Tensor factorization has been steadily used to represent high-dimensional data. In particular, the canonical polyadic decomposition (CPD) is very appreciated for its remarkable uniqueness properties. However, computing the high-order CPD is challenging: numerical issues and high needs for storage and processing can make algorithms diverge. Furthermore, the recovery of the CP factors is an ill-posed problem. One way to circumvent this limitation is to exploit the equivalence between the CPD and the Tensor Train Decomposition (TTD). This paper formulates the CPD as a dimension reduction using a TTD followed by a global marginally convex optimization problem . This global optimization scheme estimates the CP factors with minimal error. The resulting approach, Dimensionality Reduction, joint Estimation of the Ambiguity Matrices and the CP FACtors (DREAMFAC), relies on a block-coordinate descent that reaches a first-order stationary point when estimating the CP factors. DREAMFAC is also shown to be an optimal estimator that reaches the corresponding constrained Cramér–Rao bound. It therefore appears as a state-of-the-art solution to estimate the best rank- K CPD of a tensor (when it exists). Its performance is illustrated on the problem of parameter estimation in a dual-polarized MIMO system . Numerical experiments show the excellent practical performance of DREAMFAC, even with very low SNR.

Abstract Normalizing flows (NF) use a continuous generator to map a simple latent (e.g. Gaussian) distribution, towards an empirical target distribution associated with a training data set. Once trained by minimizing a variational objective, the learnt map provides an approximate generative model of the target distribution. Since standard NF implement differentiable maps, they may suffer from pathological behaviors when targeting complex distributions. For instance, such problems may appear for distributions on multi-component topologies or characterized by multiple modes with high probability regions separated by very unlikely areas. A typical symptom is the explosion of the Jacobian norm of the transformation in very low probability areas. This paper proposes to overcome this issue thanks to a new Markov chain Monte Carlo algorithm to sample from the target distribution in the latent domain before transporting it back to the target domain. The approach relies on a Metropolis adjusted Langevin algorithm whose dynamics explicitly exploits the Jacobian of the transformation. Contrary to alternative approaches, the proposed strategy preserves the tractability of the likelihood and it does not require a specific training. Notably, it can be straightforwardly used with any pre-trained NF network, regardless of the architecture. Experiments conducted on synthetic and high-dimensional real data sets illustrate the efficiency of the method.

This paper introduces a stochastic plug-and-play (PnP) sampling algorithm that leverages variable splitting to efficiently sample from a posterior distribution. The algorithm based on split Gibbs sampling (SGS) draws inspiration from the half quadratic splitting method (HQS) and the alternating direction method of multipliers (ADMM). It divides the challenging task of posterior sampling into two simpler sampling problems. The first problem depends on the likelihood function, while the second is interpreted as a Bayesian denoising problem that can be readily carried out by a deep generative model. Specifically, for an illustrative purpose, the proposed method is implemented in this paper using state-of-the-art diffusion-based generative models. Akin to its deterministic PnP-based counterparts, the proposed method exhibits the great advantage of not requiring an explicit choice of the prior distribution, which is rather encoded into a pre-trained generative model. However, unlike optimization methods (e.g., PnP-ADMM and PnP-HQS) which generally provide only point estimates, the proposed approach allows conventional Bayesian estimators to be accompanied by confidence intervals at a reasonable additional computational cost. Experiments on commonly studied image processing problems illustrate the efficiency of the proposed sampling strategy. Its performance is compared to recent state-of-the-art optimization and sampling methods.

Despite their advantages, normalizing flows generally suffer from several shortcomings including their tendency to generate unrealistic data (e.g., images) and their failing to detect out-of-distribution data.One reason for these deficiencies lies in the training strategy which traditionally exploits a maximum likelihood principle only.This paper proposes a new training paradigm based on a hybrid objective function combining the maximum likelihood principle (MLE) and a sliced-Wasserstein distance.Results obtained on synthetic toy examples and real image data sets show better generative abilities in terms of both likelihood and visual aspects of the generated samples.Reciprocally, the proposed approach leads to a lower likelihood of out-of-distribution data, demonstrating a greater data fidelity of the resulting flows.

Hyperspectral super-resolution based on coupled Tucker decomposition has been recently considered in the remote sensing community. The state-of-the-art approaches did not fully exploit the coupling of information contained in hyperspectral and multispectral images of the same scene. This paper proposes a new algorithm that overcomes this limitation. It accounts for both the high-resolution and the low-resolution information in the model by solving a set of least-squares problems. In addition, we provide exact recovery conditions for the super-resolution image in the noiseless case. Our simulations show that the proposed algorithm achieves very good reconstruction quality with a very low computational complexity.

A fundamental task in kernel methods is to pick nodes and weights, so as to approximate a given function from an RKHS by the weighted sum of kernel translates located at the nodes. This is the crux of kernel density estimation, kernel quadrature, or interpolation from discrete samples. Furthermore, RKHSs offer a convenient mathematical and computational framework. We introduce and analyse continuous volume sampling (VS), the continuous counterpart -for choosing node locations- of a discrete distribution introduced in (Deshpande & Vempala, 2006). Our contribution is theoretical: we prove almost optimal bounds for interpolation and quadrature under VS. While similar bounds already exist for some specific RKHSs using ad-hoc node constructions, VS offers bounds that apply to any Mercer kernel and depend on the spectrum of the associated integration operator. We emphasize that, unlike previous randomized approaches that rely on regularized leverage scores or determinantal point processes, evaluating the pdf of VS only requires pointwise evaluations of the kernel. VS is thus naturally amenable to MCMC samplers.

Two popular approaches to dimensionality reduction are principal component analysis, which projects onto a small number of well-chosen but non-interpretable directions, and feature selection, which selects a small number of the original features. Feature selection can be abstracted as selecting the subset of columns of a matrix $X \in \mathbb{R}^{N \times d}$ which minimize the approximation error, i.e., the norm of the residual after projecting $X$ onto the space spanned by the selected columns. Such a combinatorial optimization is usually impractical, and there has been interest in polynomial-cost, random subset selection algorithms that favour small values of this approximation error. We propose sampling from a projection determinantal point process, a repulsive distribution over column indices that favours diversity among the selected columns. We bound the ratio of the expected approximation error over the optimal error of PCA. These bounds improve over the state-of-the-art bounds of volume sampling when some realistic structural assumptions are satisfied for $X$. Numerical experiments suggest that our bounds are tight, and that our algorithms have comparable performance with the double phase algorithm, often considered the practical state-of-the-art.

In recent years, much research has been devoted to the restoration of Poissonian images using optimization-based methods. On the other hand, the derivation of efficient and general fully Bayesian approaches is still an active area of research and especially if standard regularization functions are used, e.g. the total variation (TV) norm. This paper proposes to use the recent split-and-augmented Gibbs sampler (SPA) to sample efficiently from an approximation of the initial target distribution when log-concave prior distributions are used. SPA embeds proximal Markov chain Monte Carlo (MCMC) algorithms to sample from possibly non-smooth log-concave full conditionals. The benefit of the proposed approach is illustrated on several experiments including different regularizers, intensity levels and with both analysis and synthesis approaches.

Markov chain Monte Carlo (MCMC) methods are an important class of computation techniques to solve Bayesian inference problems. Much recent research has been dedicated to scale these algorithms in high-dimensional settings by relying on powerful optimization tools such as gradient information or proximity operators. In a similar vein, this paper proposes a new Bayesian hierarchical model to solve large scale inference problems by taking inspiration from variable splitting methods. Similarly to the latter, the derived Gibbs sampler permits to divide the initial sampling task into simpler ones. As a result, the proposed Bayesian framework can lead to a faster sampling scheme than state-of-the-art methods by embedding them. The strength of the proposed methodology is illustrated on two often-studied image processing problems.

We study quadrature rules for functions living in an RKHS, using nodes sampled from a projection determinantal point process (DPP). DPPs are parametrized by a kernel, and we use a truncated and saturated version of the RKHS kernel. This natural link between the two kernels, along with DPP machinery, leads to relatively tight bounds on the quadrature error, that depends on the spectrum of the RKHS kernel. Finally, we experimentally compare DPPs to existing kernel-based quadratures such as herding, Bayesian quadrature, or continuous leverage score sampling. Numerical results confirm the interest of DPPs, and even suggest faster rates than our bounds in particular cases.

Principal component analysis is a widely used technique to perform dimension reduction. However, selecting a finite number of significant components is essential and remains a crucial issue. Only few attempts have proposed a probabilistic approach to adaptively select this number. This paper introduces a Bayesian nonparametric model to jointly estimate the principal components and the corresponding intrinsic dimension. More precisely, the observations are projected onto a random orthogonal basis which is assigned a prior distribution defined on the Stiefel manifold. Then the factor scores take benefit of an Indian buffet process prior to model the uncertainty related to the number of components. The parameters of interest as well as the nuisance parameters are finally inferred within a fully Bayesian framework via Monte Carlo sampling. The performances of the proposed approach are assessed thanks to experiments conducted on various examples.

Bivariate signals are commonly processed with the usual Fourier transform, using methods such as the rotary spectrum analysis. We show that bivariate signals can be efficiently processed using the Quaternion Fourier transform. A bivariate counterpart of the analytic signal is introduced, the quaternion embedding of a complex signal. It leads to identify natural parameters describing polarization properties, amplitude and phase of the signal. The properties of the quaternion short-term Fourier transform are studied and the polarization spectrogram is introduced. A synthetic example illustrates the relevance of the proposed approach.

The registration process is a key step for super-resolution (SR) reconstruction. More and more devices permit to overcome this bottleneck using a controlled positioning system, e.g., sensor shifting using a piezoelectric stage. This makes possible to acquire multiple images of the same scene at different controlled positions. Then, a fast SR algorithm can be used for efficient SR reconstruction. In this case, the optimal use of r(2) images for a resolution enhancement factor r is generally not enough to obtain satisfying results due to the random inaccuracy of the positioning system. Thus, we propose to take several images around each reference position. We study the error produced by the SR algorithm due to spatial uncertainty as a function of the number of images per position. We obtain a lower bound on the number of images that is necessary to ensure a given error upper bound with probability higher than some desired confidence level. Such results give precious hints to the design of SR systems.

This paper presents a novel approach to visual object tracking based on particle filtering. The tracked object is modelled by a sparse representation provided by dictionary learning. Such an approach permits to describe the target by a model of reduced dimension. The likelihood of a candidate region is built on a similarity measure between the sparse representations of a set of patches (at known positions) in the dictionary learnt from the reference template. Experimental validation is performed on various video sequences and shows the robustness of the proposed approach.

While the registration step is often problematic for superresolution, many microscopes and telescopes are now equipped with a piezoelectric mechanical system which permits to accurately control their motion (down to nanometers). Therefore one can use such devices to acquire multiple images of the same scene at various controlled positions. Then a fast super-resolution algorithm [1] can be used for efficient super-resolution. However the minimal use of r2images for a resolution enhancement factor r is generally not sufficient to obtain good results. We propose to take several images at positions randomly distributed close to each reference position. We study the number of images necessary to control the error resulting from the super-resolution algorithm by [1] due to the uncertainty on positions. The main result is a lower bound on the number of images to respect a given error upper bound with probability higher than a desired confidence level.

The mystery of the dynamics of the Sun is even deeper when it is quiet, i.e. when and where there are no coherent areas structured by strong magnetic field and called active regions. The corona that is part of the solar atmosphere is observed by extreme UV spatial telescopes for which higher resolutions are always desired: the best pixel resolution currently achievable is 500 km which still leaves much room for crucial hidden details. We present our work on the statistical analysis of quiet Sun images of the corona. These images exhibit multifractal properties and a model based on scale invariant stochastic processes, called fractionally integrated compound Poisson cascades, is able to reproduce their main statistical properties. We show that it can be used to develop a virtual super-resolution method that proposes plausible predictions of the high resolution information hidden below the pixel size of present observations. Such images may help physicists to test and calibrate on-board processing algorithms (compression, detection of events) for future missions.

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Multifractal random walks are defined as integrals of infinitely divisible stationary multifractal cascades with respect to fractional Brownian motion. Their key properties are studied, such as finiteness of moments and scaling, with respect to the chosen values of the self-similarity and infinite divisibility parameters. The range of these parameters is larger than that considered previously in the literature, and the cases of both exact and nonexact scale invariance are considered. Special attention is paid to various types of definitions of multifractal random walks. The resulting random walks are of interest in modeling multifractal processes whose marginals exhibit stationarity and symmetry.

We propose to model the statistics of natural images thanks to the large class of stochastic processes called Infinitely Divisible Cascades (IDC). IDC were first introduced in one dimension to provide multifractal time series to model the so-called intermittency phenomenon in hydrodynamical turbulence. We have extended the definition of scalar infinitely divisible cascades from 1 to N dimensions and commented on the relevance of such a model in fully developed turbulence in [1]. In this article, we focus on the particular 2 dimensional case. IDC appear as good candidates to model the statistics of natural images. They share most of their usual properties and appear to be consistent with several independent theoretical and experimental approaches of the literature. We point out the interest of IDC for applications to procedural texture synthesis.

Infinitely divisible cascades (IDC) were first introduced in one dimension to provide multifractal time series to model the so-called intermittency phenomenon in hydrodynamical turbulence. This work extends the definition of infinitely divisible cascades from 1 to N dimensions, and focuses on the 2D case in particular. 2D IDC appear as good candidates to model the statistics of natural images since they share most of their usual properties. The potential of IDC for applications to image processing (texture synthesis, denoising, etc) is emphasized.

Multiplicative processes, multifractals, and more recently also infinitely divisible cascades have seen increased popularity in a host of applications requiring versatile multiscale models, ranging from hydrodynamic turbulence to computer network traffic, from image processing to economics. The methodologies prevalent as of today rely to a large extent on iterative schemes used to produce infinite detail and repetitive structure across scales. While appealing, due to their simplicity, these constructions have limited applicability as they lead by default to power-law progression of moments through scales, to nonstationary increments and often to inherent log-periodic scaling which favors an exponential set of scales. This paper studies and develops a wide class of infinitely divisible cascades (IDC), thereby establishing the first reported cases of controllable scaling of moments in non-power-law form. Embedded in the framework of IDC, these processes exhibit stationary increments and scaling over a continuous range of scales. Criteria for convergence, further statistical properties, as well as MATLAB routines are provided.

We study the statistical performance of multiresolution (wavelet based) estimators commonly used for the estimation of the scaling exponents, /spl zeta/(q), of multifractal processes. So far, such studies have been conducted exclusively using the celebrated Mandelbrot's cascades. A new class of processes, compound Poisson cascades, with better statistical properties -stationary increments and continuous scale invariance - has recently been proposed in the literature. Making use of this new type of process, we show that the multiresolution estimators are characterised by a generic and systematic feature: beyond a critical order, q, (which is determined analytically), they fail to estimate the /spl zeta/(q) and present instead a linear behaviour in q. We study in detail this linearisation effect and show that it does not disappear in the limit of infinite observation duration, n, and that the parameters characterising it do not depend on n. We comment on its major practical consequences and on its having been mostly overlooked in applications.

This work is a contribution to the analysis of the procedure, based on wavelet coefficient partition functions, commonly used to estimate the Legendre multifractal spectrum. The procedure is applied to two examples, a fractional Brownian motion in multifractal time and a self-similar /spl alpha/-stable process, whose sample paths exhibit irregularities that by eye appear very close. We observe that, for the second example, this analysis results in a qualitatively inaccurate estimation of its multifractal spectrum, and a related masking of the /spl alpha/-stable nature of the process. We explain the origin of this error through a detailed analysis of the partition functions of the self-similar /spl alpha/-stable process. Such a study is made possible by the specific properties of the wavelet coefficients of such processes. We indicate how the estimation procedure might be modified to avoid such errors.

Infinitely divisible cascades are a model class previously introduced in the field of turbulence to describe the statistics of velocity fields. In this paper, using a wavelet reformulation of the cascades, we investigate their ability to analyze band model scaling properties of data and compare their fundamental ingredients to those of other scaling model classes such as self-similar and multifractal processes. We also propose an estimation procedure for the propagator or kernel of the cascades. Finally the cascade model is successfully applied to describe Internet TCP network traffic data, bringing new insights into their scaling properties and revealing a pitfall in existing techniques.