Showing papers on "Singular value decomposition published in 2006"
21 Oct 2006
TL;DR: In this paper, the authors present a (1 + ∆)-approximation algorithm for the singular value decomposition of an m? n matrix A with M non-zero entries that requires 2 passes over the data and runs in time O(n 2 ).
Abstract: Recently several results appeared that show significant reduction in time for matrix multiplication, singular value decomposition as well as linear (\ell_ 2) regression, all based on data dependent random sampling. Our key idea is that low dimensional embeddings can be used to eliminate data dependence and provide more versatile, linear time pass efficient matrix computation. Our main contribution is summarized as follows. --Independent of the recent results of Har-Peled and of Deshpande and Vempala, one of the first -- and to the best of our knowledge the most efficient -- relative error (1 + \in) \parallel A - A_k \parallel _F approximation algorithms for the singular value decomposition of an m ? n matrix A with M non-zero entries that requires 2 passes over the data and runs in time O\left( {\left( {M(\frac{k} { \in } + k\log k) + (n + m)(\frac{k} { \in } + k\log k)^2 } \right)\log \frac{1} {\delta }} \right) --The first o(nd^{2}) time (1 + \in) relative error approximation algorithm for n ? d linear (\ell_2) regression. --A matrix multiplication and norm approximation algorithm that easily applies to implicitly given matrices and can be used as a black box probability boosting tool.
852 citations
TL;DR: Two simple and intuitive algorithms are presented which compute a description of a low-rank approximation of a singular value decomposition (SVD) to an matrix of rank not greater than a specified rank, and which are qualitatively faster than the SVD.
Abstract: In many applications, the data consist of (or may be naturally formulated as) an $m \times n$ matrix $A$. It is often of interest to find a low-rank approximation to $A$, i.e., an approximation $D$ to the matrix $A$ of rank not greater than a specified rank $k$, where $k$ is much smaller than $m$ and $n$. Methods such as the singular value decomposition (SVD) may be used to find an approximation to $A$ which is the best in a well-defined sense. These methods require memory and time which are superlinear in $m$ and $n$; for many applications in which the data sets are very large this is prohibitive. Two simple and intuitive algorithms are presented which, when given an $m \times n$ matrix $A$, compute a description of a low-rank approximation $D^{*}$ to $A$, and which are qualitatively faster than the SVD. Both algorithms have provable bounds for the error matrix $A-D^{*}$. For any matrix $X$, let $\|{X}\|_F$ and $\|{X}\|_2$ denote its Frobenius norm and its spectral norm, respectively. In the first algorithm, $c$ columns of $A$ are randomly chosen. If the $m \times c$ matrix $C$ consists of those $c$ columns of $A$ (after appropriate rescaling), then it is shown that from $C^TC$ approximations to the top singular values and corresponding singular vectors may be computed. From the computed singular vectors a description $D^{*}$ of the matrix $A$ may be computed such that $\mathrm{rank}(D^{*}) \le k$ and such that $$ \left\|A-D^{*}\right\|_{\xi}^{2} \le \min_{D:\mathrm{rank}(D)\le k} \left\|A-D\right\|_{\xi}^{2} + poly(k,1/c) \left\|{A}\right\|^2_F $$ holds with high probability for both $\xi = 2,F$. This algorithm may be implemented without storing the matrix $A$ in random access memory (RAM), provided it can make two passes over the matrix stored in external memory and use $O(cm+c^2)$ additional RAM. The second algorithm is similar except that it further approximates the matrix $C$ by randomly sampling $r$ rows of $C$ to form a $r \times c$ matrix $W$. Thus, it has additional error, but it can be implemented in three passes over the matrix using only constant additional RAM. To achieve an additional error (beyond the best rank $k$ approximation) that is at most $\epsilon\|{A}\|^2_F$, both algorithms take time which is polynomial in $k$, $1/\epsilon$, and $\log(1/\delta)$, where $\delta>0$ is a failure probability; the first takes time linear in $\mbox{max}(m,n)$ and the second takes time independent of $m$ and $n$. Our bounds improve previously published results with respect to the rank parameter $k$ for both the Frobenius and spectral norms. In addition, the proofs for the error bounds use a novel method that makes important use of matrix perturbation theory. The probability distribution over columns of $A$ and the rescaling are crucial features of the algorithms which must be chosen judiciously.
603 citations
TL;DR: A model (the pass-efficient model) is presented in which the efficiency of these and other approximate matrix algorithms may be studied and which is argued is well suited to many applications involving massive data sets.
Abstract: Motivated by applications in which the data may be formulated as a matrix, we consider algorithms for several common linear algebra problems. These algorithms make more efficient use of computational resources, such as the computation time, random access memory (RAM), and the number of passes over the data, than do previously known algorithms for these problems. In this paper, we devise two algorithms for the matrix multiplication problem. Suppose $A$ and $B$ (which are $m\times n$ and $n\times p$, respectively) are the two input matrices. In our main algorithm, we perform $c$ independent trials, where in each trial we randomly sample an element of $\{ 1,2,\ldots, n\}$ with an appropriate probability distribution ${\cal P}$ on $\{ 1,2,\ldots, n\}$. We form an $m\times c$ matrix $C$ consisting of the sampled columns of $A$, each scaled appropriately, and we form a $c\times n$ matrix $R$ using the corresponding rows of $B$, again scaled appropriately. The choice of ${\cal P}$ and the column and row scaling are crucial features of the algorithm. When these are chosen judiciously, we show that $CR$ is a good approximation to $AB$. More precisely, we show that $$ \left\|AB-CR\right\|_F = O(\left\|A\right\|_F \left\|B\right\|_F /\sqrt c) , $$ where $\|\cdot\|_F$ denotes the Frobenius norm, i.e., $\|A\|^2_F=\sum_{i,j}A_{ij}^2$. This algorithm can be implemented without storing the matrices $A$ and $B$ in RAM, provided it can make two passes over the matrices stored in external memory and use $O(c(m+n+p))$ additional RAM to construct $C$ and $R$. We then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm. In addition, we present a model (the pass-efficient model) in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets. In this model, the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm. The input matrices may be presented in any order of the entries (and not just row or column order), as is the case in many applications where, e.g., the data has been written in by multiple agents. In addition, the input matrices may be presented in a sparse representation, where only the nonzero entries are written.
535 citations
TL;DR: An identity for additive modivations of a singular value decomposition (SVD) to reflect updates, downdates, shifts, and edits of the data matrix is developed to set the stage for fast and ememory-efficient sequential algorithms for tracking singular values and subspaces.
509 citations
Book•
01 Jan 2006TL;DR: In this paper, the authors introduce a randomized procedure that, given an m × n matrix A and a positive integer k, approximates A with a matrix Z of rank k. The algorithm relies on applying a structured l × m random matrix R to each column of A,w herel is an integer near to, but greater than, k. And the resulting procedure can construct a rank-k approximation Z from the entries of A at a cost proportional to mn log(k) + l 2 (m + n).
Abstract: We introduce a randomized procedure that, given an m × n matrix A and a positive integer k, approximates A with a matrix Z of rank k. The algorithm relies on applying a structured l × m random matrix R to each column of A ,w herel is an integer near to, but greater than, k. The structure of R allows us to apply it to an arbitrary m × 1 vector at a cost proportional to m log(l); the resulting procedure can construct a rank-k approximation Z from the entries of A at a cost proportional to mn log(k) + l 2 (m + n). We prove
385 citations
01 Aug 2006
TL;DR: Experimental results show that the difference of the average recognition accuracy between the proposed incremental method and the batch-mode method is less than 1%.
Abstract: Principal component analysis (PCA) has been proven to be an efficient method in pattern recognition and image analysis. Recently, PCA has been extensively employed for face-recognition algorithms, such as eigenface and fisherface. The encouraging results have been reported and discussed in the literature. Many PCA-based face-recognition systems have also been developed in the last decade. However, existing PCA-based face-recognition systems are hard to scale up because of the computational cost and memory-requirement burden. To overcome this limitation, an incremental approach is usually adopted. Incremental PCA (IPCA) methods have been studied for many years in the machine-learning community. The major limitation of existing IPCA methods is that there is no guarantee on the approximation error. In view of this limitation, this paper proposes a new IPCA method based on the idea of a singular value decomposition (SVD) updating algorithm, namely an SVD updating-based IPCA (SVDU-IPCA) algorithm. In the proposed SVDU-IPCA algorithm, we have mathematically proved that the approximation error is bounded. A complexity analysis on the proposed method is also presented. Another characteristic of the proposed SVDU-IPCA algorithm is that it can be easily extended to a kernel version. The proposed method has been evaluated using available public databases, namely FERET, AR, and Yale B, and applied to existing face-recognition algorithms. Experimental results show that the difference of the average recognition accuracy between the proposed incremental method and the batch-mode method is less than 1%. This implies that the proposed SVDU-IPCA method gives a close approximation to the batch-mode PCA method
221 citations
18 Sep 2006
TL;DR: This paper describes a matrix decomposition formulation for Boolean data, the Discrete Basis Problem, and gives a simple greedy algorithm for solving it and shows how it can be solved using existing methods.
Abstract: Matrix decomposition methods represent a data matrix as a product of two smaller matrices: one containing basis vectors that represent meaningful concepts in the data, and another describing how the observed data can be expressed as combinations of the basis vectors. Decomposition methods have been studied extensively, but many methods return real-valued matrices. If the original data is binary, the interpretation of the basis vectors is hard. We describe a matrix decomposition formulation, the Discrete Basis Problem. The problem seeks for a Boolean decomposition of a binary matrix, thus allowing the user to easily interpret the basis vectors. We show that the problem is computationally difficult and give a simple greedy algorithm for solving it. We present experimental results for the algorithm. The method gives intuitively appealing basis vectors. On the other hand, the continuous decomposition methods often give better reconstruction accuracies. We discuss the reasons for this behavior.
200 citations
TL;DR: In this article, it was shown that a large class of fast recursive matrix multiplication algorithms are stable in a norm-wise sense, including LU decomposition, QR decomposition and linear equation solving, matrix inversion, solving least squares problems, eigenvalue problems and singular value decomposition.
Abstract: In an earlier paper, we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of $n$-by-$n$ matrices can be done by any algorithm in $O(n^{\omega + \eta})$ operations for any $\eta > 0$, then it can be done stably in $O(n^{\omega + \eta})$ operations for any $\eta > 0$. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in $O(n^{\omega + \eta})$ operations.
168 citations
TL;DR: It is found that Gaussian elimination can be used reliably to solve the MFS equations and the use of the singular value decomposition shows no improvement overGaussian elimination provided that the boundary condition is non-noisy.
Abstract: In this paper, we consider the accuracy and stability of implementing the method of fundamental solutions. In contrast to the results shown in [5], we find that Gaussian elimination can be used reliably to solve the MFS equations and the use of the singular value decomposition shows no improvement over Gaussian elimination provided that the boundary condition is non-noisy. However, for noisy boundary conditions, there is evidence that the singular value decomposition with truncation is more accurate than Gaussian elimination.
153 citations
TL;DR: This paper presents a robust watermarking approach for hiding grayscale watermarks into digital images by plugging the codebook concept into the singular value decomposition (SVD) and embeds the singular values of the original image into the watermark one to attain the lossless objective.
142 citations
TL;DR: Through this letter, some well-known properties of the SVD are extended in evaluating the generalizability of single hidden layer feedforward networks (SLFNs) with respect to the number of hidden layer neurons.
Abstract: In this letter, we attempt to quantify the significance of increasing the number of neurons in the hidden layer of a feedforward neural network architecture using the singular value decomposition (SVD). Through this, we extend some well-known properties of the SVD in evaluating the generalizability of single hidden layer feedforward networks (SLFNs) with respect to the number of hidden layer neurons. The generalization capability of the SLFN is measured by the degree of linear independency of the patterns in hidden layer space, which can be indirectly quantified from the singular values obtained from the SVD, in a postlearning step. A pruning/growing technique based on these singular values is then used to estimate the necessary number of neurons in the hidden layer. More importantly, we describe in detail properties of the SVD in determining the structure of a neural network particularly with respect to the robustness of the selected model
TL;DR: A model for representing and predicting distances in large-scale networks by matrix factorization is presented which can model suboptimal and asymmetric routing policies, an improvement on previous approaches and a scalable system is designed and implemented that predicts large numbers of network distances from limited samples of Internet measurements.
Abstract: The responsiveness of networked applications is limited by communications delays, making network distance an important parameter in optimizing the choice of communications peers. Since accurate global snapshots are difficult and expensive to gather and maintain, it is desirable to use sampling techniques in the Internet to predict unknown network distances from a set of partially observed measurements. This paper makes three contributions. First, we present a model for representing and predicting distances in large-scale networks by matrix factorization which can model suboptimal and asymmetric routing policies, an improvement on previous approaches. Second, we describe two algorithms-singular value decomposition and non-negative matrix factorization-for representing a matrix of network distances as the product of two smaller matrices. Third, based on our model and algorithms, we have designed and implemented a scalable system-Internet Distance Estimation Service (IDES)-that predicts large numbers of network distances from limited samples of Internet measurements. Extensive simulations on real-world data sets show that IDES leads to more accurate, efficient and robust predictions of latencies in large-scale networks than existing approaches
TL;DR: The maximum-likelihood estimates of a principal component analysis on the logit or probit scale are computed using majorization algorithms that iterate a sequence of weighted or unweighted singular value decompositions.
TL;DR: In this article, a damage detection method of mechanical system based on subspace identification concepts and statistical process techniques is presented, where measured time-responses of structures subjected to artificial or environmental vibrations are assembled to form the Hankel matrix, which is further factorised by performing singular value decomposition to obtain characteristic subspaces.
01 Sep 2006-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: The method discussed here is a version of the SVD-matching proposed by Scott and Longuet-Higgins and later modified by Pilu, that is elaborate in order to cope with large scale variations, and includes a keypoint descriptor that is robust to large scale and view-point changes.
Abstract: The paper tackles the problem of feature points matching between pair of images of the same scene. This is a key problem in computer vision. The method we discuss here is a version of the SVD-matching proposed by Scott and Longuet-Higgins and later modified by Pilu, that we elaborate in order to cope with large scale variations. To this end we add to the feature detection phase a keypoint descriptor that is robust to large scale and view-point changes. Furthermore, we include this descriptor in the equations of the proximity matrix that is central to the SVD-matching. At the same time we remove from the proximity matrix all the information about the point locations in the image, that is the source of mismatches when the amount of scene variation increases. The main contribution of this work is in showing that this compact and easy algorithm can be used for severe scene variations. We present experimental evidence of the improved performance with respect to the previous versions of the algorithm.
TL;DR: In this paper, it is shown that the singular value decomposition is not needed to derive the approximation error, the polar decomposition of the homogeneous transformation matrix being sufficient, and that the approximation does not have a minimum that would allow us to define naturally the characteristic length.
01 Jan 2006
TL;DR: It is shown that the EnKF can be implemented without access to the observation matrix, and only an observation function is needed; this greatly simplifies software design.
Abstract: We present several methods for the efficient implementation of the Ensemble Kalman Filter (EnKF) of Evensen. It is shown that the EnKF can be implemented without access to the observation matrix, and only an observation function is needed; this greatly simplifies software design. New implementations of the EnKF formulas are proposed, with linear computational complexity in the number of data points. These implementations are possible when the data covariance matrix is easy to decompose, such as a diagonal or a banded matrix, or given in a factored form as sample covariance. Unlike previous methods, our method for the former case uses Choleski decomposition on a small matrix from the Sherman-Morrison-Woodbury formula instead of SVD on a large matrix, and our method in the latter case does not impose any constraints on data randomization. One version of the EnKF formulas was implemented in a distributed parallel environment, using SCALAPACK and MPI.
Book•
01 Jan 2006TL;DR: A version of the fast multipole method (FMM) is described for charge distributions on the line, utilizing the matrix compression scheme described in [H. Cheng, Z. Gimbutas, P. Martinsson, and V. Rokhlin].
Abstract: A version of the fast multipole method (FMM) is described for charge distributions on the line. Previously published schemes of this type relied either on analytical representations of the potentials to be evaluated (multipoles, Legendre expansions, Taylor series, etc.) or on tailored representations that were constructed numerically (using, e.g., the singular value decomposition (SVD), artificial charges, etc.). The algorithm of this paper belongs to the second category, utilizing the matrix compression scheme described in [H. Cheng, Z. Gimbutas, P. G. Martinsson, and V. Rokhlin, SIAM J. Sci. Comput. 26 (2005), pp. 1389-1404]. The resulting scheme exhibits substantial improvements in the CPU time requirements. Furthermore, the scheme is applicable to a wide variety of potentials; in this respect, it is similar to the SVD-based FMMs. The performance of the method is illustrated with several numerical examples.
TL;DR: The algorithms proposed are able to identify algebraically a UDM using the second characteristic function of the observations, without any need of sparsity assumption on sources, and the blind identification problem is shown to reduce to a tensor decomposition.
TL;DR: In this paper, a model-based alternative to the preceding approach by providing prior distributions and posterior estimation for the rank of M and the components of its singular value decomposition is proposed.
Abstract: Many multivariate data-analysis techniques for an m × n matrix Y are related to the model Y = M + E, where Y is an m × n matrix of full rank and M is an unobserved mean matrix of rank K < (m ∧ n) Typically the rank of M is estimated in a heuristic way and then the least-squares estimate of M is obtained via the singular value decomposition of Y, yielding an estimate that can have a very high variance In this article we suggest a model-based alternative to the preceding approach by providing prior distributions and posterior estimation for the rank of M and the components of its singular value decomposition In addition to providing more accurate inference, such an approach has the advantage of being extendable to more general data-analysis situations, such as inference in the presence of missing data and estimation in a generalized linear modeling framework
Proceedings Article•
01 Apr 2006TL;DR: An algorithm based on the Generalized Hebbian Algorithm is described that allows the singular value decomposition of a dataset to be learned based on single observation pairs presented serially, which is interesting in the natural language domain, where very large datasets are often used, and datasets quickly become intractable.
Abstract: An algorithm based on the Generalized Hebbian Algorithm is described that allows the singular value decomposition of a dataset to be learned based on single observation pairs presented serially. The algorithm has minimal memory requirements, and is therefore interesting in the natural language domain, where very large datasets are often used, and datasets quickly become intractable. The technique is demonstrated on the task of learning word and letter bigram pairs from text.
03 Apr 2006
TL;DR: This paper presents the proposed Symbolic Aggregate Approximation of Singular Value Decomposition (SVD), the Discrete Fourier transform (DFT), the Adaptive Piecewise Constant Approximating (APCA), and the recently proposed SSAX for efficient similarity searching of time series data.
Abstract: Efficient and accurate similarity searching for a large amount of time series data set is an important but non-trivial problem. Many dimensionality reduction techniques have been proposed for effective representation of time series data in order to realize such similarity searching, including Singular Value Decomposition (SVD), the Discrete Fourier transform (DFT), the Adaptive Piecewise Constant Approximation (APCA), and the recently proposed Symbolic Aggregate Approximation (SAX).
TL;DR: It is shown in the paper that left and right quaternionic Householder transformations are different because of the non-commutative multiplication of quaternions and formulae for computing the Householder vector and matrix in each case are presented.
TL;DR: In this paper, the authors study a standard method of regularization by projections of the linear inverse problem Y = Af + ∈, where ∈ is a white Gaussian noise, and A is a known compact operator with singular values converging to zero with polynomial decay.
Abstract: We study a standard method of regularization by projections of the linear inverse problem Y = Af + ∈, where ∈ is a white Gaussian noise, and A is a known compact operator with singular values converging to zero with polynomial decay. The unknown function f is recovered by a projection method using the singular value decomposition of A. The bandwidth choice of this projection regularization is governed by a data-driven procedure which is based on the principle of risk hull minimization. We provide nonasymptotic upper bounds for the mean square risk of this method and we show, in particular, that in numerical simulations this approach may substantially improve the classical method of unbiased risk estimation.
Posted Content•
TL;DR: In this paper, a model-based alternative for multivariate data analysis is proposed by providing prior distributions and posterior estimation for the rank of a matrix and the components of its singular value decomposition.
Abstract: Many multivariate data analysis techniques for an $m\times n$ matrix $\m Y$ are related to the model $\m Y = \m M +\m E$, where $\m Y$ is an $m\times n$ matrix of full rank and $\m M$ is an unobserved mean matrix of rank $K< (m\wedge n)$. Typically the rank of $\m M$ is estimated in a heuristic way and then the least-squares estimate of $\m M$ is obtained via the singular value decomposition of $\m Y$, yielding an estimate that can have a very high variance. In this paper we suggest a model-based alternative to the above approach by providing prior distributions and posterior estimation for the rank of $\m M$ and the components of its singular value decomposition. In addition to providing more accurate inference, such an approach has the advantage of being extendable to more general data-analysis situations, such as inference in the presence of missing data and estimation in a generalized linear modeling framework.
TL;DR: A robust identification algorithm has been developed for linear, time-invariant, multiple-input single-output systems, with an emphasis on how this algorithm can be used to estimate the dynamic relationship between a set of neural recordings and related physiological signals.
Abstract: A robust identification algorithm has been developed for linear, time-invariant, multiple-input single-output systems, with an emphasis on how this algorithm can be used to estimate the dynamic relationship between a set of neural recordings and related physiological signals. The identification algorithm provides a decomposition of the system output such that each component is uniquely attributable to a specific input signal, and then reduces the complexity of the estimation problem by discarding those input signals that are deemed to be insignificant. Numerical difficulties due to limited input bandwidth and correlations among the inputs are addressed using a robust estimation technique based on singular value decomposition. The algorithm has been evaluated on both simulated and experimental data. The latter involved estimating the relationship between up to 40 simultaneously recorded motor cortical signals and peripheral electromyograms (EMGs) from four upper limb muscles in a freely moving primate. The algorithm performed well in both cases: it provided reliable estimates of the system output and significantly reduced the number of inputs needed for output prediction. For example, although physiological recordings from up to 40 different neuronal signals were available, the input selection algorithm reduced this to 10 neuronal signals that made significant contributions to the recorded EMGs.
TL;DR: In this paper, an analytical evaluation of a single-iteration Jacobian matrix based on a previously derived nodal adjoint representation is performed to assess the impact of important system parameters on the expected image quality.
Abstract: For non-linear inverse scattering problems utilizing Gauss-Newton methods, the Jacobian matrix encodes rich information concerning the system performance and algorithm efficiency. In this paper, we perform an analytical evaluation of a single-iteration Jacobian matrix based on a previously derived nodal adjoint representation. Concepts for studying linear ill-posed problems, such as the degree-of-ill-posedness, are used to assess the impact of important system parameters on the expected image quality. Analytical singular value decomposition (SVD) of the Jacobian matrix for a circular imaging domain is derived along with the numerical SVD for optimizing imaging system configurations. The results show significant reductions in the degree-of-ill-posedness when signal frequency, antenna array density and property parameter sampling are increased. Specifically, the decay rate in the singular spectrum of the Jacobian decreases monotonically with signal frequency being approximately 1/3 of its 0.1 GHz value at 3 GHz, is improved with antenna array density up to about 35 equally-spaced circumferentially positioned elements and drops significantly with increased property parameter sampling to more than twice the amount of measurement data. These results should serve as useful guidelines in the development of design specifications for an optimized hardware installation
TL;DR: A method is described for denoising multiple-echo data sets using singular value decomposition (SVD), which results in a reduction in noise in the individual TE images with minimal degradation of the spatial resolution and contrast.
TL;DR: A new variant of the gradzone analysis, which tries to address the problem of selecting the most relevant parameters for a history matching, is proposed using the singular value decomposition of the sensitivity matrix.
Abstract: Automatic history matching is based on minimizing an objective function that quantifies the mismatch between observed and simulated data. When using gradient-based methods for solving this optimization problem, a key point for the overall procedure is how the simulator delivers the necessary derivative information. In this paper, forward and adjoint methods for derivative calculation are discussed. Procedures for sensitivity matrix building, sensitivity matrix and transpose sensitivity matrix vector products are fully described. To show the usefulness of the derivative calculation algorithms, a new variant of the gradzone analysis, which tries to address the problem of selecting the most relevant parameters for a history matching, is proposed using the singular value decomposition of the sensitivity matrix. Application to a simple synthetic case shows that this procedure can reveal important information about the nature of the history-matching problem.