The task of transfer learning using pretrained convolutional neural networks is considered. We propose a convolution-SVD layer to analyze the convolution operators with a singular value decomposition computed in the Fourier domain. Singular vectors extracted from the source domain are transferred to the target domain, whereas the singular values are fine-tuned with a target data set. In this way, dimension reduction is achieved to avoid overfitting, while some flexibility to fine-tune the convolution kernels is maintained. We extend an existing convolution kernel reconstruction algorithm to allow for a reconstruction from an arbitrary set of learned singular values. A generalization bound for a single convolution-SVD layer is devised to show the consistency between training and testing errors. We further introduce a notion of transfer learning gap. We prove that the testing error for a single convolution-SVD layer is bounded in terms of the gap, which motivates us to develop a regularization model with the gap as the regularizer. Numerical experiments are conducted to demonstrate the superiority of the proposed model in solving classification problems and the influence of various parameters. In particular, the regularization is shown to yield a significantly higher prediction accuracy.

We study a multi-instance (MI) learning dimensionality-reduction algorithm through sparsity and orthogonality, which is especially useful for high-dimensional MI data sets. We develop a novel algorithm to handle both sparsity and orthogonality constraints that existing methods do not handle well simultaneously. Our main idea is to formulate an optimization problem where the sparse term appears in the objective function and the orthogonality term is formed as a constraint. The resulting optimization problem can be solved by using approximate augmented Lagrangian iterations as the outer loop and inertial proximal alternating linearized minimization (iPALM) iterations as the inner loop. The main advantage of this method is that both sparsity and orthogonality can be satisfied in the proposed algorithm. We show the global convergence of the proposed iterative algorithm. We also demonstrate that the proposed algorithm can achieve high sparsity and orthogonality requirements, which are very important for dimensionality reduction. Experimental results on both synthetic and real data sets show that the proposed algorithm can obtain learning performance comparable to that of other tested MI learning algorithms.