting unusual bursts in discrete-time signals extracted from Twitter. This method only requires a one-off semisupervised initialization and can be scaled to track multiple signals in a robust manner. We also show empirically how our proposed approach, which was envisioned for generic event detection, can be adapted for worldwide earthquake detection, where we compare the proposed model to the state of the art for earthquake tracking using social media. Experimental results validate our approach as a competitive alternative in terms of precision and recall to leading solutions, with the advantage of implementation simplicity and worldwide scalability.

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.

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.

A widely linear quaternion recursive total least squares (WL-QRTLS) algorithm is introduced for the processing of Q-improper processes contaminated by noise. The total least squares for quaternions (QTLS) is a generalisation of the real-valued total least squares and is introduced rigorously, starting from the existence condition for low-rank approximation of quaternion matrices. Then, a quaternion Rayleigh quotient (QRQ) is defined to establish the link between the QTLS solution and the minimisation of the QRQ. Finally, the rank-one update formula is employed to allow for fast iterative solution based on the QRQ. Through simulations, the WL-QRTLS was shown to exhibit superior performance, under perturbations on both input and output signals, to other adaptive filtering of the same class - the widely linear quaternion least mean squares (WL-QLMS) and the widely linear quaternion recursive least squares (WL-QRLS). The experiments on both synthetic and real-world $\mathbb{Q}$-improper processes supported the analysis.

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.

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 nonparametric-window 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 free-energy 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.

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.

A class of stochastic volatility models (SVMs) with time-varying parameters is presented for online volatility estimation in nonstationary environments. This is achieved by modelling both the volatility and model parameters as states of a hidden Markov model (HMM), thus allowing for the use of particle filters to estimate the resulting posterior densities. The proposed models, based on the logarithmic SVM and the unobserved GARCH model, are evaluated for the estimation of the volatility of the NASDAQ-C and the Chilean IGPA financial indices between June 2007 and January 2010, where the late-2000s financial crisis is included. Simulations show that the proposed time-varying models are well suited for online volatility estimation as (i) they achieve an accuracy comparable to those of offline (batch) algorithms, and (ii) their parameters can be used to identify market changes.

A novel kernel algorithm is proposed for nonlinear prediction whereby the signal is modelled as a state of a hidden Markov model (HMM). The transition function of the HMM is approximated using kernels, whose weights are also part of the state of the system and are learnt in an unsupervised fashion by a sample importance resampling (SIR) particle filter. The SIR proposal density is designed so as to maintain a diverse population of particles, thus avoiding particle degeneracy arising from inaccuracies of early model estimates. The kernel HMM algorithm is further equipped with a sparsification criterion based on approximate linear dependence and its performance is evaluated against the KNLMS and KRLS algorithms for the prediction of synthetic signals and real world point-of-gaze data.

The existence and uniqueness conditions of quaternion reproducing kernel Hilbert spaces (QRKHS) are established in order to provide a mathematical foundation for the development of quaternion-valued kernel learning algorithms. This is achieved through a rigorous account of left quaternion Hilbert spaces, which makes it possible to generalise standard RKHS to quaternion RKHS. Quaternion versions of the Riesz representation and Moore-Aronszajn theorems are next introduced, thus underpinning kernel estimation algorithms operating on quaternion-valued feature spaces. The difference between the proposed quaternion kernel concept and the existing real and vector approaches is also established in terms of both theoretical advantages and computational complexity. The enhanced estimation ability of the so-introduced quaternion-valued kernels over their real- and vector-valued counterparts is validated through kernel ridge regression applications. Simulations on real world 3D inertial body sensor data and nonlinear channel equalisation using novel quaternion cubic and Gaussian kernels support the approach.

The multikernel least-mean-square algorithm is introduced for adaptive estimation of vector-valued nonlinear and nonstationary signals. This is achieved by mapping the multivariate input data to a Hilbert space of time-varying vector-valued functions, whose inner products (kernels) are combined in an online fashion. The proposed algorithm is equipped with novel adaptive sparsification criteria ensuring a finite dictionary, and is computationally efficient and suitable for nonstationary environments. We also show the ability of the proposed vector-valued reproducing kernel Hilbert space to serve as a feature space for the class of multikernel least-squares algorithms. The benefits of adaptive multikernel (MK) estimation algorithms are illuminated in the nonlinear multivariate adaptive prediction setting. Simulations on nonlinear inertial body sensor signals and nonstationary real-world wind signals of low, medium, and high dynamic regimes support the approach.

The quaternion kernel least squares algorithm (QKLS) is introduced as a generic kernel framework for the estimation of multivariate quaternion valued signals. This is achieved based on the concepts of quaternion inner product and quaternion positive definiteness, allowing us to define quaternion kernel regression. Next, the least squares solution is derived using the recently introduced Hℝ calculus. We also show that QKLS is a generic extension of standard kernel least squares, and their equivalence is established for real valued kernels. The superiority of the quaternion-valued linear kernel with respect to its real-valued counterpart is illustrated for both synthetic and real-world prediction applications, in terms of accuracy and robustness to overfitting.standard kernel least squares,quaternion-valued linear kernelreal-world prediction applications,real-world 3D inertial body sensor signals.synthetic autoregressive processes