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.
We focus on an interpolation method referred to Bayesian reconstruction in this paper. Whereas in standard interpolation methods missing data are interpolated deterministically, in Bayesian reconstruction, missing data are interpolated probabilistically using a Bayesian treatment. In this paper, we address the framework of Bayesian reconstruction and its application to the traffic data reconstruction problem in the field of traffic engineering. In the latter part of this paper, we describe the evaluation of the statistical performance of our Bayesian traffic reconstruction model using a statis…
We revisit Rahimi and Recht (2007)'s kernel random Fourier features (RFF) method through the lens of the PAC-Bayesian theory. While the primary goal of RFF is to approximate a kernel, we look at the Fourier transform as a prior distribution over trigonometric hypotheses. It naturally suggests learning a posterior on these hypotheses. We derive generalization bounds that are optimized by learning a pseudo-posterior obtained from a closed-form expression. Based on this study, we consider two learning strategies: The first one finds a compact landmarks-based representation of the data where each …
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Amazon Web Services（亚马逊云科技）
Gaussian ODE filtering is a probabilistic numerical method to solve ordinary differential equations (ODEs). It computes a Bayesian posterior over the solution from evaluations of the vector field defining the ODE. Its most popular version, which employs an integrated Brownian motion prior, uses Taylor expansions of the mean to extrapolate forward and has the same convergence rates as classical numerical methods. As the solution of many important ODEs are periodic functions (oscillators), we raise the question whether Fourier expansions can also be brought to bear within the framework of Gaussi…
This paper studies a new Bayesian algorithm for the joint reconstruction and classification of reflectance confocal microscopy (RCM) images, with application to the identification of human skin lentigo. The proposed Bayesian approach takes advantage of the distribution of the multiplicative speckle noise affecting the true reflectivity of these images and of appropriate priors for the unknown model parameters. A Markov chain Monte Carlo (MCMC) algorithm is proposed to jointly estimate the model parameters and the image of true reflectivity while classifying images according to the distribution…
We propose a two stage procedure for the estimation of the parameters of a fairly general, continuous-time stochastic volatility. An important ingredient of the proposed method is the Cuchiero-Teichmann volatility estimator, which is based on Fourier transforms and provides a continuous time estimate of the latent process. This estimate is then used to construct an approximate likelihood for the parameters of interest, whose restrictions are taken into account through prior distributions. The procedure is shown to be highly successful for constructing the posterior distribution of the paramete…
Complex network reconstruction is a hot topic in many fields. Currently, the most popular data-driven reconstruction framework is based on lasso. However, it is found that, in the presence of noise, lasso loses efficiency for weighted networks. This paper builds a new framework to cope with this problem. The key idea is to employ a series of linear regression problems to model the relationship between network nodes, and then to use an efficient variational Bayesian algorithm to infer the unknown coefficients. The numerical experiments conducted on both synthetic and real data demonstrate that …
We present a principled Bayesian framework for signal reconstruction, in which the signal is modelled by basis functions whose number (and form, if required) is determined by the data themselves. This approach is based on a Bayesian interpretation of conventional sparse reconstruction and regularisation techniques, in which sparsity is imposed through priors via Bayesian model selection. We demonstrate our method for noisy 1- and 2-dimensional signals, including astronomical images. Furthermore, by using a product-space approach, the number and type of basis functions can be treated as integer…
Max Planck Institute for Molecular Genetics（马克斯·普朗克分子遗传学研究所）
We present a Bayesian method for feature selection in the presence of grouping information with sparsity on the between- and within group level. Instead of using a stochastic algorithm for parameter inference, we employ expectation propagation, which is a deterministic and fast algorithm. Available methods for feature selection in the presence of grouping information have a number of short-comings: on one hand, lasso methods, while being fast, underestimate the regression coefficients and do not make good use of the grouping information, and on the other hand, Bayesian approaches, while accura…
Parameter reconstruction is a common problem in optical nano metrology. It generally involves a set of measurements, to which one attempts to fit a numerical model of the measurement process. The model evaluation typically involves to solve Maxwell's equations and is thus time consuming. This makes the reconstruction computationally demanding. Several methods exist for fitting the model to the measurements. On the one hand, Bayesian optimization methods for expensive black-box optimization enable an efficient reconstruction by training a machine learning model of the squared sum of deviations.…
Parameter reconstructions are indispensable in metrology. Here, the objective is to to explain $K$ experimental measurements by fitting to them a parameterized model of the measurement process. The model parameters are regularly determined by least-square methods, i.e., by minimizing the sum of the squared residuals between the $K$ model predictions and the $K$ experimental observations, $χ^2$. The model functions often involve computationally demanding numerical simulations. Bayesian optimization methods are specifically suited for minimizing expensive model functions. However, in contrast to…
University of Oxford（牛津大学）；University College London（伦敦大学学院）；Inria（法国国家信息与自动化研究所）
Despite its wide use and empirical successes, the theoretical understanding and study of the behaviour and performance of the variational autoencoder (VAE) have only emerged in the past few years. We contribute to this recent line of work by analysing the VAE's reconstruction ability for unseen test data, leveraging arguments from the PAC-Bayes theory. We provide generalisation bounds on the theoretical reconstruction error, and provide insights on the regularisation effect of VAE objectives. We illustrate our theoretical results with supporting experiments on classical benchmark datasets.
This paper analyzes DONE, an online optimization algorithm that iteratively minimizes an unknown function based on costly and noisy measurements. The algorithm maintains a surrogate of the unknown function in the form of a random Fourier expansion (RFE). The surrogate is updated whenever a new measurement is available, and then used to determine the next measurement point. The algorithm is comparable to Bayesian optimization algorithms, but its computational complexity per iteration does not depend on the number of measurements. We derive several theoretical results that provide insight on how…
Statistical machine learning methods, especially nonparametric Bayesian methods, have become increasingly popular to infer clonal population structure of tumors. Here we describe the treeCRP, an extension of the Chinese restaurant process (CRP), a popular construction used in nonparametric mixture models, to infer the phylogeny and genotype of major subclonal lineages represented in the population of cancer cells. We also propose new split-merge updates tailored to the subclonal reconstruction problem that improve the mixing time of Markov chains. In comparisons with the tree-structured stick …
Sparse signal reconstruction algorithms have attracted research attention due to their wide applications in various fields. In this paper, we present a simple Bayesian approach that utilizes the sparsity constraint and a priori statistical information (Gaussian or otherwise) to obtain near optimal estimates. In addition, we make use of the rich structure of the sensing matrix encountered in many signal processing applications to develop a fast sparse recovery algorithm. The computational complexity of the proposed algorithm is relatively low compared with the widely used convex relaxation meth…
We propose a Bayesian nonparametric mixture model for the reconstruction and prediction from observed time series data, of discretized stochastic dynamical systems, based on Markov Chain Monte Carlo methods (MCMC). Our results can be used by researchers in physical modeling interested in a fast and accurate estimation of low dimensional stochastic models when the size of the observed time series is small and the noise process (perhaps) is non-Gaussian. The inference procedure is demonstrated specifically in the case of polynomial maps of arbitrary degree and when a Geometric Stick Breaking mix…
The advent of comprehensive synaptic wiring diagrams of large neural circuits has created the field of connectomics and given rise to a number of open research questions. One such question is whether it is possible to reconstruct the information stored in a recurrent network of neurons, given its synaptic connectivity matrix. Here, we address this question by determining when solving such an inference problem is theoretically possible in specific attractor network models and by providing a practical algorithm to do so. The algorithm builds on ideas from statistical physics to perform approxima…
This paper presents a novel Block Iterative Bayesian Algorithm (Block-IBA) for reconstructing block-sparse signals with unknown block structures. Unlike the existing algorithms for block sparse signal recovery which assume the cluster structure of the nonzero elements of the unknown signal to be independent and identically distributed (i.i.d.), we use a more realistic Bernoulli-Gaussian hidden Markov model (BGHMM) to characterize the non-i.i.d. block-sparse signals commonly encountered in practice. The Block-IBA iteratively estimates the amplitudes and positions of the block-sparse signal usin…
3D Lidar imaging can be a challenging modality when using multiple wavelengths, or when imaging in high noise environments (e.g., imaging through obscurants). This paper presents a hierarchical Bayesian algorithm for the robust reconstruction of multispectral single-photon Lidar data in such environments. The algorithm exploits multi-scale information to provide robust depth and reflectivity estimates together with their uncertainties to help with decision making. The proposed weight-based strategy allows the use of available guide information that can be obtained by using state-of-the-art lea…
We provide a complete framework for performing infinite-dimensional Bayesian inference and uncertainty quantification for image reconstruction with Poisson data. In particular, we address the following issues to make the Bayesian framework applicable in practice. We first introduce a positivity-preserving reparametrization, and we prove that under the reparametrization and a hybrid prior, the posterior distribution is well-posed in the infinite dimensional setting. Second we provide a dimension-independent MCMC algorithm, based on the preconditioned Crank-Nicolson Langevin method, in which we …
This paper presents a hierarchical Bayesian model to reconstruct sparse images when the observations are obtained from linear transformations and corrupted by an additive white Gaussian noise. Our hierarchical Bayes model is well suited to such naturally sparse image applications as it seamlessly accounts for properties such as sparsity and positivity of the image via appropriate Bayes priors. We propose a prior that is based on a weighted mixture of a positive exponential distribution and a mass at zero. The prior has hyperparameters that are tuned automatically by marginalization over the hi…
Computed tomography is a method for synthesizing volumetric or cross-sectional images of an object from a collection of projections. Popular reconstruction methods for computed tomography are based on idealized models and assumptions that may not be valid in practice. One such assumption is that the exact projection geometry is known. The projection geometry describes the relative location of the radiation source, object, and detector for each projection. However, in practice, the geometric parameters used to describe the position and orientation of the radiation source, object, and detector a…
A fast matching pursuit method using a Bayesian approach is introduced for sparse signal recovery. This method, referred to as nGpFBMP, performs Bayesian estimates of sparse signals even when the signal prior is non-Gaussian or unknown. It is agnostic on signal statistics and utilizes a priori statistics of additive noise and the sparsity rate of the signal, which are shown to be easily estimated from data if not available. nGpFBMP utilizes a greedy approach and order-recursive updates of its metrics to find the most dominant sparse supports to determine the approximate minimum mean square err…
Fossil-based palaeoclimate reconstruction is an important area of ecological science that has gained momentum in the backdrop of the global climate change debate. The hierarchical Bayesian paradigm provides an interesting platform for studying such important scientific issue. However, our cross-validation based assessment of the existing Bayesian hierarchical models with respect to two modern proxy data sets based on chironomid and pollen, respectively, revealed that the models are inadequate for the data sets. In this paper, we model the species assemblages (compositional data) by the zero-in…
A central goal of modern magnetic resonance imaging (MRI) is to reduce the time required to produce high-quality images. Efforts have included hardware and software innovations such as parallel imaging, compressed sensing, and deep learning-based reconstruction. Here, we propose and demonstrate a Bayesian method to build statistical libraries of magnetic resonance (MR) images in k-space and use these libraries to identify optimal subsampling paths and reconstruction processes. Specifically, we compute a multivariate normal distribution based upon Gaussian processes using a publicly available l…
Trustworthy MRI Reconstruction via Bayesian Uncertainty Quantification with Sparsity Prior Models
Department of Computer Science, University of Exeter（埃克塞特大学计算机科学系）；UCL Hawkes Institute, Department of Computer Science, University College London（伦敦大学学院计算机科学系霍克斯研究所）
M/EEG source localization is an open research issue. To solve it, it is important to have good knowledge of several physical parameters to build a reliable head operator. Amongst them, the value of the conductivity of the human skull has remained controversial. This report introduces a novel hierarchical Bayesian framework to estimate the skull conductivity jointly with the brain activity from the M/EEG measurements to improve the reconstruction quality. A partially collapsed Gibbs sampler is used to draw samples asymptotically distributed according to the associated posterior. The generated s…
A non-destructive testing (NDT) application of X-ray computed tomography (CT) is inspection of subsea pipes in operation via 2D cross-sectional scans. Data acquisition is time-consuming and costly due to the challenging subsea environment. Reducing the number of projections in a scan can yield time and cost savings, but compromises the reconstruction quality, if conventional reconstruction methods are used. In this work we take a Bayesian approach to CT reconstruction and focus on designing an effective prior to make use of available structural information about the pipe geometry. We propose a…
We present a Bayesian hierarchical modeling approach to paleoclimate reconstruction using borehole temperature profiles. The approach relies on modeling heat conduction in solids via the heat equation with step function, surface boundary conditions. Our analysis includes model error and assumes that the boundary conditions are random processes. The formulation also enables separation of measurement error and model error. We apply the analysis to data from nine borehole temperature records from the San Rafael region in Utah. We produce ground surface temperature histories with uncertainty estim…
Holocene (the last 12,000 years) temperature variation, including the transition out of the last Ice Age to a warmer climate, is reconstructed at multiple locations in southern Finland, Sweden and Estonia based on pollen fossil data from lake sediment cores. A novel Bayesian statistical approach is proposed that allows the reconstructed temperature histories to interact through shared environmental response parameters and spatial dependence. The prior distribution for past temperatures is partially based on numerical climate simulation. The features in the reconstructions are consistent with t…
In recent investigations, the problem of detecting edges given non-uniform Fourier data was reformulated as a sparse signal recovery problem with an l1-regularized least squares cost function. This result can also be derived by employing a Bayesian formulation. Specifically, reconstruction of an edge map using l1 regularization corresponds to a so-called type-I (maximum a posteriori) Bayesian estimate. In this paper, we use the Bayesian framework to design an improved algorithm for detecting edges from non-uniform Fourier data. In particular, we employ what is known as type-II Bayesian estimat…
Fourier domain methods are fast algorithms for SAR imaging. They typically involve an interpolation in the frequency domain to re-grid non-uniform data so inverse fast Fourier transforms can be performed. In this paper, we apply a frame reconstruction algorithm, extending the non-uniform fast Fourier transform, to stripmap SAR data. Further, we present an improved thresholded frame reconstruction algorithm for robust performance and improved computational efficiency. We demonstrate compelling results on real stripmap SAR data.
The paper observes the Hermite and the Fourier Transform domains in terms of Frequency Hopping Spread Spectrum signals sparsification. Sparse signals can be recovered from a reduced set of samples by using the Compressive Sensing approach. The under-sampling and the reconstruction of those signals are also analyzed in this paper. The number of measurements (available signal samples) is varied and reconstruction performance is tested in all considered cases and for both observed domains. The signal recovery is done using an adaptive gradient based algorithm. The theory is verified with the expe…
Optimal Sampling and Reconstruction of Graph Signals in the Fractional Fourier Domain
Nanjing University of Information Science and Technology（南京信息工程大学）；Hubei University（湖北大学）；Shanghai Jiao Tong University（上海交通大学）；Nanjing Institute of Technology（南京工程学院）
Over the past several decades, many different types of computational imaging approaches have been proposed for improving MRI. In this paper, we provide an overview of methods that assume that MRI Fourier data is linearly predictable. Linear prediction is well known in signal processing and may be most recognizable for its usefulness in speech processing and spectrum estimation applications. In MRI, linear predictability implies that data can be sampled below the conventional Nyquist rate, since unmeasured data may be imputed as a shift-invariant linear combination of measured samples. Linear p…
Purpose: To develop a deep learning-based Bayesian inference for MRI reconstruction. Methods: We modeled the MRI reconstruction problem with Bayes's theorem, following the recently proposed PixelCNN++ method. The image reconstruction from incomplete k-space measurement was obtained by maximizing the posterior possibility. A generative network was utilized as the image prior, which was computationally tractable, and the k-space data fidelity was enforced by using an equality constraint. The stochastic backpropagation was utilized to calculate the descent gradient in the process of maximum a pos…
Light detection and ranging (Lidar) single-photon devices capture range and intensity information from a 3D scene. This modality enables long range 3D reconstruction with high range precision and low laser power. A multispectral single-photon Lidar system provides additional spectral diversity, allowing the discrimination of different materials. However, the main drawback of such systems can be the long acquisition time needed to collect enough photons in each spectral band. In this work, we tackle this problem in two ways: first, we propose a Bayesian 3D reconstruction algorithm that is able …
Indian Institute of Technology Bombay（印度理工学院孟买分校）；Stanford University（斯坦福大学）
In this paper, we present two variations of an algorithm for signal reconstruction from one-bit or two-bit noisy observations of the discrete Fourier transform (DFT). The one-bit observations of the DFT correspond to the sign of its real part, whereas, the two-bit observations of the DFT correspond to the signs of both the real and imaginary parts of the DFT. We focus on images for analysis and simulations, thus using the sign of the 2D-DFT. This choice of the class of signals is inspired by previous works on this problem. For our algorithm, we show that the expected mean squared error (MSE) i…
There is growing interest in learning Fourier domain sampling strategies (particularly for magnetic resonance imaging, MRI) using optimization approaches. For non-Cartesian sampling patterns, the system models typically involve non-uniform FFT (NUFFT) operations. Commonly used NUFFT algorithms contain frequency domain interpolation, which is not differentiable with respect to the sampling pattern, complicating the use of gradient methods. This paper describes an efficient and accurate approach for computing approximate gradients involving NUFFTs. Multiple numerical experiments validated the im…
This thesis consists of original contributions in the area of digital signal processing. The reconstruction of signals sparse (highly concentrated) in various transform domains is the primary problem analyzed in the thesis. The considered domains include Fourier, discrete Hermite, one-dimensional and two-dimensional discrete cosine transform, as well as various time-frequency representations. Sparse signals are reconstructed using sparsity measures, being, in fact, the measures of signal concentration in the considered domains. The thesis analyzes the compressive sensing reconstruction algorit…
