We study a deep learning inspired formulation for the blind demodulation problem, which is the task of recovering two unknown vectors from their entrywise multiplication.

In safety-critical deep learning applications, robustness measures the ability of neural models that handle imperceptible perturbations in input data, which may lead to potential safety hazards. Existing pre-deployment robustness assessment methods typically suffer from significant trade-offs between computational cost and measurement precision, limiting their practical utility. To address these limitations, this paper conducts a comprehensive comparative analysis of existing robustness definitions and associated assessment methodologies. We propose tower robustness to evaluate robustness, which is a novel, practical metric based on hypothesis testing to quantitatively evaluate probabilistic robustness, enabling more rigorous and efficient pre-deployment assessments. Our extensive comparative evaluation illustrates the advantages and applicability of our proposed approach, thereby advancing the systematic understanding and enhancement of model robustness in safety-critical deep learning applications.

The paper considers the problem of mixed linear regression: in this problem, an algorithm is given access to n data samples (x_i, y_i) with possibly corrupted labels, where each y_i is one of the m possible linear functions of x_i, i.e., y_i x_i T theta_j for j in {1,...,m} (but the algorithm does not know which one). The goal of the algorithm is to determine vectors theta_1,..., theta_m. A straightforward but computationally inefficient (the complexity is exponential in d) approach to solving this problem is by using Least Trimmed Squares (LTS), which tries to identify the best fit vector (in terms of least squares) over all possible subsets of the data points of a particular, predefined size. To address this issue, the paper proposes using an alternative, simple, algorithm called Iterative Least Trimmed Squares (ILTS), which is similar to other algorithms that have been used for related problems in the literature, as acknowledged in the paper. The algorithm is essentially alternating minimization: it alternates between (1) finding the best set of a given size tau * n, given the least squares solution from the previous iteration and (2) solving least squares over the set determined in (1).

We study a deep learning inspired formulation for the blind demodulation problem, which is the task of recovering two unknown vectors from their entrywise multiplication. In the case when the networks corresponding to the generative models are expansive, the weight matrices are random and the dimension of the unknown vectors satisfy \ell \Omega(n 2 p 2), up to log factors, we show that the empirical risk objective has a favorable landscape for optimization. That is, the objective function has a descent direction at every point outside of a small neighborhood around four hyperbolic curves. We also characterize the local maximizers of the empirical risk objective and, hence, show that there does not exist any other stationary points outside of these neighborhood around four hyperbolic curves and the set of local maximizers. We also implement a gradient descent scheme inspired by the geometry of the landscape of the objective function.