Learning Permutations with Exponential Weights

@article{Helmbold2007LearningPW,
  title={Learning Permutations with Exponential Weights},
  author={David P. Helmbold and Manfred K. Warmuth},
  journal={J. Mach. Learn. Res.},
  year={2007},
  volume={10},
  pages={1705-1736}
}
We give an algorithm for learning a permutation on-line. The algorithm maintains its uncertainty about the target permutation as a doubly stochastic matrix. This matrix is updated by multiplying the current matrix entries by exponential factors. These factors destroy the doubly stochastic property of the matrix and an iterative procedure is needed to re-normalize the rows and columns. Even though the result of the normalization procedure does not have a closed form, we can still bound the… 
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121 Citations
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121 Citations

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References

SHOWING 1-10 OF 57 REFERENCES

Online kernel PCA with entropic matrix updates

TLDR
The main problem is the kernelization of an online PCA algorithm which belongs to this family of updates for density matrices, which involve a softmin calculation based on matrix logs and matrix exponentials.

Fourier Theoretic Probabilistic Inference over Permutations

TLDR
This paper uses the "low-frequency" terms of a Fourier decomposition to represent distributions over permutations compactly, and presents Kronecker conditioning, a novel approach for maintaining and updating these distributions directly in the Fourier domain, allowing for polynomial time bandlimited approximations.

Randomized PCA Algorithms with Regret Bounds that are Logarithmic in the Dimension

TLDR
An on-line algorithm for Principal Component Analysis that calculates the total expected quadratic approximation error of the best subspace chosen in hindsight plus some additional term that grows linearly in dimension of the subspace but logarithmically in the dimensions of the instances.

Quadratic Convergence for Scaling of Matrices

TLDR
A scaling algorithm is proposed that is conjectured to run much faster than any previous scaling algorithm, and it is shown that this algorithm converges quadratically for strictly scalable matrices, suggesting that the algorithm might always be fast.

Randomized Online PCA Algorithms with Regret Bounds that are Logarithmic in the Dimension

TLDR
The methodology in the expert setting of online learning is developed by giving an algorithm for learning as well as the best subset of experts of a certain size and then lifted to the matrix setting where the subsets of experts correspond to subspaces.

A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries

TLDR
A polynomial-time randomized algorithm for estimating the permanent of an arbitrary n × n matrix with nonnegative entries computes an approximation that is within arbitrarily small specified relative error of the true value of the permanent.

A Deterministic Strongly Polynomial Algorithm for Matrix Scaling and Approximate Permanents

TLDR
This work develops the first strongly polynomial-time algorithm for matrix scaling –– an important nonlinear optimization problem with many applications and suggests a simple new (slow)Polynomial time decision algorithm for bipartite perfect matching.

ON THE COMPLEXITY OF NONNEGATIVE-MATRIX SCALING

Polynomial approximation algorithms for belief matrix maintenance in identity management

TLDR
It is proved that even in cases in which the matrices are not exactly scalable, the problem can be solved to e-optimality in strongly polynomial time, improving the best known bound for the problem of scaling arbitrary nonnegative rectangular matrices to prescribed row and column sums.

Relative Loss Bounds for On-Line Density Estimation with the Exponential Family of Distributions

TLDR
This work considers on-line density estimation with a parameterized density from the exponential family and uses a Bregman divergence to derive and analyze each algorithm to design algorithms with the best possible relative loss bounds.
...