Fundamental limits of Non-Linear Low-Rank Matrix Estimation
This work addresses fundamental limits in matrix estimation for researchers in statistics and machine learning, providing theoretical insights but is incremental as it extends linear results to non-linear settings.
The paper tackles the problem of estimating a low-rank matrix from non-linear noisy observations, proving that Bayes-optimal performance is characterized by an equivalent Gaussian model and deriving a required signal-to-noise ratio scaling as N^(1/2(1-1/k_F)) for accurate reconstruction, with k_F being a Fisher information coefficient.
We consider the task of estimating a low-rank matrix from non-linear and noisy observations. We prove a strong universality result showing that Bayes-optimal performances are characterized by an equivalent Gaussian model with an effective prior, whose parameters are entirely determined by an expansion of the non-linear function. In particular, we show that to reconstruct the signal accurately, one requires a signal-to-noise ratio growing as $N^{\frac 12 (1-1/k_F)}$, where $k_F$ is the first non-zero Fisher information coefficient of the function. We provide asymptotic characterization for the minimal achievable mean squared error (MMSE) and an approximate message-passing algorithm that reaches the MMSE under conditions analogous to the linear version of the problem. We also provide asymptotic errors achieved by methods such as principal component analysis combined with Bayesian denoising, and compare them with Bayes-optimal MMSE.