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We describe an algorithm that learns two-layer residual units using rectified linear unit (ReLU) activation: suppose the input $\mathbf{x}$ is from a distribution with support space $\mathbb{R}^d$ and the ground-truth generative model is a residual unit of this type, given by $\mathbf{y} = \boldsymbol{B}^\ast\left[\left(\boldsymbol{A}^\ast\mathbf{x}\right)^+ + \mathbf{x}\right]$, where ground-truth network parameters $\boldsymbol{A}^\ast \in \mathbb{R}^{d\times d}$ represent a full-rank matrix with nonnegative entries and $\boldsymbol{B}^\ast \in \mathbb{R}^{m\times d}$ is full-rank with $m \geq d$ and for $\boldsymbol{c} \in \mathbb{R}^d$, $[\boldsymbol{c}^{+}]_i = \max\{0, c_i\}$. We design layer-wise objectives as functionals whose analytic minimizers express the exact ground-truth network in terms of its parameters and nonlinearities. Following this objective landscape, learning residual units from finite samples can be formulated using convex optimization of a nonparametric function: for each layer, we first formulate the corresponding empirical risk minimization (ERM) as a positive semi-definite quadratic program (QP), then we show the solution space of the QP can be equivalently determined by a set of linear inequalities, which can then be efficiently solved by linear programming (LP). We further prove the strong statistical consistency of our algorithm, and demonstrate its robustness and sample efficiency through experimental results on synthetic data and a set of benchmark regression datasets., Comment: Published in Transactions on Machine Learning Research (11/2022), slightly typographically revised

Image deconvolution is the process of recovering convolutional degraded images, which is always a hard inverse problem because of its mathematically ill-posed property. On the success of the recently proposed deep image prior (DIP), we build an image deconvolution model with deep image and kernel priors (DIKP). DIP is a learning-free representation which uses neural net structures to express image prior information, and it showed great success in many energy-based models, e.g. denoising, super-resolution, inpainting. Instead, our DIKP model uses such priors in image deconvolution to model not only images but also kernels, combining the ideas of traditional learning-free deconvolution methods with neural nets. In this paper, we show that DIKP improve the performance of learning-free image deconvolution, and we experimentally demonstrate this on the standard benchmark of six standard test images in terms of PSNR and visual effects., Comment: In Proceedings of the 2019 IEEE International Conference on Computer Vision Workshops (ICCVW)