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Many recent works have studied the eigenvalue spectrum of the Conjugate Kernel (CK) defined by the nonlinear feature map of a feedforward neural network. However, existing results only establish weak convergence of the empirical eigenvalue distribution, and fall short of providing precise quantitative characterizations of the “spike” eigenvalues and eigenvectors that often capture the low-dimensional signal structure of the learning problem. In this work, we characterize these signal eigenvalues and eigenvectors for a nonlinear version of the spiked covariance model, including the CK as a special case. Using this general result, we give a quantitative description of how spiked eigenstructure in the input data propagates through the hidden layers of a neural network with random weights. As a second application, we study a simple regime of representation learning where the weight matrix develops a rank-one signal component over training and characterize the alignment of the target function with the spike eigenvector of the CK on test data. 

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Given a part design, the task of manufacturing process selection chooses an appropriate manufacturing process to fabricate it. Prior research has traditionally determined manufacturing processes through direct classification. However, an alternative approach to select a manufacturing process for a new design involves identifying previously produced parts with comparable shapes and materials and learning from them. Finding similar designs from a large dataset of previously manufactured parts is a challenging problem. To solve this problem, researchers have proposed different spatial and spectral shape descriptors to extract shape features including the D2 distribution, spherical harmonics (SH), and the Fast Fourier Transform (FFT), as well as the application of different machine learning methods on various representations of 3D part models like multi-view images, voxel, triangle mesh, and point cloud. However, there has not been a comprehensive analysis of these different shape descriptors, especially for part similarity search aimed at manufacturing process selection. To remedy this gap, this paper presents an in-depth comparative study of these shape descriptors for part similarity search. While we acknowledge the importance of factors like part size, tolerance, and cost in manufacturing process selection, this paper focuses on part shape and material properties only. Our findings show that SH performs the best among non-machine learning methods for manufacturing process selection, yielding 97.96% testing accuracy using the proposed quantitative evaluation metric. For machine learning methods, deep learning on multi-view image representations is best, yielding 99.85% testing accuracy when rotational invariance is not a primary concern. Deep learning on point cloud representations excels, yielding 99.44% testing accuracy when considering rotational invariance.