Generative Topographic Map (GTM) Generative topographic map (GTM) is a machine learning method that is a probabilistic counterpart of the self-organizing map (SOM), is provably convergent and does not require a shrinking neighborhood or a decreasing step size. It is a generative model: the data is assumed to arise by first probabilistically picking a point in a low-dimensional space, mapping the point to the observed high-dimensional input space (via a smooth function), then adding noise in that space. The parameters of the low-dimensional probability distribution, the smooth map and the noise are all learned from the training data using the expectation-maximization (EM) algorithm. GTM was introduced in 1996 in a paper by Christopher M. Bishop, Markus Svensen, and Christopher K. I. Williams. …

Approximate Message Passing (AMP) The standard linear regression (SLR) problem is to recover a vector $\mathbf{x}^0$ from noisy linear observations $\mathbf{y}=\mathbf{Ax}^0+\mathbf{w}$. The approximate message passing (AMP) algorithm recently proposed by Donoho, Maleki, and Montanari is a computationally efficient iterative approach to SLR that has a remarkable property: for large i.i.d.\ sub-Gaussian matrices $\mathbf{A}$, its per-iteration behavior is rigorously characterized by a scalar state-evolution whose fixed points, when unique, are Bayes optimal. AMP, however, is fragile in that even small deviations from the i.i.d.\ sub-Gaussian model can cause the algorithm to diverge. This paper considers a ‘vector AMP’ (VAMP) algorithm and shows that VAMP has a rigorous scalar state-evolution that holds under a much broader class of large random matrices $\mathbf{A}$: those that are right-rotationally invariant. After performing an initial singular value decomposition (SVD) of $\mathbf{A}$, the per-iteration complexity of VAMP can be made similar to that of AMP. In addition, the fixed points of VAMP’s state evolution are consistent with the replica prediction of the minimum mean-squared error recently derived by Tulino, Caire, Verd\’u, and Shamai. The effectiveness and state evolution predictions of VAMP are confirmed in numerical experiments. …

Explain to Fix (E2X) Explaining predictions of deep neural networks (DNNs) is an important and nontrivial task. In this paper, we propose a practical approach to interpret decisions made by a DNN object detector that has fidelity comparable to state-of-the-art methods and sufficient computational efficiency to process large datasets. Our method relies on recent theory and approximates Shapley feature importance values. We qualitatively and quantitatively show that the proposed explanation method can be used to find image features which cause failures in DNN object detection. The developed software tool combined into the ‘Explain to Fix’ (E2X) framework has a factor of 10 higher computational efficiency than prior methods and can be used for cluster processing using graphics processing units (GPUs). Lastly, we propose a potential extension of the E2X framework where the discovered missing features can be added into training dataset to overcome failures after model retraining. …