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We consider the rate-limited quantum-to-classical optimal transport in terms of output-constrained rate-distortion coding for discrete quantum measurement systems with limited classical common randomness. The main coding theorem provides the achievable rate region of a lossy measurement source coding for an exact construction of the destination distribution (or the equivalent quantum state) while maintaining a threshold of distortion from the source state according to a generally defined distortion observable. The constraint on the output space fixes the output distribution to an i.i.d. predefined probability mass function. Therefore, this problem can also be viewed as information-constrained optimal transport which finds the optimal cost of transporting the source quantum state to the destination state via an entanglement-breaking channel with limited communication rate and common randomness.
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Estimating the Rate-Distortion Function by Wasserstein Gradient Descent 37th Conference on Neural Information Processing Systems (NeurIPS)
In the theory of lossy compression, the rate-distortion (R-D) function R(D) describes how much a data source can be compressed (in bit-rate) at any given level of fidelity (distortion). Obtaining R(D) for a given data source establishes the fundamental performance limit for all compression algorithms. We propose a new method to estimate R(D) from the perspective of optimal transport. Unlike the classic Blahut--Arimoto algorithm which fixes the support of the reproduction distribution in advance, our Wasserstein gradient descent algorithm learns the support of the optimal reproduction distribution by moving particles. We prove its local convergence and analyze the sample complexity of our R-D estimator based on a connection to entropic optimal transport. Experimentally, we obtain comparable or tighter bounds than state-of-the-art neural network methods on low-rate sources while requiring considerably less tuning and computation effort. We also highlight a connection to maximum-likelihood deconvolution and introduce a new class of sources that can be used as test cases with known solutions to the R-D problem.
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- PAR ID:
- 10505296
- Editor(s):
- A. Oh; T. Naumann; A. Globerson; K. Saenko; M. Hardt; S. Levine
- Publisher / Repository:
- Curran Associates, Inc.
- Date Published:
- Journal Name:
- 37th Conference on Neural Information Processing Systems (NeurIPS)
- Volume:
- 36
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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