On geodesically convex formulations for the brascamp-lieb constant
© 2018 Aditya Bhaskara and Srivatsan Kumar. We consider two non-convex formulations for computing the optimal constant in the Brascamp-Lieb inequality corresponding to a given datum and show that they are geodesically log-concave on the manifold of positive definite matrices endowed with the Riemann...
| Format: | Article |
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| Language: | English |
| Published: |
2021-11-04T15:02:47Z.
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| Online Access: | Get fulltext |
| Summary: | © 2018 Aditya Bhaskara and Srivatsan Kumar. We consider two non-convex formulations for computing the optimal constant in the Brascamp-Lieb inequality corresponding to a given datum and show that they are geodesically log-concave on the manifold of positive definite matrices endowed with the Riemannian metric corresponding to the Hessian of the log-determinant function. The first formulation is present in the work of Lieb [15] and the second is new and inspired by the work of Bennett et al. [5]. Recent work of Garg et al. [12] also implies a geodesically log-concave formulation of the Brascamp-Lieb constant through a reduction to the operator scaling problem. However, the dimension of the arising optimization problem in their reduction depends exponentially on the number of bits needed to describe the Brascamp-Lieb datum. The formulations presented here have dimensions that are polynomial in the bit complexity of the input datum. |
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