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...or. Thus, estimation of F from a family of matching points {xi ↔ x′i; i = 1, . . . , n} is a problem of linear regression. Typically, matches are computed by comparing local descriptors (such as SIFT =-=[16]-=-) and, for images of reasonable resolution, hundreds of matching points are found. The computation of the fundamental matrix would not be a problem in this context of large sample size / low dimension...

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...indexes i ∈ I ⊂ {1, ..., n}, called inliers. The indexes does not belonging to I will be referred to as outliers. The setting we are interested in is the one frequently encountered in computer vision =-=[13, 25]-=-: the dimensionality k of θ∗ is small as compared to n but the presence of outliers causes the complete failure of the least squares estimator. In what follows, we use the standard assumption that the...

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... be transformed into linear inequalities, except the last one which is a second order cone constraint. The problems of this type can be efficiently solved by various standard toolboxes such as SeDuMi =-=[22]-=- or TFOCS [1]. 2.4 Finite sample risk bound To provide theoretical guarantees for our estimator, we impose the by now usual assumption of restricted eigenvalues on a suitably chosen matrix. This assum...

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... Lasso and the Dantzig Selector (DS), rely on convex relaxation of `0-norm penalty leading to a convex program that involves the `1-norm of β. More precisely, for a given λ̄ > 0, the Lasso and the DS =-=[26, 4, 5, 3]-=- are defined as β̂ L = arg min β∈Rp { 1 2 ‖Y −Xβ‖22 + λ̄‖β‖1 } (Lasso) β̂ DS = arg min ‖β‖1 subject to ‖X>(Y −Xβ)‖∞ ≤ λ̄. (DS) The performance of these algorithms depends heavily on the choice of the ...

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... Lasso and the Dantzig Selector (DS), rely on convex relaxation of `0-norm penalty leading to a convex program that involves the `1-norm of β. More precisely, for a given λ̄ > 0, the Lasso and the DS =-=[26, 4, 5, 3]-=- are defined as β̂ L = arg min β∈Rp { 1 2 ‖Y −Xβ‖22 + λ̄‖β‖1 } (Lasso) β̂ DS = arg min ‖β‖1 subject to ‖X>(Y −Xβ)‖∞ ≤ λ̄. (DS) The performance of these algorithms depends heavily on the choice of the ...

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... Lasso and the Dantzig Selector (DS), rely on convex relaxation of `0-norm penalty leading to a convex program that involves the `1-norm of β. More precisely, for a given λ̄ > 0, the Lasso and the DS =-=[26, 4, 5, 3]-=- are defined as β̂ L = arg min β∈Rp { 1 2 ‖Y −Xβ‖22 + λ̄‖β‖1 } (Lasso) β̂ DS = arg min ‖β‖1 subject to ‖X>(Y −Xβ)‖∞ ≤ λ̄. (DS) The performance of these algorithms depends heavily on the choice of the ...

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... Lasso and the Dantzig Selector (DS), rely on convex relaxation of `0-norm penalty leading to a convex program that involves the `1-norm of β. More precisely, for a given λ̄ > 0, the Lasso and the DS =-=[26, 4, 5, 3]-=- are defined as β̂ L = arg min β∈Rp { 1 2 ‖Y −Xβ‖22 + λ̄‖β‖1 } (Lasso) β̂ DS = arg min ‖β‖1 subject to ‖X>(Y −Xβ)‖∞ ≤ λ̄. (DS) The performance of these algorithms depends heavily on the choice of the ...

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...indexes i ∈ I ⊂ {1, ..., n}, called inliers. The indexes does not belonging to I will be referred to as outliers. The setting we are interested in is the one frequently encountered in computer vision =-=[13, 25]-=-: the dimensionality k of θ∗ is small as compared to n but the presence of outliers causes the complete failure of the least squares estimator. In what follows, we use the standard assumption that the...

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...obust estimation of F are crucial steps for performing 3D reconstruction. Here, we apply the SRDS to the problem of estimation of F for 10 pairs of consecutive images provided by the fountain dataset =-=[21]-=-: the 11 images are shown at the bottom of Fig. 1. Using SIFT descriptors, we found more than 17.000 point matches in most pairs of images among the 10 pairs we are considering. The CPU time for compu...

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...d into linear inequalities, except the last one which is a second order cone constraint. The problems of this type can be efficiently solved by various standard toolboxes such as SeDuMi [22] or TFOCS =-=[1]-=-. 2.4 Finite sample risk bound To provide theoretical guarantees for our estimator, we impose the by now usual assumption of restricted eigenvalues on a suitably chosen matrix. This assumption, stated...

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..., `2 and `∞ norms, corresponding respectively to the sum of absolute values, the square root of the sum of squares and the maximum of the coefficients of v. 1 The term “fused” sparsity, introduced by =-=[27]-=-, originates from the case where Mβ is the discrete derivative of a signal β and the aim is to minimize the total variation, see [12, 19] for a recent overview and some asymptotic results. For general...

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...ceived special attention in last years, cf. [10] and the references therein, with the introduction of computationally efficient and theoretically justified σ-adaptive procedures the square-root Lasso =-=[2]-=- (a.k.a. scaled Lasso [24]) and the `1 penalized log-likelihood minimization [20]. In the present work, we are interested in the setting where β∗ is not necessarily sparse, but for a known q × p matri...

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... causes the complete failure of the least squares estimator. In what follows, we use the standard assumption that the matrix 1nA >A has diagonal entries equal to one. Following the ideas developed in =-=[6, 7, 8, 18, 15]-=-, we introduce a new vector ω ∈ Rn that serves to characterize the outliers. If an entry ωi of ω is nonzero, then the corresponding observation Yi is an outlier. This leads to the model: Y = Aθ∗ + √ n...

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...n the problem of multiple change-point detection, which is an important particular instance of fused sparsity. There are some workarounds to circumvent this limitation in that particular setting, see =-=[17, 11]-=-. The extension of these kind of arguments to the case of unknown σ∗ is an open problem we intend to tackle in the near future. 3 Application to robust estimation This methodology can be applied in th...

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... the coefficients of v. 1 The term “fused” sparsity, introduced by [27], originates from the case where Mβ is the discrete derivative of a signal β and the aim is to minimize the total variation, see =-=[12, 19]-=- for a recent overview and some asymptotic results. For general matrices M, tight risk bounds were proved in [14]. We adopt here this framework of general M and aim at designing a computationally effi...

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...n the problem of multiple change-point detection, which is an important particular instance of fused sparsity. There are some workarounds to circumvent this limitation in that particular setting, see =-=[17, 11]-=-. The extension of these kind of arguments to the case of unknown σ∗ is an open problem we intend to tackle in the near future. 3 Application to robust estimation This methodology can be applied in th...

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... causes the complete failure of the least squares estimator. In what follows, we use the standard assumption that the matrix 1nA >A has diagonal entries equal to one. Following the ideas developed in =-=[6, 7, 8, 18, 15]-=-, we introduce a new vector ω ∈ Rn that serves to characterize the outliers. If an entry ωi of ω is nonzero, then the corresponding observation Yi is an outlier. This leads to the model: Y = Aθ∗ + √ n...

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... the arguments used to prove analogous results for ordinary sparsity, but contains some qualitatively novel ideas. More precisely, the cornerstone of the proof of risk bounds for the Dantzig selector =-=[4, 3, 9]-=- is that the true parameter β∗ is a feasible solution. In our case, this argument cannot be used anymore. Our proposal is then to specify another vector β̃ that simultaneously satisfies the following ...

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...h the introduction of computationally efficient and theoretically justified σ-adaptive procedures the square-root Lasso [2] (a.k.a. scaled Lasso [24]) and the `1 penalized log-likelihood minimization =-=[20]-=-. In the present work, we are interested in the setting where β∗ is not necessarily sparse, but for a known q × p matrix M, the vector Mβ∗ is sparse. We call this setting “fused sparsity scenario”. 1W...

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... causes the complete failure of the least squares estimator. In what follows, we use the standard assumption that the matrix 1nA >A has diagonal entries equal to one. Following the ideas developed in =-=[6, 7, 8, 18, 15]-=-, we introduce a new vector ω ∈ Rn that serves to characterize the outliers. If an entry ωi of ω is nonzero, then the corresponding observation Yi is an outlier. This leads to the model: Y = Aθ∗ + √ n...

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...most applications, the latter is unavailable. It is therefore vital to design statistical procedures that estimate β and σ in a joint fashion. This topic received special attention in last years, cf. =-=[10]-=- and the references therein, with the introduction of computationally efficient and theoretically justified σ-adaptive procedures the square-root Lasso [2] (a.k.a. scaled Lasso [24]) and the `1 penali...

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... causes the complete failure of the least squares estimator. In what follows, we use the standard assumption that the matrix 1nA >A has diagonal entries equal to one. Following the ideas developed in =-=[6, 7, 8, 18, 15]-=-, we introduce a new vector ω ∈ Rn that serves to characterize the outliers. If an entry ωi of ω is nonzero, then the corresponding observation Yi is an outlier. This leads to the model: Y = Aθ∗ + √ n...

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...in last years, cf. [10] and the references therein, with the introduction of computationally efficient and theoretically justified σ-adaptive procedures the square-root Lasso [2] (a.k.a. scaled Lasso =-=[24]-=-) and the `1 penalized log-likelihood minimization [20]. In the present work, we are interested in the setting where β∗ is not necessarily sparse, but for a known q × p matrix M, the vector Mβ∗ is spa...

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...sumption of restricted eigenvalues on a suitably chosen matrix. This assumption, stated in Definition 2.1 below, was introduced and thoroughly discussed by [3]; we also refer the interested reader to =-=[28]-=-. Définition 2.1. We say that a n× q matrix A satisfies the restricted eigenvalue condition RE(s, 1), if κ(s, 1) ∆ = min |J|≤s min ‖δJc‖1≤‖δJ‖1 ‖Aδ‖2√ n‖δJ‖2 > 0. We say that A satisfies the strong r...

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...eally an estimator but rather an oracle since it exploits the knowledge of the true σ∗. This is why the accuracy in estimating σ∗ is not reported in Table 1. To reduce the well known bias toward zero =-=[4, 23]-=-, we performed a post-processing for all of three procedures. It consisted in computing least squares estimators after removing all the covariates corresponding to vanishing coefficients of the estima...

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... causes the complete failure of the least squares estimator. In what follows, we use the standard assumption that the matrix 1nA >A has diagonal entries equal to one. Following the ideas developed in =-=[6, 7, 8, 18, 15]-=-, we introduce a new vector ω ∈ Rn that serves to characterize the outliers. If an entry ωi of ω is nonzero, then the corresponding observation Yi is an outlier. This leads to the model: Y = Aθ∗ + √ n...

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...crete derivative of a signal β and the aim is to minimize the total variation, see [12, 19] for a recent overview and some asymptotic results. For general matrices M, tight risk bounds were proved in =-=[14]-=-. We adopt here this framework of general M and aim at designing a computationally efficient procedure capable to handle the situation of unknown noise level and for which we are able to provide theor...