Metric Violation Distance: Hardness and Approximation

Fan, Chenglin; Raichel, Benjamin; Buskirk, Gregory Van

doi:10.1137/1.9781611975031.14

Cited by 11 publications

(42 citation statements)

References 20 publications

(17 reference statements)

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“…Hence, we would need a general metric repair algorithm. Fan, et al [18] present an algorithm that runs in θ(n 6 ) and Gilbert and Sonthalia [14] present an alternative algorithm that runs in O(n 5 ). Both of these algorithms are impractical and cannot be used on large data sets.…”

Section: Discussionmentioning

confidence: 99%

Unsupervised Metric Learning in Presence of Missing Data

Gilbert

Sonthalia

2018

2018 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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For many machine learning tasks, the input data lie on a low-dimensional manifold embedded in a highdimensional space and, because of this high-dimensional structure, most algorithms inefficient. The typical solution is to reduce the dimension of the input data using a standard dimension reduction algorithms such as ISOMAP, LAPLACIAN EIGENMAPS or LLES. This approach, however, does not always work in practice as these algorithms require that we have somewhat ideal data. Unfortunately, most data sets either have missing entries or unacceptably noisy values. That is, real data are far from ideal and we cannot use these algorithms directly.In this paper, we focus on the case when we have missing data. Some techniques, such as matrix completion, can be used to fill in missing data but these methods do not capture the non-linear structure of the manifold. Here, we present a new algorithm MR-MISSING that extends these previous algorithms and can be used to compute low dimensional representation on data sets with missing entries. We demonstrate the effectiveness of our algorithm by running three different experiments. We visually verify the effectiveness of our algorithm on synthetic manifolds, we numerically compare our projections against those computed by first filling in data using nlPCA and mDRUR on the MNIST data set, and we also show that we can do classification on MNIST with missing data. We also provide a theoretical guarantee for MR-MISSING under some simplifying assumptions.

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Section: Discussionmentioning

confidence: 99%

Unsupervised Metric Learning in Presence of Missing Data

Gilbert

Sonthalia

2018

2018 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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“…This kind of approach makes assumptions of certain Euclidean distance properties, e.g., symmetry and triangle inequality. However, such assumptions may not always hold, e.g., triangle inequality violations (TIVs) are commonly found in practice [9], [10]. To overcome these problems, matrix factorization are used [11], [12].…”

Section: Introductionmentioning

confidence: 99%

Matrix completion based adaptive sampling for measuring network delay with online support

2020

KSII TIIS

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End-to-end network delay plays an vital role in distributed services. This delay is used to measure QoS (Quality-of-Service). It would be beneficial to know all node-pair delay information, but unfortunately it is not feasible in practice because the use of active probing will cause a quadratic growth in overhead. Alternatively, using the measured network delay to estimate the unknown network delay is an economical method. In this paper, we adopt the state-of-the-art matrix completion technology to better estimate the network delay from limited measurements. Although the number of measurements required for an exact matrix completion is theoretically bounded, it is practically less helpful. Therefore, we propose an online adaptive sampling algorithm to measure network delay in which statistical leverage scores are used to select potential matrix elements. The basic principle behind is to sample the elements with larger leverage scores to keep the traits of important rows or columns in the matrix. The amount of samples is adaptively decided by a proposed stopping condition. Simulation results based on real delay matrix show that compared with the traditional sampling algorithm, our proposed sampling algorithm can provide better performance (smaller estimation error and less convergence pressure) at a lower cost (fewer samples and shorter processing time).

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“…However, due to noise, missing data, and other corruptions, in practice, these distances do not often adhere to a metric. Motivated by these observations and early work by Brickell et al [5], Fan et al [8] and, independently, Gilbert and Jain [11] respectively formulated the Metric Violation Distance (MVD) and the sparse metric repair (SMR) problems. Formally, the problem both authors studied was given a distance matrix, modify as few entries as possible so that the repaired distances satisfy a metric.…”

Section: Introductionmentioning

confidence: 99%

“…For example, Tenenbaum et al [19] showed that Isomap can be approximated by looking at only a few key points. In many of these cases, using the approximation algorithms in [8,11] may not produce meaningful results. As a more algorithmic example, let us consider traveling salesman problem (TSP).…”

Section: Introductionmentioning

confidence: 99%

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Generalized Metric Repair on Graphs

C.¹,

Sonthalia²

2018

Preprint

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Many modern data analysis algorithms either assume that or are considerably more efficient if the distances between the data points satisfy a metric. These algorithms include metric learning, clustering, and dimensionality reduction. Because real data sets are noisy, the similarity measures often fail to satisfy a metric. For this reason, Gilbert and Jain [11] and Fan, et al. [8] introduce the closely related problems of sparse metric repair and metric violation distance. The goal of each problem is to repair as few distances as possible to ensure that the distances between the data points satisfy a metric.We generalize these problems so as to no longer require all the distances between the data points. That is, we consider a weighted graph G with corrupted weights w and our goal is to find the smallest number of modifications to the weights so that the resulting weighted graph distances satisfy a metric. This problem is a natural generalization of the sparse metric repair problem and is more flexible as it takes into account different relationships amongst the input data points. As in previous work, we distinguish amongst the types of repairs permitted (decrease, increase, and general repairs). We focus on the increase and general versions and establish hardness results and show the inherent combinatorial structure of the problem. We then show that if we restrict to the case when G is a chordal graph, then the problem is fixed parameter tractable. We also present several classes of approximation algorithms. These include and improve upon previous metric repair algorithms for the special case when G = K n

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Metric Violation Distance: Hardness and Approximation

Cited by 11 publications

References 20 publications

Unsupervised Metric Learning in Presence of Missing Data

Unsupervised Metric Learning in Presence of Missing Data

Matrix completion based adaptive sampling for measuring network delay with online support

Generalized Metric Repair on Graphs

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