Consider a data set of vector-valued observations that consists of noisy inliers, which are explained well by a low-dimensional subspace, along with some number of outliers. This work describes a convex optimization problem, called reaper, that can reliably fit a low-dimensional model to this type of data. This approach parameterizes linear subspaces using orthogonal projectors and uses a relaxation of the set of orthogonal projectors to reach the convex formulation. The paper provides an efficient algorithm for solving the reaper problem, and it documents numerical experiments that confirm that reaper can dependably find linear structure in synthetic and natural data. In addition, when the inliers lie near a low-dimensional subspace, there is a rigorous theory that describes when reaper can approximate this subspace.

/pdf/robust-computation-of-linear-models-by-convex-relaxation-10x9997slh.pdf

Robust Computation of Linear Models by Convex Relaxation

Consider a dataset of vector-valued observations that consists of noisy inliers, which are explained well by a low-dimensional subspace, along with some number of outliers. This work describes a convex optimization problem, called REAPER, that can reliably fit a low-dimensional model to this type of data. This approach parameterizes linear subspaces using orthogonal projectors, and it uses a relaxation of the set of orthogonal projectors to reach the convex formulation. The paper provides an efficient algorithm for solving the REAPER problem, and it documents numerical experiments which confirm that REAPER can dependably find linear structure in synthetic and natural data. In addition, when the inliers lie near a low-dimensional subspace, there is a rigorous theory that describes when REAPER can approximate this subspace.

/pdf/robust-computation-of-linear-models-by-convex-relaxation-2ufvxh3pbq.pdf

Robust computation of linear models by convex relaxation

Robust ellipse and spheroid fitting

Given a set of points in the plane, the problem of existence and finding the least absolute deviations line is considered. The most important properties are stated and proved and two efficient methods for finding the best least absolute deviations line are proposed. Compared to other known methods, our proposed methods proved to be considerably more efficient.

/pdf/the-best-least-absolute-deviations-line-properties-and-two-4yvwxkqiuu.pdf

The best least absolute deviations line-properties and two efficient methods for its derivation

Piecewise smooth (PS) functions are perhaps the best-known examples of semismooth functions, which play key roles in the solution of nonsmooth equations and nonsmooth optimization. Recently, there have emerged other examples of semismooth functions, including the p -norm function ( 1 p ∞ ) defined on R n with n ≥ 2 , NCP functions, smoothing/penalty functions, and integral functions. These semismooth functions share the special property that their smooth point sets are locally connected around their nonsmooth points. By extending a result of Rockafellar, we show that the smooth point set of a PS function cannot have such a property. This shows that the above functions, though semismooth, are not PS. We call such functions almost smooth (AS). We show that the B-subdifferential of an AS function at a point has either one or infinitely many elements, which contrasts with PS functions whose B-subdifferential at a point has only a finite number of elements. We derive other useful properties of AS functions and sufficient conditions for a function to be AS. These results are then applied to various smoothing/penalty functions and integral functions.

On almost smooth functions and piecewise smooth functions

The problem of fitting a curve or surface to data has many applications. There are also many fitting criteria which can be used, and one which is widely used in metrology, for example, is that of minimizing the least squares norm of the orthogonal distances from the data points to the curve or surface. The Gauss–Newton method, in correct separated form, is a popular method for solving this problem. There is also interest in alternatives to least squares, and here we focus on the use of the l
 1 norm, which is traditionally regarded as important when the data contain wild points. The effectiveness of the Gauss–Newton method in this case is studied, with particular attention given to the influence of zero distances. Different aspects of the computation are illustrated by consideration of two particular fitting problems.

On the Gauss-Newton method for l1 orthogonal distance regression

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On the Gauss-Newton method for l1 orthogonal distance regression