Solving a Special Type of Optimal Transport Problem by a Modified Hungarian Algorithm

• URL: http://arxiv.org/abs/2210.16645v1
• Date: Sat, 29 Oct 2022 16:28:46 GMT
• Title: Solving a Special Type of Optimal Transport Problem by a Modified Hungarian Algorithm
• Authors: Yiling Xie, Yiling Luo, Xiaoming Huo
• Abstract summary: We study a special type of transport (OT) problem and propose a modified Hungarian algorithm to solve it exactly. For an OT problem between marginals with $m$ and $n$ atoms, the computational complexity of the proposed algorithm is $O(m2n)$.
• Score: 2.1485350418225244
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We observe that computing empirical Wasserstein distance in the independence test is an optimal transport (OT) problem with a special structure. This observation inspires us to study a special type of OT problem and propose a modified Hungarian algorithm to solve it exactly. For an OT problem between marginals with $m$ and $n$ atoms ($m\geq n$), the computational complexity of the proposed algorithm is $O(m^2n)$. Computing the empirical Wasserstein distance in the independence test requires solving this special type of OT problem, where we have $m=n^2$. The associate computational complexity of our algorithm is $O(n^5)$, while the order of applying the classic Hungarian algorithm is $O(n^6)$. Numerical experiments validate our theoretical analysis. Broader applications of the proposed algorithm are discussed at the end.

Related papers

• Sum-of-Squares inspired Quantum Metaheuristic for Polynomial Optimization with the Hadamard Test and Approximate Amplitude Constraints [76.53316706600717]
  Recently proposed quantum algorithm arXiv:2206.14999 is based on semidefinite programming (SDP) We generalize the SDP-inspired quantum algorithm to sum-of-squares. Our results show that our algorithm is suitable for large problems and approximate the best known classicals.
  arXiv  Detail & Related papers  (2024-08-14T19:04:13Z)
• Robust Approximation Algorithms for Non-monotone $k$-Submodular Maximization under a Knapsack Constraint [0.0]
  Two deterministic approximation algorithms are presented for the problem of non-monotone $k$-submodular complexity under a knapsack constraint. Our algorithms provide constant approximation ratios within only $O(nk)$ query complexity for the non-monotone objective.
  arXiv  Detail & Related papers  (2023-09-21T12:42:52Z)
• Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials [50.90125395570797]
  We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $mathbbRd$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/epsilon)O(k)$, whereepsilon>0$ is the target accuracy.
  arXiv  Detail & Related papers  (2023-07-24T14:37:22Z)
• Information-Computation Tradeoffs for Learning Margin Halfspaces with Random Classification Noise [50.64137465792738]
  We study the problem of PAC $gamma$-margin halfspaces with Random Classification Noise. We establish an information-computation tradeoff suggesting an inherent gap between the sample complexity of the problem and the sample complexity of computationally efficient algorithms.
  arXiv  Detail & Related papers  (2023-06-28T16:33:39Z)
• Distributed Quantum-classical Hybrid Shor's Algorithm [1.7396274240172125]
  Shor's algorithm is considered as one of the most significant quantum algorithms of the Noisy Intermediate Quantum era. To reduce the resources required for Shor's algorithm, we propose a new distributed quantum-classical hybrid order-finding algorithm.
  arXiv  Detail & Related papers  (2023-04-24T13:52:05Z)
• Deterministic Nonsmooth Nonconvex Optimization [94.01526844386977]
  We show that randomization is necessary to obtain a dimension-free dimension-free algorithm. Our algorithm yields the first deterministic dimension-free algorithm for optimizing ReLU networks.
  arXiv  Detail & Related papers  (2023-02-16T13:57:19Z)
• Private estimation algorithms for stochastic block models and mixture models [63.07482515700984]
  General tools for designing efficient private estimation algorithms. First efficient $(epsilon, delta)$-differentially private algorithm for both weak recovery and exact recovery.
  arXiv  Detail & Related papers  (2023-01-11T09:12:28Z)
• Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end [114.3957763744719]
  We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems. We show that the algorithm finds the optimal solution in time $O*(2(0.5-c)n)$ for an $n$-independent constant $c$. We also show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case.
  arXiv  Detail & Related papers  (2022-12-03T02:45:23Z)
• An Accelerated Stochastic Algorithm for Solving the Optimal Transport Problem [2.1485350418225244]
  A primal-dual accelerated gradient descent with variance reduction algorithm (PDASGD) is proposed to solve linear-constrained optimization problems. PDASGD enjoys the best-known computational complexity -- $widetildemathcalO(n2/epsilon)$, where $n$ is the number of atoms, and $epsilon>0$ is the accuracy.
  arXiv  Detail & Related papers  (2022-03-02T01:16:10Z)
• Choosing the Right Algorithm With Hints From Complexity Theory [16.33500498939925]
  We show that the Metropolis algorithm is clearly the best of all algorithms regarded for reasonable problem sizes. An artificial algorithm of this type having an $O(n log n)$ runtime leads to the result that the significance-based compact genetic algorithm (sig-cGA) can solve the DLB problem in time $O(n log n)$ with high probability.
  arXiv  Detail & Related papers  (2021-09-14T11:12:32Z)
• Time and Query Optimal Quantum Algorithms Based on Decision Trees [2.492300648514128]
  We show that a quantum algorithm can be implemented in time $tilde O(sqrtGT)$. Our algorithm is based on non-binary span programs and their efficient implementation.
  arXiv  Detail & Related papers  (2021-05-18T06:51:11Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.