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Recently, the precise performance of the Generalized LASSO algorithm forrecovering structured signals from compressed noisy measurements, obtained viai.i.d. Gaussian matrices, has been characterized. The analysis is based on aframework introduced by Stojnic and heavily relies on the use of Gordon'sGaussian min-max theorem (GMT), a comparison principle on Gaussian processes.As a result, corresponding characterizations for other ensembles of measurementmatrices have not been developed. In this work, we analyze the correspondingperformance of the ensemble of isotropically random orthogonal (i.r.o.)measurements. We consider the constrained version of the Generalized LASSO andderive a sharp characterization of its normalized squared error in thelarge-system limit. When compared to its Gaussian counterpart, our resultanalytically confirms the superiority in performance of the i.r.o. ensemble.Our second result, derives an asymptotic lower bound on the minimum conicsingular values of i.r.o. matrices. This bound is larger than the correspondingbound on Gaussian matrices. To prove our results we express i.r.o. matrices interms of Gaussians and show that, with some modifications, the GMT framework isstill applicable.

Isotropically Random Orthogonal Matrices: Performance of LASSO and Minimum Conic Singular Values