Estimating a binary vector from noisy linear measurements is a prototypical problem for MIMO systems. A popular algorithm, called the box-relaxation decoder, estimates the target signal by solving a least squares problem with convex constraints. This paper shows that the performance of the algorithm, measured by the number of incorrectly-decoded bits, has a limiting Poisson law. This occurs when the sampling ratio and noise variance, two key parameters of the problem, follow certain scalings as the system dimension grows. Moreover, at a well-defined threshold, the probability of perfect recovery is shown to undergo a phase transition that can be characterized by the Gumbel distribution. Numerical simulations corroborate these theoretical predictions, showing that they match the actual performance of the algorithm even in moderate system dimensions.
We study a spectral initialization method that serves as a key ingredient in recent work on using efficient iterative algorithms for estimating signals in nonconvex settings. Unlike previous analysis in the literature, which is restricted to the phase retrieval setting and which provides only performance bounds, we consider arbitrary generalized linear sensing models and present a precise asymptotic characterization of the performance of the spectral method in the high-dimensional regime. Our analysis reveals a phase transition phenomenon that depends on the sampling ratio. When the ratio is below a minimum threshold, the estimates given by the spectral method are no better than a random guess drawn uniformly from the hypersphere; above a maximum threshold, however, the estimates become increasingly aligned with the target signal. The computational complexity of the spectral method is also markedly different in the two phases. Worked examples and numerical results are provided to illustrate and verify the analytical predictions. In particular, simulations show that our asymptotic formulas provide accurate predictions even at moderate signal dimensions.
We consider a commonly studied supervised classification of a synthetic dataset whose labels are generated by feeding a one-layer neural network with random iid inputs. We study the generalization performances of standard classifiers in the high-dimensional regime where α=n/d is kept finite in the limit of a high dimension d and number of samples n. Our contribution is three-fold: First, we prove a formula for the generalization error achieved by ℓ2 regularized classifiers that minimize a convex loss. This formula was first obtained by the heuristic replica method of statistical physics. Secondly, focussing on commonly used loss functions and optimizing the ℓ2 regularization strength, we observe that while ridge regression performance is poor, logistic and hinge regression are surprisingly able to approach the Bayes-optimal generalization error extremely closely. As α→∞ they lead to Bayes-optimal rates, a fact that does not follow from predictions of margin-based generalization error bounds. Third, we design an optimal loss and regularizer that provably leads to Bayes-optimal generalization error.
We consider a high-dimensional mixture of two Gaussians in the noisy regime where even an oracle knowing the centers of the clusters misclassifies a small but finite fraction of the points. We provide a rigorous analysis of the generalization error of regularized convex classifiers, including ridge, hinge and logistic regression, in the high-dimensional limit where the number n of samples and their dimension d go to infinity while their ratio is fixed to α=n/d. We discuss surprising effects of the regularization that in some cases allows to reach the Bayes-optimal performances. We also illustrate the interpolation peak at low regularization, and analyze the role of the respective sizes of the two clusters.