This paper develops a unified framework for upper bounding the minimax risk in high-dimensional parameter estimation problems. To this end, we study a quantized maximum likelihood estimator, where the estimator computes the likelihood for all points within a discrete cover, and outputs the candidate with the maximal likelihood. While this concept is straightforward, our analysis is quite delicate. It splits the competing candidates in the cover to small shells, and controls the number of candidates in each shell, as well as the probability that a candidate in the shell outscores a candidate which is close to the true parameter. We demonstrate the utility of our bounds by applying them to different Gaussian problems, and showing that they recover the optimal minimax rate for the Gaussian location model and the spiked Wigner Model. For the multi-reference alignment problem we obtain a novel minimax upper bound, which essentially places no assumptions on the signal of interest.
|Original language||American English|
|Title of host publication||2023 IEEE International Symposium on Information Theory, ISIT 2023|
|Publisher||Institute of Electrical and Electronics Engineers Inc.|
|Number of pages||6|
|State||Published - 2023|
|Event||2023 IEEE International Symposium on Information Theory, ISIT 2023 - Taipei, Taiwan, Province of China|
Duration: 25 Jun 2023 → 30 Jun 2023
|Name||IEEE International Symposium on Information Theory - Proceedings|
|Conference||2023 IEEE International Symposium on Information Theory, ISIT 2023|
|Country/Territory||Taiwan, Province of China|
|Period||25/06/23 → 30/06/23|
Bibliographical noteFunding Information:
While the risk for MRA in high dimensions was previously studied for a given prior distribution  or in a minimax setting under the assumption of “generic” signals , to the best of our knowledge, this is the first minimax upper bound for this problem which only assumes ∥θ∥2 = eeo(d). IV. ACKNOWLEDGEMENTS This work was supported by ISF under Grant 1641/21.
© 2023 IEEE.