PHNNs: Lightweight Neural Networks via Parameterized Hypercomplex Convolutions

Abstract

Hypercomplex neural networks have proven to reduce the overall number of parameters while ensuring valuable performance by leveraging the properties of Clifford algebras. Recently, hypercomplex linear layers have been further improved by involving efficient parameterized Kronecker products. In this article, we define the parameterization of hypercomplex convolutional layers and introduce the family of parameterized hypercomplex neural networks (PHNNs) that are lightweight and efficient large-scale models. Our method grasps the convolution rules and the filter organization directly from data without requiring a rigidly predefined domain structure to follow. PHNNs are flexible to operate in any user-defined or tuned domain, from 1-D to nD regardless of whether the algebra rules are preset. Such a malleability allows processing multidimensional inputs in their natural domain without annexing further dimensions, as done, instead, in quaternion neural networks (QNNs) for 3-D inputs like color images. As a result, the proposed family of PHNNs operates with 1/n free parameters as regards its analog in the real domain. We demonstrate the versatility of this approach to multiple domains of application by performing experiments on various image datasets and audio datasets in which our method outperforms real and quaternion-valued counterparts. Full code is available at: https://github.com/eleGAN23/HyperNets.

Quaternion Quantum Neural Network for Classification

“…Future directions: As was mentioned before, N-dimensional convolution has been applied on Real-Valued CNNs for processing multichannel inputs. In a similar way, the equations presented can be extended to 8-channel inputs using an octonion or hypercomplex algebra, see for example [65], [66], [67], [68]. For larger number of inputs, a geometric algebra [69] representation can be applied.…”

“…For larger number of inputs, a geometric algebra [69] representation can be applied. To the best of our knowledge, this type of architecture has not been published to date, but a first approach in this direction can be found in [70], [71]. In these type of deep learning architectures: quaternion, hyper-complex, or geometric, a major concern is the selection of the signature of the algebra, which will embed data into different geometric spaces, and the processing will take distinct meanings accordingly.…”

“…QS(q) = S(q R ) + S(q I ) î + S(q J ) ĵ + S(q K ) k, (65) where S : R → R is the real-valued sigmoid function:…”

“…Models Website Gaudet and Maida [19] Residual https://github.com/gaudetcj/DeepQuaternionNetworks/ Grassuccii et al [97], [70], [132], [82] VAE, GAN, Hypercomplex https://github.com/eleGAN23 Parcollet et al [22], [32] QRNNs, QLSTMs, QCNNs (Con-vNets), QCAEs.…”

Automatic pavement texture recognition using lightweight few-shot learning

“…Jiang et al [28] developed a method for network compression in super-resolution, which employed weight pruning and a multi-slicing network of information to extract and integrate multi-scale features. Grassucci et al [29] introduced a family of parameterized hypercomplex neural networks that directly captured convolution rules and filter organization from data, making them flexible in any user-defined or tuned domain. Cheng et al [30] proposed a lightweight unified fusion network for multi-focus image fusion, which used guided filtering to separate the source image into basic and detailed layers and applied a gradient perception strategy to handle the fusion problem.…”