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Fast matrix completion without the condition number

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Moritz Hardt, Mary Wootters

Machine Learning · Submitted 15 Jul 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We give the first algorithm for Matrix Completion whose running time and sample complexity is polynomial in the rank of the unknown target matrix, linear in the dimension of the matrix, and logarithmic in the condition number of the matrix.

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arXiv ID1407.4070 (version 1)
Pith IDGJRWVRUL, Pith's permanent record of this paper and its review
CitationMary Wootters, Moritz Hardt. "Fast matrix completion without the condition number." arXiv:1407.4070 (2014). https://pith.science/paper/GJRWVRUL
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,992 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.96 · states limitation 0.31 · human participants 0.02 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.79 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We give the first algorithm for Matrix Completion whose running time and sample complexity is polynomial in the rank of the unknown target matrix, linear in the dimension of the matrix, and logarithmic in the condition number of the matrix. To the best of our knowledge, all previous algorithms either incurred a quadratic dependence on the condition number of the unknown matrix or a quadratic dependence on the dimension of the matrix in the running time. Our algorithm is based on a novel extension of Alternating Minimization which we show has theoretical guarantees under standard assumptions even in the presence of noise.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We give the first algorithm for Matrix Completion whose running time and sample complexity is polynomial in the rank of the unknown target matrix, linear in the dimension of the matrix, and logarithmic in the condition number of the matrix. contribution 0.96 0.96 0.89 asserted 1.00
s1 To the best of our knowledge, all previous algorithms either incurred a quadratic dependence on the condition number of the unknown matrix or a quadratic dependence on the dimension of the matrix in the running time. gap 0.80 0.11 0.09 asserted 0.79
s2 Our algorithm is based on a novel extension of Alternating Minimization which we show has theoretical guarantees under standard assumptions even in the presence of noise. method 0.93 0.94 0.10 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.98 · evidence from proof or derivation 0.99.

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  • full text
  • figures and tables
  • references and citing sentences
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We give the first algorithm for Matrix Completion whose running time and sample complexity is polynomial in the rank of the unknown target matrix, linear in the dimension of the matrix, and logarithmic in the condition number of the matrix. To the best of our knowledge, all previous algorithms either incurred a quadratic dependence on the condition number of the unknown matrix or a quadratic dependence on the dimension of the matrix in the running time. Our algorithm is based on a novel extension of Alternating Minimization which we show has theoretical guarantees under standard assumptions even in the presence of noise.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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