Classical Scaling Revisited

Shamai, Gil; Aflalo, Yonathan; Zibulevsky, Michael; Kimmel, Ron

doi:10.1109/iccv.2015.260

Cited by 13 publications

(28 citation statements)

References 30 publications

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“…On the other hand, d(z \ast , z j ) \leq \lambda for some z j \in Z m by the above algorithm, and we get from (25), (26), (27) that…”

Section: Ijmentioning

confidence: 97%

“…In this paper, we use a different approach to accelerate the Bregman projections algorithm. Inspired by the efficiency of low-rank approximations of geodesic distance matrices [26,27], we propose computing a low-rank approximation \bfitR 1 \bfitR t 2 of exp( - \bfitC /\varepsi ) where \bfitR 1 , \bfitR 2 are n \times m matrices with m \ll n. This approximation can be used to accelerate the Bregman projections algorithm by reducing the complexity of each iteration to O(mn).…”

mentioning

confidence: 99%

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Fast Entropic Regularized Optimal Transport Using Semidiscrete Cost Approximation

Tenetov¹,

Wolansky²,

Kimmel³

2018

SIAM J. Sci. Comput.

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The optimal transportation theory was successfully applied to different tasks on geometric domains as images and triangle meshes. In these applications the transport problem is defined on a Riemannian manifold with geodesic distance d(x, y). Usually, the cost function used is the geodesic distance d or the squared geodesic distance d 2. These choices result in the 1-Wasserstein distance, also known as the earth mover's distance (EMD), or the 2-Wasserstein distance. The entropy regularized optimal transport problem can be solved using the Bregman projection algorithm. This algorithm can be implemented using only matrix multiplications of matrix exp(-\bfitC /\varepsi) (pointwise exponent) and pointwise vector multiplications, where \bfitC is a cost matrix, and \varepsi is the regularization parameter. In this paper, we obtain a low-rank decomposition of this matrix and exploit it to accelerate the Bregman projection algorithm. Our low-rank decomposition is based on the semidiscrete approximation of the cost function, which is valid for a large family of cost functions, including d p (x, y), where p \geq 1. Our method requires the calculation of only a small portion of the distances.

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“…On the other hand, d(z \ast , z j ) \leq \lambda for some z j \in Z m by the above algorithm, and we get from (25), (26), (27) that…”

Section: Ijmentioning

confidence: 97%

mentioning

confidence: 99%

Fast Entropic Regularized Optimal Transport Using Semidiscrete Cost Approximation

Tenetov¹,

Wolansky²,

Kimmel³

2018

SIAM J. Sci. Comput.

Self Cite

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“…For meshes with up 15 × 10 3 vertices, the entire distance matrices were precomputed and stored in memory. For larger meshes, we used the distance approximation method suggested in [SAZK15] using a truncated geodesic distance basis of size 50 with 100 samples. This approximation tends to produce larger relative errors for smaller distances, therefore, in addition to the truncated geodesic distance basis, we pre-computed and stored distances from all vertices to their 10-ring neighbors.…”

Section: Implementation Detailsmentioning

confidence: 99%

Consistent Shape Matching via Coupled Optimization

Azencot

Dubrovina

Guibas

2019

Computer Graphics Forum

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We propose a new method for computing accurate point‐to‐point mappings between a pair of triangle meshes given imperfect initial correspondences. Unlike the majority of existing techniques, we optimize for a map while leveraging information from the inverse map, yielding results which are highly consistent with respect to composition of mappings. Remarkably, our method considers only a linear number of candidate points on the target shape, allowing us to work directly with high resolution meshes, and to avoid a delicate and possibly error‐prone up‐sampling procedure. Key to this dimensionality reduction is a novel candidate selection process, where the mapped points drift over the target shape, finalizing their location based on intrinsic distortion measures. Overall, we arrive at an iterative scheme where at each step we optimize for the map and its inverse by solving two relaxed Quadratic Assignment Problems using off‐the‐shelf optimization tools. We provide quantitative and qualitative comparison of our method with several existing techniques, and show that it provides a powerful matching tool when accurate and consistent correspondences are required.

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“…The interpolation operator H is similar but not identical to the BHA interpolation operator P. The difference lies in the fact that P does exact interpolation, meaning that the values at known points (the landmarks b) must be exactly equal to known values. In contrast, H allows some small error at the known points, where the amount of error is controlled by a scalar coefficient µ [21]. It is possible to linearly transform P into H: H = P(M bb + µI l + M bu P u ) −1 µ, where P u is defined as in Equation 6.…”

Section: Fmdsmentioning

confidence: 99%

Efficient, Sparse Representation of Manifold Distance Matrices for Classical Scaling

Turek

Huth

2018

2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition

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Geodesic distance matrices can reveal shape properties that are largely invariant to non-rigid deformations, and thus are often used to analyze and represent 3-D shapes. However, these matrices grow quadratically with the number of points. Thus for large point sets it is common to use a low-rank approximation to the distance matrix, which fits in memory and can be efficiently analyzed using methods such as multidimensional scaling (MDS). In this paper we present a novel sparse method for efficiently representing geodesic distance matrices using biharmonic interpolation. This method exploits knowledge of the data manifold to learn a sparse interpolation operator that approximates distances using a subset of points. We show that our method is 2x faster and uses 20x less memory than current leading methods for solving MDS on large point sets, with similar quality. This enables analyses of large point sets that were previously infeasible.

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Classical Scaling Revisited

Cited by 13 publications

References 30 publications

Fast Entropic Regularized Optimal Transport Using Semidiscrete Cost Approximation

Fast Entropic Regularized Optimal Transport Using Semidiscrete Cost Approximation

Consistent Shape Matching via Coupled Optimization

Efficient, Sparse Representation of Manifold Distance Matrices for Classical Scaling

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