When training overparameterized deep networks for classification tasks, it has been widely observed that the learned features exhibit a so-called "neural collapse" phenomenon. More specifically, for the output features of the penultimate layer, for each class the within-class features converge to their means, and the means of different classes exhibit a certain tight frame structure, which is also aligned with the last layer's classifier. As feature normalization in the last layer becomes a common practice in modern representation learning, in this work we theoretically justify the neural collapse phenomenon for normalized features. Based on an unconstrained feature model, we simplify the empirical loss function in a multi-class classification task into a nonconvex optimization problem over the Riemannian manifold by constraining all features and classifiers over the sphere. In this context, we analyze the nonconvex landscape of the Riemannian optimization problem over the product of spheres, showing a benign global landscape in the sense that the only global minimizers are the neural collapse solutions while all other critical points are strict saddles with negative curvature. Experimental results on practical deep networks corroborate our theory and demonstrate that better representations can be learned faster via feature normalization.
A wafer-thin chip-scale portable spectrometer suitable for wearable applications based on a reconstructive algorithm was demonstrated. A total of 16 spectral encoders that simultaneously functioned as photodetectors were monolithically integrated on a chip area of 0.16 mm2 by applying local strain engineering in compressively strained InGaN/GaN multiple quantum well heterostructures. The built-in GaN pn junction enabled a direct photocurrent measurement. A non-negative least-squares (NNLS) algorithm with total-variation regularization and a choice of a proper kernel function was shown to deliver a decent spectral reconstruction performance in the wavelength range of 400–645 nm. The accuracies of spectral peak positions and intensity ratios between peaks were found to be 0.97% and 10.4%, respectively. No external optics, such as collimation optics and apertures, were used, enabled by angle-insensitive light-harvesting structures, including an array of cone-shaped backreflectors fabricated on the underside of the sapphire substrate.
A deep learning prototype for a reconstructive spectrometer was developed using an experimentally determined spectrometer response and synthetic data generated from the response matrix. Benchmarking with synthetic and experimental data was performed.
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