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1. Learning a deep convolutional neural network via tensor decomposition

   https://doi.org/10.1093/imaiai/iaaa042  

   Oymak, Samet; Soltanolkotabi, Mahdi (February 2021, Information and Inference: A Journal of the IMA)

   Abstract In this paper, we study the problem of learning the weights of a deep convolutional neural network. We consider a network where convolutions are carried out over non-overlapping patches. We develop an algorithm for simultaneously learning all the kernels from the training data. Our approach dubbed deep tensor decomposition (DeepTD) is based on a low-rank tensor decomposition. We theoretically investigate DeepTD under a realizable model for the training data where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to planted convolutional kernels. We show that DeepTD is sample efficient and provably works as soon as the sample size exceeds the total number of convolutional weights in the network. 

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2. Convolutional Normalization: Improving Deep Convolutional Network Robustness and Training

   Liu, Sheng; Li, Xiao; Zhai, Yuexiang; You, Chong; Zhu, Zhihui; Fernandez-Granda, Carlos; Qu, Qing (January 2021, Advances in neural information processing systems)

   Normalization techniques have become a basic component in modern convolutional neural networks (ConvNets). In particular, many recent works demonstrate that promoting the orthogonality of the weights helps train deep models and improve robustness. For ConvNets, most existing methods are based on penalizing or normalizing weight matrices derived from concatenating or flattening the convolutional kernels. These methods often destroy or ignore the benign convolutional structure of the kernels; therefore, they are often expensive or impractical for deep ConvNets. In contrast, we introduce a simple and efficient Convolutional Normalization'' (ConvNorm) method that can fully exploit the convolutional structure in the Fourier domain and serve as a simple plug-and-play module to be conveniently incorporated into any ConvNets. Our method is inspired by recent work on preconditioning methods for convolutional sparse coding and can effectively promote each layer's channel-wise isometry. Furthermore, we show that our ConvNorm can reduce the layerwise spectral norm of the weight matrices and hence improve the Lipschitzness of the network, leading to easier training and improved robustness for deep ConvNets. Applied to classification under noise corruptions and generative adversarial network (GAN), we show that the ConvNorm improves the robustness of common ConvNets such as ResNet and the performance of GAN. We verify our findings via numerical experiments on CIFAR and ImageNet. Our implementation is available online at \url{https://github.com/shengliu66/ConvNorm}. 

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3. Stable tensor neural networks for efficient deep learning

   https://doi.org/10.3389/fdata.2024.1363978  

   Newman, Elizabeth; Horesh, Lior; Avron, Haim; Kilmer, Misha E (May 2024, Frontiers in Big Data)

   Zhang, Yanqing (Ed.)

   Learning from complex, multidimensional data has become central to computational mathematics, and among the most successful high-dimensional function approximators are deep neural networks (DNNs). Training DNNs is posed as an optimization problem to learn network weights or parameters that well-approximate a mapping from input to target data. Multiway data or tensors arise naturally in myriad ways in deep learning, in particular as input data and as high-dimensional weights and features extracted by the network, with the latter often being a bottleneck in terms of speed and memory. In this work, we leverage tensor representations and processing to efficiently parameterize DNNs when learning from high-dimensional data. We propose tensor neural networks (t-NNs), a natural extension of traditional fully-connected networks, that can be trained efficiently in a reduced, yet more powerful parameter space. Our t-NNs are built upon matrix-mimetic tensor-tensor products, which retain algebraic properties of matrix multiplication while capturing high-dimensional correlations. Mimeticity enables t-NNs to inherit desirable properties of modern DNN architectures. We exemplify this by extending recent work on stable neural networks, which interpret DNNs as discretizations of differential equations, to our multidimensional framework. We provide empirical evidence of the parametric advantages of t-NNs on dimensionality reduction using autoencoders and classification using fully-connected and stable variants on benchmark imaging datasets MNIST and CIFAR-10. 

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4. Correlating Time Series With Interpretable Convolutional Kernels

   https://doi.org/10.1109/TKDE.2025.3550877  

   Chen, Xinyu; Cai, HanQin; Liu, Fuqiang; Zhao, Jinhua (June 2025, IEEE Transactions on Knowledge and Data Engineering)

   This study addresses the problem of convolutional kernel learning in univariate, multivariate, and multidimensional time series data, which is crucial for interpreting temporal patterns in time series and supporting downstream machine learning tasks. First, we propose formulating convolutional kernel learning for univariate time series as a sparse regression problem with a non-negative constraint, leveraging the properties of circular convolution and circulant matrices. Second, to generalize this approach to multivariate and multidimensional time series data, we use tensor computations, reformulating the convolutional kernel learning problem in the form of tensors. This is further converted into a standard sparse regression problem through vectorization and tensor unfolding operations. In the proposed methodology, the optimization problem is addressed using the existing non-negative subspace pursuit method, enabling the convolutional kernel to capture temporal correlations and patterns. To evaluate the proposed model, we apply it to several real-world time series datasets. On the multidimensional ridesharing and taxi trip data from New York City and Chicago, the convolutional kernels reveal interpretable local correlations and cyclical patterns, such as weekly seasonality. For the monthly temperature time series data in North America, the proposed model can quantify the yearly seasonality and make it comparable across different decades. In the context of multidimensional fluid flow data, both local and nonlocal correlations captured by the convolutional kernels can reinforce tensor factorization, leading to performance improvements in fluid flow reconstruction tasks. Thus, this study lays an insightful foundation for automatically learning convolutional kernels from time series data, with an emphasis on interpretability through sparsity and non-negativity constraints. 

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5. Generalizability test of a deep learning-based CT image denoising method

   Zeng, Rongping; Lin, Claire Yilin; Li, Qin; Lu, Jiang; Fessler, Jeffrey A.; Myers, Kyle J (January 2020, The 6th International Conference on Image Formation in X-Ray Computed Tomography)

   Deep learning (DL) has been increasingly explored in low-dose CT image denoising. DL products have also been submitted to the FDA for premarket clearance. While having the potential to improve image quality over the filtered back projection method (FBP) and produce images quickly, generalizability of DL approaches is a major concern because the performance of a DL network can depend highly on the training data. In this work we take a residual encoder-decoder convolutional neural network (REDCNN)-based CT denoising method as an example. We investigate the effect of the scan parameters associated with the training data on the performance of this DL-based CT denoising method and identifies the scan parameters that may significantly impact its performance generalizability. This abstract particularly examines these three parameters: reconstruction kernel, dose level and slice thickness. Our preliminary results indicate that the DL network may not generalize well between FBP reconstruction kernels, but is insensitive to slice thickness for slice-wise denoising. The results also suggest that training with mixed dose levels improves denoising performance. 

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