arXiv subjects

Christian Jutten

Publications and source records attributed to Christian Jutten.

At least 19 recordsLinked to original sources

arXiv2023

Improving embedding of graphs with missing data by soft manifolds

Embedding graphs in continous spaces is a key factor in designing and developing algorithms for automatic information extraction to be applied in diverse tasks (e.g., learning, inferring, predicting). The reliability of graph embeddings directly depends on how much the geometry of the continuous space matches the graph structure. Manifolds are mathematical structure that can enable to incorporate in their topological spaces the graph characteristics, and in particular nodes distances. State-of-the-art of manifold-based graph embedding algorithms take advantage of the assumption that the projection on a tangential space of each point in the manifold (corresponding to a node in the graph) would locally resemble a Euclidean space. Although this condition helps in achieving efficient analytical solutions to the embedding problem, it does not represent an adequate set-up to work with modern real life graphs, that are characterized by weighted connections across nodes often computed over sparse datasets with missing records. In this work, we introduce a new class of manifold, named soft manifold, that can solve this situation. In particular, soft manifolds are mathematical structures with spherical symmetry where the tangent spaces to each point are hypocycloids whose shape is defined according to the velocity of information propagation across the data points. Using soft manifolds for graph embedding, we can provide continuous spaces to pursue any task in data analysis over complex datasets. Experimental results on reconstruction tasks on synthetic and real datasets show how the proposed approach enable more accurate and reliable characterization of graphs in continuous spaces with respect to the state-of-the-art.

cs.LG↗

arXiv2023

Electrode Selection for Noninvasive Fetal Electrocardiogram Extraction using Mutual Information Criteria

Blind source separation (BSS) techniques have revealed to be promising approaches for, among other, biomedical signal processing applications. Specifically, for the noninvasive extraction of fetal cardiac signals from maternal abdominal recordings, where conventional filtering schemes have failed to extract the complete fetal ECG components. From previous studies, it is now believed that a carefully selected array of electrodes well-placed over the abdomen of a pregnant woman contains the required `information' for BSS, to extract the complete fetal components. Based on this idea, in previous works array recording systems and sensor selection strategies based on the Mutual Information (MI) criterion have been developed. In this paper the previous works have been extended, by considering the 3-dimensional aspects of the cardiac electrical activity. The proposed method has been tested on simulated and real maternal abdominal recordings. The results show that the new sensor selection strategy together with the MI criterion, can be effectively used to select the channels containing the most `information' concerning the fetal ECG components from an array of 72 recordings. The method is hence believed to be useful for the selection of the most informative channels in online applications, considering the different fetal positions and movements.

eess.SP↗

arXiv2022

A graph representation based on fluid diffusion model for data analysis: theoretical aspects and enhanced community detection

Representing data by means of graph structures identifies one of the most valid approach to extract information in several data analysis applications. This is especially true when multimodal datasets are investigated, as records collected by means of diverse sensing strategies are taken into account and explored. Nevertheless, classic graph signal processing is based on a model for information propagation that is configured according to heat diffusion mechanism. This system provides several constraints and assumptions on the data properties that might be not valid for multimodal data analysis, especially when large scale datasets collected from heterogeneous sources are considered, so that the accuracy and robustness of the outcomes might be severely jeopardized. In this paper, we introduce a novel model for graph definition based on fluid diffusion. The proposed approach improves the ability of graph-based data analysis to take into account several issues of modern data analysis in operational scenarios, so to provide a platform for precise, versatile, and efficient understanding of the phenomena underlying the records under exam, and to fully exploit the potential provided by the diversity of the records in obtaining a thorough characterization of the data and their significance. In this work, we focus our attention to using this fluid diffusion model to drive a community detection scheme, i.e., to divide multimodal datasets into many groups according to similarity among nodes in an unsupervised fashion. Experimental results achieved by testing real multimodal datasets in diverse application scenarios show that our method is able to strongly outperform state-of-the-art schemes for community detection in multimodal data analysis.

cs.SI↗

arXiv2021

Temporally Nonstationary Component Analysis; Application to Noninvasive Fetal Electrocardiogram Extraction

Objective: Mixtures of temporally nonstationary signals are very common in biomedical applications. The nonstationarity of the source signals can be used as a discriminative property for signal separation. Herein, a semi-blind source separation algorithm is proposed for the extraction of temporally nonstationary components from linear multichannel mixtures of signals and noises. Methods: A hypothesis test is proposed for the detection and fusion of temporally nonstationary events, by using ad hoc indexes for monitoring the first and second order statistics of the innovation process. As proof of concept, the general framework is customized and tested over noninvasive fetal cardiac recordings acquired from the maternal abdomen, over publicly available datasets, using two types of nonstationarity detectors: 1) a local power variations detector, and 2) a model-deviations detector using the innovation process properties of an extended Kalman filter. Results: The performance of the proposed method is assessed in presence of white and colored noise, in different signal-to-noise ratios. Conclusion and Significance: The proposed scheme is general and it can be used for the extraction of nonstationary events and sample deviations from a presumed model in multivariate data, which is a recurrent problem in many machine learning applications.

eess.SP↗

arXiv2021

A Hypothesis Testing Approach to Nonstationary Source Separation

The extraction of nonstationary signals from blind and semi-blind multivariate observations is a recurrent problem. Numerous algorithms have been developed for this problem, which are based on the exact or approximate joint diagonalization of second or higher order cumulant matrices/tensors of multichannel data. While a great body of research has been dedicated to joint diagonalization algorithms, the selection of the diagonalized matrix/tensor set remains highly problem-specific. Herein, various methods for nonstationarity identification are reviewed and a new general framework based on hypothesis testing is proposed, which results in a classification/clustering perspective to semi-blind source separation of nonstationary components. The proposed method is applied to noninvasive fetal ECG extraction, as case study.

eess.SP↗

arXiv2021

Enhancing ensemble learning and transfer learning in multimodal data analysis by adaptive dimensionality reduction

Modern data analytics take advantage of ensemble learning and transfer learning approaches to tackle some of the most relevant issues in data analysis, such as lack of labeled data to use to train the analysis models, sparsity of the information, and unbalanced distributions of the records. Nonetheless, when applied to multimodal datasets (i.e., datasets acquired by means of multiple sensing techniques or strategies), the state-of-theart methods for ensemble learning and transfer learning might show some limitations. In fact, in multimodal data analysis, not all observations would show the same level of reliability or information quality, nor an homogeneous distribution of errors and uncertainties. This condition might undermine the classic assumptions ensemble learning and transfer learning methods rely on. In this work, we propose an adaptive approach for dimensionality reduction to overcome this issue. By means of a graph theory-based approach, the most relevant features across variable size subsets of the considered datasets are identified. This information is then used to set-up ensemble learning and transfer learning architectures. We test our approach on multimodal datasets acquired in diverse research fields (remote sensing, brain-computer interfaces, photovoltaic energy). Experimental results show the validity and the robustness of our approach, able to outperform state-of-the-art techniques.

cs.LG↗

arXiv2021

Spectral Variability in Hyperspectral Data Unmixing: A Comprehensive Review

The spectral signatures of the materials contained in hyperspectral images, also called endmembers (EM), can be significantly affected by variations in atmospheric, illumination or environmental conditions typically occurring within an image. Traditional spectral unmixing (SU) algorithms neglect the spectral variability of the endmembers, what propagates significant mismodeling errors throughout the whole unmixing process and compromises the quality of its results. Therefore, large efforts have been recently dedicated to mitigate the effects of spectral variability in SU. This resulted in the development of algorithms that incorporate different strategies to allow the EMs to vary within a hyperspectral image, using, for instance, sets of spectral signatures known a priori, Bayesian, parametric, or local EM models. Each of these approaches has different characteristics and underlying motivations. This paper presents a comprehensive literature review contextualizing both classic and recent approaches to solve this problem. We give a detailed evaluation of the sources of spectral variability and their effect in image spectra. Furthermore, we propose a new taxonomy that organizes existing works according to a practitioner's point of view, based on the necessary amount of supervision and on the computational cost they require. We also review methods used to construct spectral libraries (which are required by many SU techniques) based on the observed hyperspectral image, as well as algorithms for library augmentation and reduction. Finally, we conclude the paper with some discussions and an outline of possible future directions for the field.

eess.IV↗

arXiv2019

Spectral Unmixing: A Derivation of the Extended Linear Mixing Model from the Hapke Model

In hyperspectral imaging, spectral unmixing aims at decomposing the image into a set of reference spectral signatures corresponding to the materials present in the observed scene and their relative proportions in every pixel. While a linear mixing model was used for a long time, the complex nature of the physical mixing processes, led to shift the community's attention towards nonlinear models and algorithms accounting for the variability of the endmembers. Such intra class variations are due to local changes in the physico-chemical composition of the materials, and to illumination changes. In the physical remote sensing community, a popular model accounting for illumination variability is the radiative transfer model proposed by Hapke. It is however too complex to be directly used in hyperspectral unmixing in a tractable way. Instead, the Extended Linear Mixing Model (ELMM) allows to easily unmix hyperspectral data accounting for changing illumination conditions. In this letter, we show that the ELMM can be obtained from the Hapke model by successive simplifiying physical assumptions, thus theoretically confirming its relevance to handle illumination induced variability in the unmixing problem.

eess.IV↗

arXiv2019

Spectral Variability Aware Blind Hyperspectral Image Unmixing Based on Convex Geometry

Hyperspectral image unmixing has proven to be a useful technique to interpret hyperspectral data, and is a prolific research topic in the community. Most of the approaches used to perform linear unmixing are based on convex geometry concepts, because of the strong geometrical structure of the linear mixing model. However, two main phenomena lead to question this model, namely nonlinearities and the spectral variability of the materials. Many algorithms based on convex geometry are still used when considering these two limitations of the linear model. A natural question is to wonder to what extent these concepts and tools (Intrinsic Dimensionality estimation, endmember extraction algorithms, pixel purity) can be safely used in these different scenarios. In this paper, we analyze them with a focus on endmember variability, assuming that the linear model holds. In the light of this analysis, we propose an integrated unmixing chain which tries to adress the shortcomings of the classical tools used in the linear case, based on our previously proposed extended linear mixing model. We show the interest of the proposed approach on simulated and real datasets.

eess.IV↗

arXiv2019

Hyperspectral Image Unmixing with Endmember Bundles and Group Sparsity Inducing Mixed Norms

Hyperspectral images provide much more information than conventional imaging techniques, allowing a precise identification of the materials in the observed scene, but because of the limited spatial resolution, the observations are usually mixtures of the contributions of several materials. The spectral unmixing problem aims at recovering the spectra of the pure materials of the scene (endmembers), along with their proportions (abundances) in each pixel. In order to deal with the intra-class variability of the materials and the induced spectral variability of the endmembers, several spectra per material, constituting endmember bundles, can be considered. However, the usual abundance estimation techniques do not take advantage of the particular structure of these bundles, organized into groups of spectra. In this paper, we propose to use group sparsity by introducing mixed norms in the abundance estimation optimization problem. In particular, we propose a new penalty which simultaneously enforces group and within group sparsity, to the cost of being nonconvex. All the proposed penalties are compatible with the abundance sum-to-one constraint, which is not the case with traditional sparse regression. We show on simulated and real datasets that well chosen penalties can significantly improve the unmixing performance compared to the naive bundle approach.

eess.IV↗

arXiv2019

Optimal Measurement Times for a Small Number of Measures of a Brownian Motion over a Finite Period

The measure timetable plays a critical role for the accuracy of the estimator. This article deals with the optimization of the schedule of measures for observing a random process in time using a Kalman filter, when the length of the process is finite and fixed, and a fixed number of measures are available. The measuring devices are allowed to differ. The mean variance of the estimator is chosen as criterion for optimality. The cases of $1$ or $2$ measures are studied in detail, and analytical formulas are provided.

eess.SP↗

arXiv2018

Schur's Lemma for Coupled Reducibility and Coupled Normality

Let $\mathcal A = \{A_{ij} \}_{i, j \in \mathcal I}$, where $\mathcal I$ is an index set, be a doubly indexed family of matrices, where $A_{ij}$ is $n_i \times n_j$. For each $i \in \mathcal I$, let $\mathcal V_i$ be an $n_i$-dimensional vector space. We say $\mathcal A$ is reducible in the coupled sense if there exist subspaces, $\mathcal U_i \subseteq \mathcal V_i$, with $\mathcal U_i \neq \{0\}$ for at least one $i \in \mathcal I$, and $\mathcal U_i \neq \mathcal V_i$ for at least one $i$, such that $A_{ij} (\mathcal U_j) \subseteq \mathcal U_i$ for all $i, j$. Let $\mathcal B = \{B_{ij} \}_{i, j \in \mathcal I}$ also be a doubly indexed family of matrices, where $B_{ij}$ is $m_i \times m_j$. For each $i \in \mathcal I$, let $X_i$ be a matrix of size $n_i \times m_i$. Suppose $A_{ij} X_j = X_i B_{ij}$ for all~$i, j$. We prove versions of Schur's Lemma for $\mathcal A, \mathcal B$ satisfying coupled irreducibility conditions. We also consider a refinement of Schur's Lemma for sets of normal matrices and prove corresponding versions for $\mathcal A, \mathcal B$ satisfying coupled normality and coupled irreducibility conditions.

math.RA↗

arXiv2015

Dynamical spectral unmixing of multitemporal hyperspectral images

In this paper, we consider the problem of unmixing a time series of hyperspectral images. We propose a dynamical model based on linear mixing processes at each time instant. The spectral signatures and fractional abundances of the pure materials in the scene are seen as latent variables, and assumed to follow a general dynamical structure. Based on a simplified version of this model, we derive an efficient spectral unmixing algorithm to estimate the latent variables by performing alternating minimizations. The performance of the proposed approach is demonstrated on synthetic and real multitemporal hyperspectral images.

cs.CV↗

arXiv2013

Recovery of Low-Rank Matrices under Affine Constraints via a Smoothed Rank Function

In this paper, the problem of matrix rank minimization under affine constraints is addressed. The state-of-the-art algorithms can recover matrices with a rank much less than what is sufficient for the uniqueness of the solution of this optimization problem. We propose an algorithm based on a smooth approximation of the rank function, which practically improves recovery limits on the rank of the solution. This approximation leads to a non-convex program; thus, to avoid getting trapped in local solutions, we use the following scheme. Initially, a rough approximation of the rank function subject to the affine constraints is optimized. As the algorithm proceeds, finer approximations of the rank are optimized and the solver is initialized with the solution of the previous approximation until reaching the desired accuracy. On the theoretical side, benefiting from the spherical section property, we will show that the sequence of the solutions of the approximating function converges to the minimum rank solution. On the experimental side, it will be shown that the proposed algorithm, termed SRF standing for Smoothed Rank Function, can recover matrices which are unique solutions of the rank minimization problem and yet not recoverable by nuclear norm minimization. Furthermore, it will be demonstrated that, in completing partially observed matrices, the accuracy of SRF is considerably and consistently better than some famous algorithms when the number of revealed entries is close to the minimum number of parameters that uniquely represent a low-rank matrix.

cs.IT↗

arXiv2013

Sparse Channel Estimation by Factor Graphs

The problem of estimating a sparse channel, i.e. a channel with a few non-zero taps, appears in various areas of communications. Recently, we have developed an algorithm based on iterative alternating minimization which iteratively detects the location and the value of the taps. This algorithms involves an approximate Maximum A Posteriori (MAP) probability scheme for detection of the location of taps, while a least square method is used for estimating the values at each iteration. In this work, based on the method of factor graphs and message passing algorithms, we will compute an exact solution for the MAP estimation problem. Indeed, we first find a factor graph model of this problem, and then perform the well-known min-sum algorithm on the edges of this graph. Consequently, we will find an exact estimator for the MAP problem that its complexity grows linearly with respect to the channel memory. By substituting this estimator in the mentioned alternating minimization method, we will propose an estimator that will nearly achieve the Cramer-Rao bound of the genie-aided estimation of sparse channels (estimation based on knowing the location of non-zero taps of the channel), while it can perform faster than most of the proposed algorithms in literature.

cs.IT↗

arXiv2013

On the Achievability of Cramér-Rao Bound In Noisy Compressed Sensing

Recently, it has been proved in Babadi et al. that in noisy compressed sensing, a joint typical estimator can asymptotically achieve the Cramer-Rao lower bound of the problem.To prove this result, this paper used a lemma,which is provided in Akcakaya et al,that comprises the main building block of the proof. This lemma is based on the assumption of Gaussianity of the measurement matrix and its randomness in the domain of noise. In this correspondence, we generalize the results obtained in Babadi et al by dropping the Gaussianity assumption on the measurement matrix. In fact, by considering the measurement matrix as a deterministic matrix in our analysis, we find a theorem similar to the main theorem of Babadi et al for a family of randomly generated (but deterministic in the noise domain) measurement matrices that satisfy a generalized condition known as The Concentration of Measures Inequality. By this, we finally show that under our generalized assumptions, the Cramer-Rao bound of the estimation is achievable by using the typical estimator introduced in Babadi et al.

cs.IT↗

arXiv2011

On the error of estimating the sparsest solution of underdetermined linear systems

Let A be an n by m matrix with m>n, and suppose that the underdetermined linear system As=x admits a sparse solution s0 for which ||s0||_0 < 1/2 spark(A). Such a sparse solution is unique due to a well-known uniqueness theorem. Suppose now that we have somehow a solution s_hat as an estimation of s0, and suppose that s_hat is only `approximately sparse', that is, many of its components are very small and nearly zero, but not mathematically equal to zero. Is such a solution necessarily close to the true sparsest solution? More generally, is it possible to construct an upper bound on the estimation error ||s_hat-s0||_2 without knowing s0? The answer is positive, and in this paper we construct such a bound based on minimal singular values of submatrices of A. We will also state a tight bound, which is more complicated, but besides being tight, enables us to study the case of random dictionaries and obtain probabilistic upper bounds. We will also study the noisy case, that is, where x=As+n. Moreover, we will see that where ||s0||_0 grows, to obtain a predetermined guaranty on the maximum of ||s_hat-s0||_2, s_hat is needed to be sparse with a better approximation. This can be seen as an explanation to the fact that the estimation quality of sparse recovery algorithms degrades where ||s0||_0 grows.

cs.IT↗

arXiv2010

Fast Sparse Decomposition by Iterative Detection-Estimation

Finding sparse solutions of underdetermined systems of linear equations is a fundamental problem in signal processing and statistics which has become a subject of interest in recent years. In general, these systems have infinitely many solutions. However, it may be shown that sufficiently sparse solutions may be identified uniquely. In other words, the corresponding linear transformation will be invertible if we restrict its domain to sufficiently sparse vectors. This property may be used, for example, to solve the underdetermined Blind Source Separation (BSS) problem, or to find sparse representation of a signal in an `overcomplete' dictionary of primitive elements (i.e., the so-called atomic decomposition). The main drawback of current methods of finding sparse solutions is their computational complexity. In this paper, we will show that by detecting `active' components of the (potential) solution, i.e., those components having a considerable value, a framework for fast solution of the problem may be devised. The idea leads to a family of algorithms, called `Iterative Detection-Estimation (IDE)', which converge to the solution by successive detection and estimation of its active part. Comparing the performance of IDE(s) with one of the most successful method to date, which is based on Linear Programming (LP), an improvement in speed of about two to three orders of magnitude is observed.

stat.ML↗