Tensor Completion via Integer Optimization: A Practical Algorithm Achieving Information-Theoretic Guarantees
Centrala begrepp
This paper develops a novel tensor completion algorithm that achieves both provable convergence in a linear number of oracle steps and the information-theoretic rate for estimation error.
Sammanfattning
The main content of this paper is as follows:
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The authors define a new tensor norm using a gauge function, which is constructed from a convex polytope with rank-1 tensors as its vertices. This gauge norm is shown to be related to tensor rank and have low Rademacher complexity.
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The tensor completion problem is formulated as a convex optimization problem using the gauge norm as a constraint. The authors prove that this problem is NP-hard to solve to arbitrary accuracy.
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To solve the tensor completion problem efficiently, the authors design a Blended Conditional Gradients (BCG) algorithm that uses a weak separation oracle based on integer linear optimization. This allows the algorithm to converge in a linear number of oracle calls while achieving the information-theoretic rate.
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Numerical experiments demonstrate the efficacy and scalability of the proposed algorithm. It outperforms benchmark methods in terms of normalized mean squared error (NMSE) for tensor completion, while remaining computationally efficient for tensors with up to 10 million entries.
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Tensor Completion via Integer Optimization
Statistik
The paper does not provide specific numerical values, but makes the following claims about the performance of the proposed algorithm:
It achieves the information-theoretic rate for tensor completion.
It scales well, handling tensors with up to 10 million entries.
It outperforms benchmark algorithms in terms of NMSE for tensor completion.
Citat
"Our algorithm is a step in this direction for the case of general tensors."
"The main idea behind our algorithm is to define the tensor completion problem using a gauge norm, and then use a Frank-Wolfe-like first-order optimization algorithm to solve the newly defined convex optimization formulation."
"Our numerical experiments demonstrate that this algorithm achieves the information-theoretic rate and is efficient for tensors with as large as ten million entries."
Djupare frågor
What are some potential applications of the proposed tensor completion algorithm that require positive and negative tensor entries
The proposed tensor completion algorithm, which can handle tensors with both positive and negative entries, has various potential applications. One such application is in social computing, where the algorithm can be used to analyze and predict social interactions and behaviors based on multidimensional data. Another application could be in the moment method for multivariate distribution analysis, where the algorithm can help in estimating missing or noisy data points to improve the accuracy of the distribution model. Additionally, in healthcare applications, the algorithm can be utilized for patient data analysis and prediction, especially in scenarios where the data is incomplete or contains errors.
How could the algorithm be further accelerated to achieve computation times similar to benchmark methods while maintaining its high accuracy
To further accelerate the algorithm and achieve computation times similar to benchmark methods while maintaining high accuracy, several strategies can be implemented. One approach could involve optimizing the implementation of the integer optimization solver used in the weak separation oracle. This optimization could include fine-tuning the solver parameters, leveraging parallel processing capabilities, and implementing efficient data structures and algorithms to speed up the computation. Additionally, exploring hardware acceleration options such as GPU computing could significantly reduce the computation time of the algorithm. Furthermore, algorithmic optimizations, such as refining the alternating maximization heuristic and exploring advanced optimization techniques, could also contribute to faster convergence and reduced computation times.
What are the implications of the NP-hardness result for the tensor completion problem using the gauge norm, and how might this inform future research directions
The NP-hardness result for the tensor completion problem using the gauge norm has significant implications for the complexity of the problem. It indicates that finding an optimal solution to the tensor completion problem is computationally challenging and may require exponential time in the worst case scenario. This result underscores the inherent difficulty of the problem and highlights the need for efficient algorithms and heuristics to tackle tensor completion tasks. Moving forward, future research directions could focus on developing approximation algorithms, exploring specialized optimization techniques tailored to tensor completion, and investigating the theoretical limits of computationally tractable solutions for NP-hard tensor completion problems. Additionally, research efforts could be directed towards understanding the structural properties of tensors that impact the complexity of completion tasks and devising novel approaches to address the computational challenges posed by tensor completion.
