This paper introduces a new conservative fusion method to exploit the correlated components within the estimation errors. Fusion is the process of combining multiple estimates of a given state to produce a new estimate with a smaller MSE. To perform the optimal linear fusion, the (centralized) covariance associated with the errors of all estimates is required. If it is partially unknown, the optimal fusion cannot be computed. Instead, a solution is to perform a conservative fusion. A conservative fusion provides a gain and a bound on the resulting MSE matrix which guarantees that the error is not underestimated. A well-known conservative fusion is the Covariance Intersection fusion. It has been modified to exploit the uncorrelated components within the errors. In this paper, it is further extended to exploit the correlated components as well. The resulting fusion rule is integrated into standard distributed algorithms where it allows exploiting the process noise observed by all agents. The improvement is confirmed by simulations.

Discrete Determinantal Point Processes (DPPs) have a wide array of potential applications for subsampling datasets. They are however held back in some cases by the high cost of sampling. In the worst-case scenario, the sampling cost scales as $O(n^3)$ where n is the number of elements of the ground set. A popular workaround to this prohibitive cost is to sample DPPs defined by low-rank kernels. In such cases, the cost of standard sampling algorithms scales as $O(np^2 + nm^2)$ where m is the (average) number of samples of the DPP (usually m $\ll$ n) and p the rank of the kernel used to define the DPP (m $\leq$ p $\leq$ n). The first term, $O(np^2)$, comes from a SVD-like step. We focus here on the second term of this cost, $O(nm^2)$, and show that it can be brought down to $O(nm + m^3 log m)$ without loss on the sampling’s exactness. In practice, we observe very substantial speedups compared to the classical algorithm as soon as n $>$ 1, 000. The algorithm described here is a close variant of the standard algorithm for sampling continuous DPPs, and uses rejection sampling. In the specific case of projection DPPs, we also show that any additional sample can be drawn in time $O(m^3 log m)$. Finally, an interesting by-product of the analysis is that a realisation from a DPP is typically contained in a subset of size $O(m \log m)$ formed using leverage score i.i.d. sampling.

We introduce new smoothing estimators for complex signals on graphs, based on a recently studied Determinantal Point Process (DPP). These estimators are built from subsets of edges and nodes drawn according to this DPP, making up trees and unicycles, i.e., connected components containing exactly one cycle. We provide a Julia implementation of these estimators and study their performance when applied to a ranking problem.

Another facet of the elegant link between random processes on graphs and Laplacian-based numerical linear algebra is uncovered: based on random spanning forests, novel Monte-Carlo estimators for graph signal smoothing are proposed. These random forests are sampled efficiently via a variant of Wilson's algorithm -in time linear in the number of edges. The theoretical variance of the proposed estimators are analyzed, and their application to several problems are considered, such as Tikhonov denoising of graph signals or semi-supervised learning for node classification on graphs.

When faced with a data set too large to be processed all at once, an obvious solution is to retain only part of it. In practice this takes a wide variety of different forms, and among them “coresets” are especially appealing. A coreset is a (small) weighted sample of the original data that comes with the following guarantee: a cost function can be evaluated on the smaller set instead of the larger one, with low relative error. For some classes of problems, and via a careful choice of sampling distribution (based on the so-called “sensitivity” metric), iid random sampling has turned to be one of the most successful methods for building coresets efficiently. However, independent samples are sometimes overly redundant, and one could hope that enforcing diversity would lead to better performance. The difficulty lies in proving coreset properties in non-iid samples. We show that the coreset property holds for samples formed with determinantal point processes (DPP). DPPs are interesting because they are a rare example of repulsive point processes with tractable theoretical properties, enabling us to prove general coreset theorems. We apply our results to both the $k$-means and the linear regression problems, and give extensive empirical evidence that the small additional computational cost of DPP sampling comes with superior performance over its iid counterpart. Of independent interest, we also provide analytical formulas for the sensitivity in the linear regression and $1$-means cases.

In this study, an original experimental setup is proposed to analyze the transfer function of an electroencephalographic(EEG) measurement chain: this setup allows to control electrodes inserted into an electrolyticmedium, and observe the electrical potentials at different points with electrodes connected to an electroencephalographicrecording system. Experimental transfer functions are obtained that enable to characterize theinfluence of the electrolyte, the electrode-electrolyte interfaces, and the medium itself. A theoretical modelcontaining so-called constant phase elements is then proposed and analysed to understand the behaviour ofthe transfer function in Bode or Nyquist representations. Experimental and theoretical systems are finallycompared.

In this letter, we investigated the connection between information and estimation measures for mismatched Gaussian models. In addition to the input prior mismatch, we take into account the noise mismatch and establish a new relation between relative entropy and excess mean square error. The derived formula shows that the input prior mismatch may be canceled by the noise mismatch. Finally, an example illustrates the impact of model mismatches on estimation accuracy.

A difficult aspect of multimodal estimation is the possible discrepancy between the sampling rates and/or the noise levels of the considered data. Many algorithms cope with these dissimilarities empirically. In this paper, we propose a conceptual analysis of multimodality where we try to find the "optimal" way of combining modalities. More specifically, we consider a simple Kalman filtering framework where several noisy sensors with different sampling frequences and noise variances regularly observe a hidden state. We experimentally underline some relationships between the sampling grids and the asymptotic variance of the maximum a posteriori (MAP) estimator. However, the explicit study of the asymptotic variance seems intractable even in the simplest cases. We describe a promising idea to circumvent this difficulty: exploiting a stochastic measurement model for which one can more easily study the average asymptotic behavior.

In this study we use the crossing tree of a signal for the purpose of analysis of H-sssi processes. The crossing tree performs an ad-hoc decomposition of a signal adapted to its dynamics, and represents a natural tool for the analysis of its local fluctuations. We present here a new multifractal formalism and a novel approach for estimating the spectrum of singularities of H-sssi processes using the crossing-tree. The performance of the crossing-tree based method is demonstrated in a numerical study. Its performance is also compared with state-of-the-art techniques based on wavelets, including wavelet-leaders.

Technological improvements have led to situations where data sets are sufficiently rich that in the interests of processing speed it is desirable to throw away samples that provide little additional information. This is referred to here as data sparsification. The first contribution is a study of a recently proposed data sparsification scheme; ideas from vector quantisation are used to assess its performance. Informed by this study, a modification of the data sparsification algorithm is proposed and applied to the problem of estimating a kernel-based measure of independence of two datasets. (Given i.i.d. observations from two random variables, x and y, the underlying problem is to determine whether or not x and y are independent of each other.) The second contribution of this paper is to make recursive an existing algorithm for measuring independence and able to operate on both raw data and on sparsified data generated by the aforementioned data sparsification algorithm. Compared with the original algorithm, the recursive algorithm is significantly faster due to its lower memory and computational requirements.

Wide sense stationary processes are a mainstay of classical signal processing. It is well known that they can be obtained by solving ordinary differential equations with constant coefficients whose right-hand side is a white noise. This paper addresses the extension of this construction to random fields defined on compact Lie groups. On an underlying compact Lie group, the paper studies left invariant second-order elliptic partial differential equations whose right-hand side is a spatial white noise. Quite often, the solution of a partial differential equation is not defined as a function but as a distribution. To adapt to this situation, the paper introduces a definition of wide sense stationary distributions on a compact Lie group. This is shown to be consistent with the more restricted definition of wide sense stationary fields given in a classic paper by Yaglom. It is proved that the solution of a partial differential equation, of the kind being studied, is a wide sense stationary distribution whose covariance structure is determined by the fundamental solution of the equation. As a concrete example, this paper describes the fundamental solution of the Helmholtz equation on the rotation group and the resulting covariance structure.

Granger causality considers the question of whether two time series exert causal influences on each other. Causality testing usually relies on prediction, i.e., if the prediction error of the first time series is reduced by taking measurements from the second one into account, then the latter is said to have a causal influence on the former. In this paper, a nonparametric framework based on functional estimation is proposed. Nonlinear prediction is performed via the Bayesian paradigm, using Gaussian processes. Some experiments illustrate the efficiency of the approach.

Iterated function systems (EFS) is a relevant model to produce fractal functions, whether deterministic (with strict self-similarity) or random (self-similar up to probability distribution). The basic idea of such a construction is to start with an initial function and then compress, dilate and translate it such that by doing so over and over again, we end up with a self-similar signal. This construction relies on a construction tree which has always been deterministic in the literature for signals. Here we introduce new fractals, called Galton Watson fractals, as fixed points of IFS with a random underlying construction tree and deterministic operators. We give a proof of the existence and uniqueness of a fixed point at the random and distribution level.

We study the link between networks of the pooling type and the problem of quantification. Pooling networks consist in parallel processors that are summed after having processed the same information. If the processors are simple threshold model of noisy neurons, the behavior of the network has the behavior of a quantizer. Using the compander approach to quantizers as well as the notion of density of levels, we study these networks and show that they are asymptotically equivalent to quantizers. Furthermore, we show how these devices can be infomax processors

The paper proposes a sequential Monte Carlo simulation method for equalizing a satellite communication channel. The main difficulties encountered are the nonlinear distortions due to the amplifier stage in the satellite. The aim of the method is to restore blindly the emitted message by considering a Bayesian approach. Thus, prior knowledge on the modeling of the nonlinearity is taken into account in the posterior distribution of the input sequence. Such a distribution is very difficult to study and thus motivates the implementation of Monte Carlo techniques. This approach makes it possible to solve the problem for a simplified model. The simulation scheme dealing with the complete transmission chain uses the method developed for the simplified model. Performance of the equalization algorithm is evaluated using bit error rate versus signal-to-noise ratio curves.

A definition of stochastic discrete scale invariance (DSI) is proposed and its properties studied. It is shown how the Lamperti (1962) transformation, which transforms stationarity in self-similarity, is also a means to connect processes deviating from stationarity and processes which are not exactly scale invariant: in particular we interpret DSI as the image of cyclostationarity. This theoretical result is employed to introduce a multiplicative spectral representation of DSI processes based on the Mellin transform, and preliminary remarks are given about estimation issues.

The Fourier transform of the output of a Volterra filter is given in terms of Schwartz distributions. This theory is developed in order to handle input signals that do not possess a Fourier transform in the usual sense. To illustrate this, the authors apply the result to the case of a sinusoidal input and to the case of a sampled and periodized signal.>

The authors address the problem of the optimality of the Volterra filters, in comparison with the optimal estimator in a minimum mean square error sense: the conditional mean. Notation and definition concerning cumulants, moments, generalized cumulants and Hermite tensors are introduced. The best Volterra estimation of a random variable x from a random observation vector Y is calculated, and an Edgeworth series expansion of the conditional mean, the optimal estimator, is given. In the case of a Gaussian observation vector, it is shown that the Volterra estimator and the expansion of the conditional mean are equal, providing a result concerning the optimality of Volterra filters.>