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We overview several properties—old and new—of training overparameterized deep networks under the square loss. We first consider a model of the dynamics of gradient flow under the square loss in deep homogeneous rectified linear unit networks. We study the convergence to a solution with the absolute minimum ρ, which is the product of the Frobenius norms of each layer weight matrix, when normalization by Lagrange multipliers is used together with weight decay under different forms of gradient descent. A main property of the minimizers that bound their expected error for a specific network architecture is ρ. In particular, we derive novel norm-based bounds for convolutional layers that are orders of magnitude better than classical bounds for dense networks. Next, we prove that quasi-interpolating solutions obtained by stochastic gradient descent in the presence of weight decay have a bias toward low-rank weight matrices, which should improve generalization. The same analysis predicts the existence of an inherent stochastic gradient descent noise for deep networks. In both cases, we verify our predictions experimentally. We then predict neural collapse and its properties without any specific assumption—unlike other published proofs. Our analysis supports the idea that the advantage of deep networks relative to other classifiers is greater for problems that are appropriate for sparse deep architectures such as convolutional neural networks. The reason is that compositionally sparse target functions can be approximated well by “sparse” deep networks without incurring in the curse of dimensionality.

Compressive video measurements can save bandwidth and data storage. However, conventional approaches to target detection require the compressive measurements to be reconstructed before any detectors are applied. This is not only time consuming but also may lose information in the reconstruction process. In this paper, we summarized the application of a recent approach to vehicle detection and classification directly in the compressive measurement domain to human targets. The raw videos were collected using a pixel-wise code exposure (PCE) camera, which condensed multiple frames into one frame. A combination of two deep learning-based algorithms (you only look once (YOLO) and residual network (ResNet)) was used for detection and confirmation. Optical and mid-wave infrared (MWIR) videos from a well-known database (SENSIAC) were used in our experiments. Extensive experiments demonstrated that the proposed framework was feasible for target detection up to 1500 m, but target confirmation needs more research.

Compressive sensing has seen many applications in recent years. One type of compressive sensing device is the Pixel-wise Code Exposure (PCE) camera, which has low power consumption and individual control of pixel exposure time. In order to use PCE cameras for practical applications, a time consuming and lossy process is needed to reconstruct the original frames. In this paper, we present a deep learning approach that directly performs target tracking and classification in the compressive measurement domain without any frame reconstruction. In particular, we propose to apply You Only Look Once (YOLO) to detect and track targets in the frames and we propose to apply Residual Network (ResNet) for classification. Extensive simulations using low quality optical and mid-wave infrared (MWIR) videos in the SENSIAC database demonstrated the efficacy of our proposed approach.