Fenglei Fan

Fenglei Fan, Juntong Fan, Dayang Wang et al.

The rapid growth of large models' size has far outpaced that of computing resources. To bridge this gap, encouraged by the parsimonious relationship between genotype and phenotype in the brain's growth and development, we propose the so-called Hyper-Compression that turns the model compression into the issue of parameter representation via a hyperfunction. Specifically, it is known that the trajectory of some low-dimensional dynamic systems can fill the high-dimensional space eventually. Thus, Hyper-Compression, using these dynamic systems as the hyperfunctions, represents the parameters of the target network by their corresponding composition number or trajectory length. This suggests a novel mechanism for model compression, substantially different from the existing pruning, quantization, distillation, and decomposition. Along this direction, we methodologically identify a suitable dynamic system with the irrational winding as the hyperfunction and theoretically derive its associated error bound. Next, guided by our theoretical insights, we propose several engineering twists to make the Hyper-Compression pragmatic and effective. Lastly, systematic and comprehensive experiments on \textcolor{black}{NLP models such as LLaMA and Qwen series and vision models} confirm that Hyper-Compression enjoys the following \textbf{PNAS} merits: 1) \textbf{P}referable compression ratio; 2) \textbf{N}o post-hoc retraining; 3) \textbf{A}ffordable inference time; and 4) \textbf{S}hort compression time. It compresses LLaMA2-7B in an hour and achieves close-to-int4-quantization performance, without retraining and with a performance drop of less than 1\%. We have open-sourced our code in https://github.com/Juntongkuki/Hyper-Compression.git for free download and evaluation.

Xiaoge Zhang, Xiao-Lin Wang, Fenglei Fan et al.

In this paper, we develop a generic methodology to encode hierarchical causality structure among observed variables into a neural network in order to improve its predictive performance. The proposed methodology, called causality-informed neural network (CINN), leverages three coherent steps to systematically map the structural causal knowledge into the layer-to-layer design of neural network while strictly preserving the orientation of every causal relationship. In the first step, CINN discovers causal relationships from observational data via directed acyclic graph (DAG) learning, where causal discovery is recast as a continuous optimization problem to avoid the combinatorial nature. In the second step, the discovered hierarchical causality structure among observed variables is systematically encoded into neural network through a dedicated architecture and customized loss function. By categorizing variables in the causal DAG as root, intermediate, and leaf nodes, the hierarchical causal DAG is translated into CINN with a one-to-one correspondence between nodes in the causal DAG and units in the CINN while maintaining the relative order among these nodes. Regarding the loss function, both intermediate and leaf nodes in the DAG graph are treated as target outputs during CINN training so as to drive co-learning of causal relationships among different types of nodes. As multiple loss components emerge in CINN, we leverage the projection of conflicting gradients to mitigate gradient interference among the multiple learning tasks. Computational experiments across a broad spectrum of UCI data sets demonstrate substantial advantages of CINN in predictive performance over other state-of-the-art methods. In addition, an ablation study underscores the value of integrating structural and quantitative causal knowledge in enhancing the neural network's predictive performance incrementally.

Cheng Yang, Hui Jin, Xinlei Yu et al.

Diagnosing lung cancer typically involves physicians identifying lung nodules in Computed tomography (CT) scans and generating diagnostic reports based on their morphological features and medical expertise. Although advancements have been made in using multimodal large language models for analyzing lung CT scans, challenges remain in accurately describing nodule morphology and incorporating medical expertise. These limitations affect the reliability and effectiveness of these models in clinical settings. Collaborative multi-agent systems offer a promising strategy for achieving a balance between generality and precision in medical applications, yet their potential in pathology has not been thoroughly explored. To bridge these gaps, we introduce LungNoduleAgent, an innovative collaborative multi-agent system specifically designed for analyzing lung CT scans. LungNoduleAgent streamlines the diagnostic process into sequential components, improving precision in describing nodules and grading malignancy through three primary modules. The first module, the Nodule Spotter, coordinates clinical detection models to accurately identify nodules. The second module, the Radiologist, integrates localized image description techniques to produce comprehensive CT reports. Finally, the Doctor Agent System performs malignancy reasoning by using images and CT reports, supported by a pathology knowledge base and a multi-agent system framework. Extensive testing on two private datasets and the public LIDC-IDRI dataset indicates that LungNoduleAgent surpasses mainstream vision-language models, agent systems, and advanced expert models. These results highlight the importance of region-level semantic alignment and multi-agent collaboration in diagnosing nodules. LungNoduleAgent stands out as a promising foundational tool for supporting clinical analyses of lung nodules.

Fengzhi Xu, Ziyuan Yang, Zexin Lu et al.

Non-ideal measurement computed tomography (NICT), which lowers radiation at the cost of image quality, is expanding the clinical use of CT. Although unified models have shown promise in NICT enhancement, most methods require paired data, which is an impractical demand due to inevitable organ motion. Unsupervised approaches attempt to overcome this limitation, but their assumption of homogeneous noise neglects the variability of scanning protocols, leading to poor generalization and potential model collapse. We further observe that distinct scanning protocols, which correspond to different physical imaging processes, produce discrete sub-manifolds in the feature space, contradicting these assumptions and limiting their effectiveness. To address this, we propose an Uncertainty-Guided Manifold Smoothing (UMS) framework to bridge the gaps between sub-manifolds. A classifier in UMS identifies sub-manifolds and predicts uncertainty scores, which guide the generation of diverse samples across the entire manifold. By leveraging the classifier's capability, UMS effectively fills the gaps between discrete sub-manifolds, and promotes a continuous and dense feature space. Due to the complexity of the global manifold, it's hard to directly model it. Therefore, we propose to dynamically incorporate the global- and sub-manifold-specific features. Specifically, we design a global- and sub-manifold-driven architecture guided by the classifier, which enables dynamic adaptation to subdomain variations. This dynamic mechanism improves the network's capacity to capture both shared and domain-specific features, thereby improving reconstruction performance. Extensive experiments on public datasets are conducted to validate the effectiveness of our method across different generation paradigms.

Yifei Sun, Zhanghao Chen, Hao Zheng et al.

Chest X-Ray (CXR) imaging for pulmonary diagnosis raises significant challenges, primarily because bone structures can obscure critical details necessary for accurate diagnosis. Recent advances in deep learning, particularly with diffusion models, offer significant promise for effectively minimizing the visibility of bone structures in CXR images, thereby improving clarity and diagnostic accuracy. Nevertheless, existing diffusion-based methods for bone suppression in CXR imaging struggle to balance the complete suppression of bones with preserving local texture details. Additionally, their high computational demand and extended processing time hinder their practical use in clinical settings. To address these limitations, we introduce a Global-Local Latent Consistency Model (GL-LCM) architecture. This model combines lung segmentation, dual-path sampling, and global-local fusion, enabling fast high-resolution bone suppression in CXR images. To tackle potential boundary artifacts and detail blurring in local-path sampling, we further propose Local-Enhanced Guidance, which addresses these issues without additional training. Comprehensive experiments on a self-collected dataset SZCH-X-Rays, and the public dataset JSRT, reveal that our GL-LCM delivers superior bone suppression and remarkable computational efficiency, significantly outperforming several competitive methods. Our code is available at https://github.com/diaoquesang/GL-LCM.

ZeYu Li, FengLei Fan, TieYong Zeng

This work addresses two fundamental limitations in neural network approximation theory. We demonstrate that a three-dimensional network architecture enables a significantly more efficient representation of sawtooth functions, which serves as the cornerstone in the approximation of analytic and $L^p$ functions. First, we establish substantially improved exponential approximation rates for several important classes of analytic functions and offer a parameter-efficient network design. Second, for the first time, we derive a quantitative and non-asymptotic approximation of high orders for general $L^p$ functions. Our techniques advance the theoretical understanding of the neural network approximation in fundamental function spaces and offer a theoretically grounded pathway for designing more parameter-efficient networks.

ZeYu Li, ShiJun Zhang, TieYong Zeng et al.

Universal approximation theory offers a foundational framework to verify neural network expressiveness, enabling principled utilization in real-world applications. However, most existing theoretical constructions are established by uniformly dividing the input space into tiny hypercubes without considering the local regularity of the target function. In this work, we investigate the universal approximation capabilities of ReLU networks from a view of polytope decomposition, which offers a more realistic and task-oriented approach compared to current methods. To achieve this, we develop an explicit kernel polynomial method to derive an universal approximation of continuous functions, which is characterized not only by the refined Totik-Ditzian-type modulus of continuity, but also by polytopical domain decomposition. Then, a ReLU network is constructed to approximate the kernel polynomial in each subdomain separately. Furthermore, we find that polytope decomposition makes our approximation more efficient and flexible than existing methods in many cases, especially near singular points of the objective function. Lastly, we extend our approach to analytic functions to reach a higher approximation rate.

Zelin Zhao, Fenglei Fan, Wenlong Liao et al.

Many contemporary studies utilize grid-based models for neural field representation, but a systematic analysis of grid-based models is still missing, hindering the improvement of those models. Therefore, this paper introduces a theoretical framework for grid-based models. This framework points out that these models' approximation and generalization behaviors are determined by grid tangent kernels (GTK), which are intrinsic properties of grid-based models. The proposed framework facilitates a consistent and systematic analysis of diverse grid-based models. Furthermore, the introduced framework motivates the development of a novel grid-based model named the Multiplicative Fourier Adaptive Grid (MulFAGrid). The numerical analysis demonstrates that MulFAGrid exhibits a lower generalization bound than its predecessors, indicating its robust generalization performance. Empirical studies reveal that MulFAGrid achieves state-of-the-art performance in various tasks, including 2D image fitting, 3D signed distance field (SDF) reconstruction, and novel view synthesis, demonstrating superior representation ability. The project website is available at https://sites.google.com/view/cvpr24-2034-submission/home.

Yanrui Du, Fenglei Fan, Sendong Zhao et al.

Instruction Fine-Tuning (IFT) has been widely adopted as an effective post-training strategy to enhance various abilities of Large Language Models (LLMs). However, prior studies have shown that IFT can significantly compromise LLMs' safety, particularly their ability to refuse malicious instructions, raising significant concerns. Recent research into the internal mechanisms of LLMs has identified the refusal direction (r-direction) in the hidden states, which plays a pivotal role in governing refusal behavior. Building on this insight, our study reveals that the r-direction tends to drift during training, which we identify as one of the causes of the associated safety risks. To mitigate such drift, our proposed ProCon method introduces a projection-constrained loss term that regularizes the projection magnitude of each training sample's hidden state onto the r-direction. Our initial analysis shows that applying an appropriate constraint can effectively mitigate the refusal direction drift and associated safety risks, but remains limited by overall performance barriers. To overcome this barrier, informed by our observation of early-stage sharp drift and a data-driven perspective, we introduce a warm-up strategy that emphasizes early-stage strong constraints and broaden the data distribution to strengthen constraint signals, leading to an enhanced ProCon method. Experimental results under various datasets, scenarios, and LLMs demonstrate that our method can significantly mitigate safety risks posed by IFT while preserving task performance gains. Even compared with strong baselines, our method consistently delivers superior overall performance. Crucially, our analysis indicates that ProCon can contribute to stabilizing the r-direction during training, while such an interpretability-driven exploration of LLMs' internal mechanisms lays a solid foundation for future safety research.

Junjie Yang, Yuhao Yan, Gang Wu et al.

As Vision-Language Models (VLMs) increasingly gain traction in medical applications, clinicians are progressively expecting AI systems not only to generate textual diagnoses but also to produce corresponding medical images that integrate seamlessly into authentic clinical workflows. Despite the growing interest, existing medical visual benchmarks present notable limitations. They often rely on ambiguous queries that lack sufficient relevance to image content, oversimplify complex diagnostic reasoning into closed-ended shortcuts, and adopt a text-centric evaluation paradigm that overlooks the importance of image generation capabilities. To address these challenges, we introduce MedGEN-Bench, a comprehensive multimodal benchmark designed to advance medical AI research. MedGEN-Bench comprises 6,422 expert-validated image-text pairs spanning six imaging modalities, 16 clinical tasks, and 28 subtasks. It is structured into three distinct formats: Visual Question Answering, Image Editing, and Contextual Multimodal Generation. What sets MedGEN-Bench apart is its focus on contextually intertwined instructions that necessitate sophisticated cross-modal reasoning and open-ended generative outputs, moving beyond the constraints of multiple-choice formats. To evaluate the performance of existing systems, we employ a novel three-tier assessment framework that integrates pixel-level metrics, semantic text analysis, and expert-guided clinical relevance scoring. Using this framework, we systematically assess 10 compositional frameworks, 3 unified models, and 5 VLMs.

Yanrui Du, Fenglei Fan, Sendong Zhao et al.

As Large Language Models (LLMs) increasingly permeate human life, their security has emerged as a critical concern, particularly their ability to maintain harmless responses to malicious instructions. Although extensive methods have improved LLMs' security, they often lead to conservative, rejection-oriented responses that compromise practical usability. This presents a key challenge: how to advance the Pareto frontier between LLMs' usability and security, rather than necessitate a trade-off between them. To address this, we propose the MoGU framework, in which the intra-layer router dynamically allocates weights by sensing hidden states, thereby balancing the contributions of security-optimized and usability-optimized variants. Despite its initial potential, the MoGU framework faces limitations such as parameter redundancy and performance bottlenecks. To overcome these, we further propose an improved MoGU_v2 framework that establishes a tighter coupling between the routers and hidden states. In MoGU_v2, routers are embedded only in layers encoding highly classifiable security features, and backbone modules are activated during router optimization to enable bidirectional adaptation. MoGU_V2 exhibits strong adaptability and stable improvements across various series of LLMs, including mainstream LLMs serving as brains in various applications, on-device LLMs optimized for resource-constrained scenarios, and reasoning LLMs tailored for user interpretability. Meanwhile, even facing risks introduced by Instruction Fine-tuning, MoGU_v2 can easily restore security without compromising the task performance gains via a simple data-mix strategy. These comprehensive improvements highlight MoGU_V2 as a robust and versatile solution for mitigating security risks in real-world applications.

Shuren Qi, Fei Wang, Tieyong Zeng et al.

Integrating invariance into data representations is a principled design in intelligent systems and web applications. Representations play a fundamental role, where systems and applications are both built on meaningful representations of digital inputs (rather than the raw data). In fact, the proper design/learning of such representations relies on priors w.r.t. the task of interest. Here, the concept of symmetry from the Erlangen Program may be the most fruitful prior -- informally, a symmetry of a system is a transformation that leaves a certain property of the system invariant. Symmetry priors are ubiquitous, e.g., translation as a symmetry of the object classification, where object category is invariant under translation. The quest for invariance is as old as pattern recognition and data mining itself. Invariant design has been the cornerstone of various representations in the era before deep learning, such as the SIFT. As we enter the early era of deep learning, the invariance principle is largely ignored and replaced by a data-driven paradigm, such as the CNN. However, this neglect did not last long before they encountered bottlenecks regarding robustness, interpretability, efficiency, and so on. The invariance principle has returned in the era of rethinking deep learning, forming a new field known as Geometric Deep Learning (GDL). In this tutorial, we will give a historical perspective of the invariance in data representations. More importantly, we will identify those research dilemmas, promising works, future directions, and web applications.

Dayang Wang, Fenglei Fan, Zhan Wu et al.

Low-dose computed tomography (LDCT) denoising is an important problem in CT research. Compared to the normal dose CT (NDCT), LDCT images are subjected to severe noise and artifacts. Recently in many studies, vision transformers have shown superior feature representation ability over convolutional neural networks (CNNs). However, unlike CNNs, the potential of vision transformers in LDCT denoising was little explored so far. To fill this gap, we propose a Convolution-free Token2Token Dilated Vision Transformer for low-dose CT denoising. The CTformer uses a more powerful token rearrangement to encompass local contextual information and thus avoids convolution. It also dilates and shifts feature maps to capture longer-range interaction. We interpret the CTformer by statically inspecting patterns of its internal attention maps and dynamically tracing the hierarchical attention flow with an explanatory graph. Furthermore, an overlapped inference mechanism is introduced to effectively eliminate the boundary artifacts that are common for encoder-decoder-based denoising models. Experimental results on Mayo LDCT dataset suggest that the CTformer outperforms the state-of-the-art denoising methods with a low computation overhead.

Chuang Niu, Mengzhou Li, Fenglei Fan et al.

Image denoising is a prerequisite for downstream tasks in many fields. Low-dose and photon-counting computed tomography (CT) denoising can optimize diagnostic performance at minimized radiation dose. Supervised deep denoising methods are popular but require paired clean or noisy samples that are often unavailable in practice. Limited by the independent noise assumption, current unsupervised denoising methods cannot process correlated noises as in CT images. Here we propose the first-of-its-kind similarity-based unsupervised deep denoising approach, referred to as Noise2Sim, that works in a nonlocal and nonlinear fashion to suppress not only independent but also correlated noises. Theoretically, Noise2Sim is asymptotically equivalent to supervised learning methods under mild conditions. Experimentally, Nosie2Sim recovers intrinsic features from noisy low-dose CT and photon-counting CT images as effectively as or even better than supervised learning methods on practical datasets visually, quantitatively and statistically. Noise2Sim is a general unsupervised denoising approach and has great potential in diverse applications.

Chuang Niu, Wenxiang Cong, Fenglei Fan et al.

Deep neural network based methods have achieved promising results for CT metal artifact reduction (MAR), most of which use many synthesized paired images for training. As synthesized metal artifacts in CT images may not accurately reflect the clinical counterparts, an artifact disentanglement network (ADN) was proposed with unpaired clinical images directly, producing promising results on clinical datasets. However, without sufficient supervision, it is difficult for ADN to recover structural details of artifact-affected CT images based on adversarial losses only. To overcome these problems, here we propose a low-dimensional manifold (LDM) constrained disentanglement network (DN), leveraging the image characteristics that the patch manifold is generally low-dimensional. Specifically, we design an LDM-DN learning algorithm to empower the disentanglement network through optimizing the synergistic network loss functions while constraining the recovered images to be on a low-dimensional patch manifold. Moreover, learning from both paired and unpaired data, an efficient hybrid optimization scheme is proposed to further improve the MAR performance on clinical datasets. Extensive experiments demonstrate that the proposed LDM-DN approach can consistently improve the MAR performance in paired and/or unpaired learning settings, outperforming competing methods on synthesized and clinical datasets.

Fenglei Fan, Jinjun Xiong, Mengzhou Li et al.

Deep learning as represented by the artificial deep neural networks (DNNs) has achieved great success in many important areas that deal with text, images, videos, graphs, and so on. However, the black-box nature of DNNs has become one of the primary obstacles for their wide acceptance in mission-critical applications such as medical diagnosis and therapy. Due to the huge potential of deep learning, interpreting neural networks has recently attracted much research attention. In this paper, based on our comprehensive taxonomy, we systematically review recent studies in understanding the mechanism of neural networks, describe applications of interpretability especially in medicine, and discuss future directions of interpretability research, such as in relation to fuzzy logic and brain science.

Fenglei Fan, Hongming Shan, Mannudeep K. Kalra et al.

Inspired by complexity and diversity of biological neurons, our group proposed quadratic neurons by replacing the inner product in current artificial neurons with a quadratic operation on input data, thereby enhancing the capability of an individual neuron. Along this direction, we are motivated to evaluate the power of quadratic neurons in popular network architectures, simulating human-like learning in the form of quadratic-neuron-based deep learning. Our prior theoretical studies have shown important merits of quadratic neurons and networks in representation, efficiency, and interpretability. In this paper, we use quadratic neurons to construct an encoder-decoder structure, referred as the quadratic autoencoder, and apply it to low-dose CT denoising. The experimental results on the Mayo low-dose CT dataset demonstrate the utility of quadratic autoencoder in terms of image denoising and model efficiency. To our best knowledge, this is the first time that the deep learning approach is implemented with a new type of neurons and demonstrates a significant potential in the medical imaging field.

Fenglei Fan, Mengzhou Li, Yueyang Teng et al.

Recently, deep learning becomes the main focus of machine learning research and has greatly impacted many important fields. However, deep learning is criticized for lack of interpretability. As a successful unsupervised model in deep learning, the autoencoder embraces a wide spectrum of applications, yet it suffers from the model opaqueness as well. In this paper, we propose a new type of convolutional autoencoders, termed as Soft Autoencoder (Soft-AE), in which the activation functions of encoding layers are implemented with adaptable soft-thresholding units while decoding layers are realized with linear units. Consequently, Soft-AE can be naturally interpreted as a learned cascaded wavelet shrinkage system. Our denoising experiments demonstrate that Soft-AE not only is interpretable but also offers a competitive performance relative to its counterparts. Furthermore, we propose a generalized linear unit (GenLU) to make an autoencoder more adaptive in nonlinearly filtering images and data, such as denoising and deblurring.

Fenglei Fan, Dayang Wang, Hengtao Guo et al.

In established network architectures, shortcut connections are often used to take the outputs of earlier layers as additional inputs to later layers. Despite the extraordinary effectiveness of shortcuts, there remain open questions on the mechanism and characteristics. For example, why are shortcuts powerful? Why do shortcuts generalize well? In this paper, we investigate the expressivity and generalizability of a novel sparse shortcut topology. First, we demonstrate that this topology can empower a one-neuron-wide deep network to approximate any univariate continuous function. Then, we present a novel width-bounded universal approximator in contrast to depth-bounded universal approximators and extend the approximation result to a family of equally competent networks. Furthermore, with generalization bound theory, we show that the proposed shortcut topology enjoys excellent generalizability. Finally, we corroborate our theoretical analyses by comparing the proposed topology with popular architectures, including ResNet and DenseNet, on well-known benchmarks and perform a saliency map analysis to interpret the proposed topology. Our work helps enhance the understanding of the role of shortcuts and suggests further opportunities to innovate neural architectures.

Fenglei Fan, Jinjun Xiong, Ge Wang

Recently, deep learning has achieved huge successes in many important applications. In our previous studies, we proposed quadratic/second-order neurons and deep quadratic neural networks. In a quadratic neuron, the inner product of a vector of data and the corresponding weights in a conventional neuron is replaced with a quadratic function. The resultant quadratic neuron enjoys an enhanced expressive capability over the conventional neuron. However, how quadratic neurons improve the expressing capability of a deep quadratic network has not been studied up to now, preferably in relation to that of a conventional neural network. Regarding this, we ask four basic questions in this paper: (1) for the one-hidden-layer network structure, is there any function that a quadratic network can approximate much more efficiently than a conventional network? (2) for the same multi-layer network structure, is there any function that can be expressed by a quadratic network but cannot be expressed with conventional neurons in the same structure? (3) Does a quadratic network give a new insight into universal approximation? (4) To approximate the same class of functions with the same error bound, is a quantized quadratic network able to enjoy a lower number of weights than a quantized conventional network? Our main contributions are the four interconnected theorems shedding light upon these four questions and demonstrating the merits of a quadratic network in terms of expressive efficiency, unique capability, compact architecture and computational capacity respectively.

Fenglei Fan, Ge Wang

Over past several years, deep learning has achieved huge successes in various applications. However, such a data-driven approach is often criticized for lack of interpretability. Recently, we proposed artificial quadratic neural networks consisting of second-order neurons in potentially many layers. In each second-order neuron, a quadratic function is used in the place of the inner product in a traditional neuron, and then undergoes a nonlinear activation. With a single second-order neuron, any fuzzy logic operation, such as XOR, can be implemented. In this sense, any deep network constructed with quadratic neurons can be interpreted as a deep fuzzy logic system. Since traditional neural networks and second-order counterparts can represent each other and fuzzy logic operations are naturally implemented in second-order neural networks, it is plausible to explain how a deep neural network works with a second-order network as the system model. In this paper, we generalize and categorize fuzzy logic operations implementable with individual second-order neurons, and then perform statistical/information theoretic analyses of exemplary quadratic neural networks.

Fenglei Fan, Ziyu Su, Yueyang Teng et al.

For manifold learning, it is assumed that high-dimensional sample/data points are embedded on a low-dimensional manifold. Usually, distances among samples are computed to capture an underlying data structure. Here we propose a metric according to angular changes along a geodesic line, thereby reflecting the underlying shape-oriented information or a topological similarity between high- and low-dimensional representations of a data cloud. Our results demonstrate the feasibility and merits of the proposed dimensionality reduction scheme.

Fenglei Fan, Ge Wang

Inspired by the fact that the neural network, as the mainstream for machine learning, has brought successes in many application areas, here we propose to use this approach for decoding hidden correlation among pseudo-random data and predicting events accordingly. With a simple neural network structure and a typical training procedure, we demonstrate the learning and prediction power of the neural network in extremely random environment. Finally, we postulate that the high sensitivity and efficiency of the neural network may allow to critically test if there could be any fundamental difference between quantum randomness and pseudo randomness, which is equivalent to the question: Does God play dice?

Fenglei Fan, Wenxiang Cong, Ge Wang

The artificial neural network is a popular framework in machine learning. To empower individual neurons, we recently suggested that the current type of neurons could be upgraded to 2nd order counterparts, in which the linear operation between inputs to a neuron and the associated weights is replaced with a nonlinear quadratic operation. A single 2nd order neurons already has a strong nonlinear modeling ability, such as implementing basic fuzzy logic operations. In this paper, we develop a general backpropagation (BP) algorithm to train the network consisting of 2nd-order neurons. The numerical studies are performed to verify of the generalized BP algorithm.

Fenglei Fan, Wenxiang Cong, Ge Wang

In machine learning, the use of an artificial neural network is the mainstream approach. Such a network consists of layers of neurons. These neurons are of the same type characterized by the two features: (1) an inner product of an input vector and a matching weighting vector of trainable parameters and (2) a nonlinear excitation function. Here we investigate the possibility of replacing the inner product with a quadratic function of the input vector, thereby upgrading the 1st order neuron to the 2nd order neuron, empowering individual neurons, and facilitating the optimization of neural networks. Also, numerical examples are provided to illustrate the feasibility and merits of the 2nd order neurons. Finally, further topics are discussed.