Multiclass Total Variation Clustering

Bresson, Xavier; Laurent, Thomas; Uminsky, David; Brecht, James von

doi:10.21236/ada612811

Cited by 54 publications

(82 citation statements)

References 14 publications

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“…On the other hand, the transportation map T n induces the transportation plan π T n ∈ Γ(µ, µ n ) defined in (18). Hence,…”

Section: The Space T L Pmentioning

confidence: 99%

“…The machine learning tasks, such as classification and clustering, can often be given in terms of minimizing a functional on the graph representing the point cloud. Some of the fundamental approaches are based on minimizing graph cuts (graph perimeter) and related functionals (normalized cut, ratio cut, balanced cut), and more generally total variation on graphs [7,12,14,17,18,19,21,35,36,40,47,49,52,53]. We focus on total variation on graphs (of which graph cuts are a special case).…”

Section: Introductionmentioning

confidence: 99%

“…cut size, cut capacity), can be effectively used as a term in functionals which give a variational description to classification and clustering [12,17,14,19,21,20,18,35,36,40,47,52,53].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Continuum Limit of Total Variation on Point Clouds

Trillos

Slepčev

2015

Arch Rational Mech Anal

149

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ABSTRACT. We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, as the number of available data points increases, of graph-based machine learning algorithms for tasks such as clustering. In particular, we study when is the cut capacity, and more generally total variation, on these graphs a good approximation of the perimeter (total variation) in the continuum setting. We address this question in the setting of Γ-convergence. We obtain almost optimal conditions on the scaling, as number of points increases, of the size of the neighborhood over which the points are connected by an edge for the Γ-convergence to hold. Taking the limit is enabled by a transportation based metric which allows to suitably compare functionals defined on different point clouds.

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“…On the other hand, the transportation map T n induces the transportation plan π T n ∈ Γ(µ, µ n ) defined in (18). Hence,…”

Section: The Space T L Pmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Continuum Limit of Total Variation on Point Clouds

Trillos

Slepčev

2015

Arch Rational Mech Anal

149

View full text Add to dashboard Cite

show abstract

“…Therefore, our work opens up a new avenue of research, which will potentially also provide insight into other problems such as image labeling [Lellmann and Schnörr 2011] and clustering [Bresson et al 2013]. …”

Section: Limitations and Future Workmentioning

confidence: 99%

Local barycentric coordinates

et al. 2014

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Figure 1: Using LBC for 3D cage-based manipulation allows for local, smooth, and shape-aware deformations. Only parts near the manipulated control points are deformed, as indicated by the logarithmic color-coding of the displacement magnitude. AbstractBarycentric coordinates yield a powerful and yet simple paradigm to interpolate data values on polyhedral domains. They represent interior points of the domain as an affine combination of a set of control points, defining an interpolation scheme for any function defined on a set of control points. Numerous barycentric coordinate schemes have been proposed satisfying a large variety of properties. However, they typically define interpolation as a combination of all control points. Thus a local change in the value at a single control point will create a global change by propagation into the whole domain. In this context, we present a family of local barycentric coordinates (LBC), which select for each interior point a small set of control points and satisfy common requirements on barycentric coordinates, such as linearity, non-negativity, and smoothness. LBC are achieved through a convex optimization based on total variation, and provide a compact representation that reduces memory footprint and allows for fast deformations. Our experiments show that LBC provide more local and finer control on shape deformation than previous approaches, and lead to more intuitive deformation results.

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“…The total variation is one of the two most popular gauges of the distinctness between a pair of probability measures together with the Relative Entropy [27]. A paper presenting a general framework for multiclass total variation clustering can be found in [28]. We have characterized each element to be detected as a vector of speeds measured when a driver goes through it.…”

Section: Introductionmentioning

confidence: 99%

Detecting Different Road Infrastructural Elements Based on the Stochastic Characterization of Speed Patterns

Muñoz-Organero

Blázquez

2017

Journal of Advanced Transportation

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The automatic detection of road related information using data from sensors while driving has many potential applications such as traffic congestion detection or automatic routable map generation. This paper focuses on the automatic detection of road elements based on GPS data from on-vehicle systems. A new algorithm is developed that uses the total variation distance instead of the statistical moments to improve the classification accuracy. The algorithm is validated for detecting traffic lights, roundabouts, and street-crossings in a real scenario and the obtained accuracy (0.75) improves the best results using previous approaches based on statistical moments based features (0.71). Each road element to be detected is characterized as a vector of speeds measured when a driver goes through it. We first eliminate the speed samples in congested traffic conditions which are not comparable with clear traffic conditions and would contaminate the dataset. Then, we calculate the probability mass function for the speed (in 1 m/s intervals) at each point. The total variation distance is then used to find the similarity among different points of interest (which can contain a similar road element or a different one). Finally, a -NN approach is used for assigning a class to each unlabelled element.

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Multiclass Total Variation Clustering

Cited by 54 publications

References 14 publications

Continuum Limit of Total Variation on Point Clouds

Continuum Limit of Total Variation on Point Clouds

Local barycentric coordinates

Detecting Different Road Infrastructural Elements Based on the Stochastic Characterization of Speed Patterns

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