Tensor-Augmented Convolutional Neural Networks: Enhancing Expressivity with Generic Tensor Kernels
Authors: Chia-Wei Hsing, Wei-Lin Tu
Organizations: blueqat Inc., 2-24-12-39F, Shibuya, Shibuya-ku, Tokyo 150-6139, Japan · Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan · Graduate School of Science and Technology, Keio University, Yokohama, Kanagawa 223-8522, Japan · Keio University Sustainable Quantum Artificial Intelligence Center (KSQAIC), Keio University, Tokyo 108-8345, Japan
Convolutional Neural Networks (CNNs) excel at extracting local features hierarchically, but their performance in capturing complex correlations hinges heavily on deep architectures, which are usually computationally demanding and difficult to interpret. To address these issues, we propose a physically-guided shallow model: tensor-augmented CNN (TACNN), which replaces conventional convolution kernels with generic tensors to enhance representational capacity. This choice is motivated by the fact that an order-N tensor naturally encodes an arbitrary quantum superposition state in the Hilbert space of dimension dN, where d is the local physical dimension, thus offering substantially richer expressivity. Furthermore, in our design the convolution output of each layer becomes a multilinear form capable of capturing high-order feature correlations, thereby equipping a shallow multilayer architecture with an expressive power competitive to that of deep CNNs. On the Fashion-MNIST benchmark, TACNN demonstrates clear advantages over conventional CNNs, achieving remarkable accuracies with only a few layers. In particular, a TACNN with only two convolution layers attains a test accuracy of 93.7%, surpassing or matching considerably deeper models such as VGG-16 (93.5%) and GoogLeNet (93.7%). These findings highlight TACNN as a promising framework that strengthens model expressivity while preserving architectural simplicity, paving the way towards more interpretable and efficient deep learning models.
Figures & tables
Figure 1: Convolution operation in (a) a conventional CNN and (b) the proposed TACNN. (a) In a standard CNN, an element-wise multiplication is performed between each local image patch and a convolution kernel, and a following summation produces a scalar output. This is equivalent to an inner product between two flattened arrays. (b) In TACNN, the input patch is first mapped into a higher‑dimensional Hilbert space, forming a product state ⟨ϕ∣ , and each convolution kernel is replaced by a generic tensor representing an arbitrary superposition state ∣ψ⟩ . Similarly, the inner product ⟨ϕ∣ψ⟩ yields a scalar, yet with substantially enhanced expressive capacity as elaborated in Section II .
Figure 2: Test accuracies of the Fashion-MNIST dataset given by TACNN (red) and comparable CNN (blue) with one convolution layer for varying kernel count 2m , where m=0,1,…,11 . For TACNN with two convolution layers (orange) the kernel counts of the second layer are 16×16 , 32×32 and 64×64 . Comparison with very deep CNNs VGG-16 (green) and GoogLeNet (purple) given by Ref. [ 28 ] further demonstrates the supremacy of TACNN in terms of parameter efficiency.
Model
Kernel count
Test accuracy
1-layer TACNN
1
89.7 %
2
90.8 %
4
91.5 %
8
91.8 %
16
92.2 %
32
92.3 %
Table 1: Test accuracies of the Fashion-MNIST dataset given by TACNN architectures with one and two convolution layers for varying number of kernels. The 1‑layer model reaches a maximum accuracy of 93.1 % , whereas the 2‑layer model further improves the performance to 93.7 % when applying 64×64 kernels in the second layer, representing the highest accuracy achieved in our experiments.
Model
Test accuracy
MPS [ 9 ]
88.0 %
PEPS [ 5 ]
88.3 %
EPS + SBS [ 11 ]
88.6 %
MPS + TTN [ 24 ]
89.0 %
Snake-SBS [ 11 ]
89.2 %
LoTeNet [ 22 ]
89.5 %
Table 2: Test accuracies of various models on the Fashion‑MNIST dataset. K-SVM and XGBoost are two of the most commonly used traditional machine-learning methods prior to the paradigm shift to deep neural networks. AlexNet and VGG-16 are typical Vanilla CNNs, whereas VGG‑16 and GoogLeNet represent CNNs with very deep network structures. The rest of the data shown here were obtained by TN-based machine-learning methods. All the abbreviations follow the conventions used in the references listed behind.
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Zhaoxiang Liu, Zhicheng Ma, Kaikai Zhao +2
Unicom Data Intelligence, China Unicom, Beijing, China
Higher-order learning is fundamentally rooted in exploiting compositional features. It clearly hinges on enriching the representation by more elaborate interactions of the data which, in turn, tends to increase the model complexity of conventional large-scale deep learning models. In this paper, a kernelized Volterra Neural Network (kVNN) is proposed. The key to the achieved efficiency lies in using a learnable multi-kernel representation, where different interaction orders are modeled by distinct polynomial-kernel components with compact, learnable centers, yielding an order-adaptive parameterization. Features are learned by the composition of layers, each of which consists of parallel branches of different polynomial orders, enabling kVNN filters to directly replace standard convolutional kernels within existing architectures. The theoretical results are substantiated by experiments on two representative tasks: video action recognition and image denoising. The results demonstrate favorable performance-efficiency trade-offs: kVNN consistently yields reduced model (parameters) and computational (GFLOPs) complexity with competitive and often improved performance. These results are maintained even when trained from scratch without large-scale pretraining. In summary, we substantiate that structured kernelized higher-order layers offer a practical path to balancing expressivity and computational cost in modern deep networks.
Haoyu Yun, Hamid Krim, Yufang Bao
Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC, USA · Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville, NC, USA
Although standard Convolutional Neural Networks (CNNs) can be mathematically reinterpreted as Self-Explainable Models (SEMs), their built-in prototypes do not on their own accurately represent the data. Replacing the final linear layer with a k-means-based classifier addresses this limitation without compromising performance. This work introduces a common formalization of k-means-based post-hoc explanations for the classifier, the encoder's final output (B4), and combinations of intermediate feature activations. The latter approach leverages the spatial consistency of convolutional receptive fields to generate concept-based explanation maps, which are supported by gradient-free feature attribution maps. Empirical evaluation with a ResNet34 shows that using shallower, less compressed feature activations, such as those from the last three blocks (B234), results in a trade-off between semantic fidelity and a slight reduction in predictive performance.
Ahcène Boubekki, Line H. Clemmensen
Department of Mathematical Sciences, University of Copenhagen, Denmark