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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

Many modern data analysis algorithms either assume or are considerably more efficient if the distances between the data points satisfy a metric. However, as real data sets are noisy, they often do not possess this fundamental property. For this reason, Gilbert and Jain [A. Gilbert and L. Jain, 2017] and Fan et al. [C. Fan et al., 2018] introduced the closely related sparse metric repair and metric violation distance problems. Given a matrix, representing all distances, the goal is to repair as few entries as possible to ensure they satisfy a metric. This problem was shown to be APX-hard, and an O(OPT^{1/3})-approximation was given, where OPT is the optimal solution size.
In this paper, we generalize the problem, by describing distances by a possibly incomplete positively weighted graph, where again our goal is to find the smallest number of weight modifications so that they satisfy a metric. This natural generalization is more flexible as it takes into account different relationships among the data points. We demonstrate the inherent combinatorial structure of the problem, and give an approximation-preserving reduction from MULTICUT, which is hard to approximate within any constant factor assuming UGC. Conversely, we show that for any fixed constant ς, for the large class of ς-chordal graphs, the problem is fixed parameter tractable, answering an open question from previous work. Call a cycle broken if it contains an edge whose weight is larger than the sum of all its other edges, and call the amount of this difference its deficit. We present approximation algorithms, one depending on the maximum number of edges in a broken cycle, and one depending on the number of distinct deficit values, both quantities which may naturally be small. Finally, we give improved analysis of previous algorithms for complete graphs.

Chenglin Fan, Anna C. Gilbert, Benjamin Raichel, Rishi Sonthalia, and Gregory Van Buskirk. Generalized Metric Repair on Graphs. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 25:1-25:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fan_et_al:LIPIcs.SWAT.2020.25, author = {Fan, Chenglin and Gilbert, Anna C. and Raichel, Benjamin and Sonthalia, Rishi and Van Buskirk, Gregory}, title = {{Generalized Metric Repair on Graphs}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {25:1--25:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.25}, URN = {urn:nbn:de:0030-drops-122727}, doi = {10.4230/LIPIcs.SWAT.2020.25}, annote = {Keywords: Approximation, FPT, Hardness, Metric Spaces} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

In this paper we consider the following sparse recovery problem. We have query access to a vector 𝐱 ∈ ℝ^N such that x̂ = 𝐅 𝐱 is k-sparse (or nearly k-sparse) for some orthogonal transform 𝐅. The goal is to output an approximation (in an 𝓁₂ sense) to x̂ in sublinear time. This problem has been well-studied in the special case that 𝐅 is the Discrete Fourier Transform (DFT), and a long line of work has resulted in sparse Fast Fourier Transforms that run in time O(k ⋅ polylog N). However, for transforms 𝐅 other than the DFT (or closely related transforms like the Discrete Cosine Transform), the question is much less settled.
In this paper we give sublinear-time algorithms - running in time poly(k log(N)) - for solving the sparse recovery problem for orthogonal transforms 𝐅 that arise from orthogonal polynomials. More precisely, our algorithm works for any 𝐅 that is an orthogonal polynomial transform derived from Jacobi polynomials. The Jacobi polynomials are a large class of classical orthogonal polynomials (and include Chebyshev and Legendre polynomials as special cases), and show up extensively in applications like numerical analysis and signal processing. One caveat of our work is that we require an assumption on the sparsity structure of the sparse vector, although we note that vectors with random support have this property with high probability.
Our approach is to give a very general reduction from the k-sparse sparse recovery problem to the 1-sparse sparse recovery problem that holds for any flat orthogonal polynomial transform; then we solve this one-sparse recovery problem for transforms derived from Jacobi polynomials. Frequently, sparse FFT algorithms are described as implementing such a reduction; however, the technical details of such works are quite specific to the Fourier transform and moreover the actual implementations of these algorithms do not use the 1-sparse algorithm as a black box. In this work we give a reduction that works for a broad class of orthogonal polynomial families, and which uses any 1-sparse recovery algorithm as a black box.

Anna Gilbert, Albert Gu, Christopher Ré, Atri Rudra, and Mary Wootters. Sparse Recovery for Orthogonal Polynomial Transforms. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 58:1-58:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gilbert_et_al:LIPIcs.ICALP.2020.58, author = {Gilbert, Anna and Gu, Albert and R\'{e}, Christopher and Rudra, Atri and Wootters, Mary}, title = {{Sparse Recovery for Orthogonal Polynomial Transforms}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {58:1--58:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.58}, URN = {urn:nbn:de:0030-drops-124653}, doi = {10.4230/LIPIcs.ICALP.2020.58}, annote = {Keywords: Orthogonal polynomials, Jacobi polynomials, sublinear algorithms, sparse recovery} }

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