These notes review six lectures given by Prof. Andrea Montanari on the topic of statistical estimation for linear models. The first two lectures cover the principles of signal recovery from linear measurements in terms of minimax risk. Subsequent lectures demonstrate the application of these principles to several practical problems in science and engineering. Specifically, these topics include denoising of error-laden signals, recovery of compressively sensed signals, reconstruction of low-rank matrices, and also the discovery of hidden cliques within large networks.
Minimizing the empirical risk is a popular training strategy, but for learning tasks where the data may be noisy or heavy-tailed, one may require many observations in order to generalize well. To achieve better performance under less stringent requirements, we introduce a procedure which constructs a robust approximation of the risk gradient for use in an iterative learning routine. We provide high-probability bounds on the excess risk of this algorithm, by showing that it does not deviate far from the ideal gradient-based update. Empirical tests show that in diverse settings, the proposed procedure can learn more efficiently, using less resources (iterations and observations) while generalizing better.
Though ordinary differential equations (ODE) are used extensively in science and engineering, the task of learning ODE parameters from noisy observations still presents challenges. To address these challenges, we propose a direct method that involves alternating minimization of an objective function over the filtered states and parameters. This objective function directly measures how well the filtered states and parameters satisfy the ODE, in contrast to many existing methods that use separate objectives over the observations, filtered states, and parameters. As we show on several ODE systems, as compared to state-of-the-art methods, the direct method exhibits increased robustness (to noise, parameter initialization, and hyperparameters), decreased training times, and improved accuracy in estimating both filtered states and parameters. The direct method involves only one hyperparameter that plays the role of an inverse step size. We show how the direct method can be used with general multistep numerical discretizations, and demonstrate its performance on systems with up to d 40 dimensions. The code of our algorithms can be found in the authors' web pages.
The unscented transformation (UT) is an efficient method to solve the state estimation problem for a non-linear dynamic system, utilizing a derivative-free higher-order approximation by approximating a Gaussian distribution rather than approximating a non-linear function. Applying the UT to a Kalman filter type estimator leads to the well-known unscented Kalman filter (UKF). Although the UKF works very well in Gaussian noises, its performance may deteriorate significantly when the noises are non-Gaussian, especially when the system is disturbed by some heavy-tailed impulsive noises. To improve the robustness of the UKF against impulsive noises, a new filter for nonlinear systems is proposed in this work, namely the maximum correntropy unscented filter (MCUF). In MCUF, the UT is applied to obtain the prior estimates of the state and covariance matrix, and a robust statistical linearization regression based on the maximum correntropy criterion (MCC) is then used to obtain the posterior estimates of the state and covariance. The satisfying performance of the new algorithm is confirmed by two illustrative examples.
We present a method for audio denoising that combines processing done in both the time domain and the time-frequency domain. Given a noisy audio clip, the method trains a deep neural network to fit this signal. Since the fitting is only partly successful and is able to better capture the underlying clean signal than the noise, the output of the network helps to disentangle the clean audio from the rest of the signal. The method is completely unsupervised and only trains on the specific audio clip that is being denoised. Our experiments demonstrate favorable performance in comparison to the literature methods, and our code and audio samples are available at https: //github.com/mosheman5/DNP. Index Terms: Audio denoising; Unsupervised learning