Where We Have Arrived in Proving the Emergence of Sparse Symbolic Concepts in AI Models

This paper aims to prove the emergence of symbolic concepts in well-trained AI models. We prove that if (1) the high-order derivatives of the model output w.r.t. the input variables are all zero, (2) the AI model can be used on occluded samples and will yield higher confidence when the input sample is less occluded, and (3) the confidence of the AI model does not significantly degrade on occluded samples, then the AI model will encode sparse interactive concepts. Each interactive concept represents an interaction between a specific set of input variables, and has a certain numerical effect on the inference score of the model. Specifically, it is proved that the inference score of the model can always be represented as the sum of the interaction effects of all interactive concepts. In fact, we hope to prove that conditions for the emergence of symbolic concepts are quite common. It means that for most AI models, we can usually use a small number of interactive concepts to mimic the model outputs on any arbitrarily masked samples.

Can the Inference Logic of Large Language Models be Disentangled into Symbolic Concepts?

In this paper, we explain the inference logic of large language models (LLMs) as a set of symbolic concepts. Many recent studies have discovered that traditional DNNs usually encode sparse symbolic concepts. However, because an LLM has much more parameters than traditional DNNs, whether the LLM also encodes sparse symbolic concepts is still an open problem. Therefore, in this paper, we propose to disentangle the inference score of LLMs for dialogue tasks into a small number of symbolic concepts. We verify that we can use those sparse concepts to well estimate all inference scores of the LLM on all arbitrarily masking states of the input sentence. We also evaluate the transferability of concepts encoded by an LLM and verify that symbolic concepts usually exhibit high transferability across similar input sentences. More crucially, those symbolic concepts can be used to explain the exact reasons accountable for the LLM's prediction errors.

Concept-Level Explanation for the Generalization of a DNN

This paper explains the generalization power of a deep neural network (DNN) from the perspective of interactive concepts. Many recent studies have quantified a clear emergence of interactive concepts encoded by the DNN, which have been observed on different DNNs during the learning process. Therefore, in this paper, we investigate the generalization power of each interactive concept, and we use the generalization power of different interactive concepts to explain the generalization power of the entire DNN. Specifically, we define the complexity of each interactive concept. We find that simple concepts can be better generalized to testing data than complex concepts. The DNN with strong generalization power usually learns simple concepts more quickly and encodes fewer complex concepts. More crucially, we discover the detouring dynamics of learning complex concepts, which explain both the high learning difficulty and the low generalization power of complex concepts.

Defects of Convolutional Decoder Networks in Frequency Representation

In this paper, we prove representation bottlenecks of a cascaded convolutional decoder network, considering the capacity of representing different frequency components of an input sample. We conduct the discrete Fourier transform on each channel of the feature map in an intermediate layer of the decoder network. Then, we introduce the rule of the forward propagation of such intermediate-layer spectrum maps, which is equivalent to the forward propagation of feature maps through a convolutional layer. Based on this, we find that each frequency component in the spectrum map is forward propagated independently with other frequency components. Furthermore, we prove two bottlenecks in representing feature spectrums. First, we prove that the convolution operation, the zero-padding operation, and a set of other settings all make a convolutional decoder network more likely to weaken high-frequency components. Second, we prove that the upsampling operation generates a feature spectrum, in which strong signals repetitively appears at certain frequencies.

Batch Normalization Is Blind to the First and Second Derivatives of the Loss

In this paper, we prove the effects of the BN operation on the back-propagation of the first and second derivatives of the loss. When we do the Taylor series expansion of the loss function, we prove that the BN operation will block the influence of the first-order term and most influence of the second-order term of the loss. We also find that such a problem is caused by the standardization phase of the BN operation. Experimental results have verified our theoretical conclusions, and we have found that the BN operation significantly affects feature representations in specific tasks, where losses of different samples share similar analytic formulas.

Why Adversarial Training of ReLU Networks Is Difficult?

This paper mathematically derives an analytic solution of the adversarial perturbation on a ReLU network, and theoretically explains the difficulty of adversarial training. Specifically, we formulate the dynamics of the adversarial perturbation generated by the multi-step attack, which shows that the adversarial perturbation tends to strengthen eigenvectors corresponding to a few top-ranked eigenvalues of the Hessian matrix of the loss w.r.t. the input. We also prove that adversarial training tends to strengthen the influence of unconfident input samples with large gradient norms in an exponential manner. Besides, we find that adversarial training strengthens the influence of the Hessian matrix of the loss w.r.t. network parameters, which makes the adversarial training more likely to oscillate along directions of a few samples, and boosts the difficulty of adversarial training. Crucially, our proofs provide a unified explanation for previous findings in understanding adversarial training.

Interpreting Representation Quality of DNNs for 3D Point Cloud Processing

In this paper, we evaluate the quality of knowledge representations encoded in deep neural networks (DNNs) for 3D point cloud processing. We propose a method to disentangle the overall model vulnerability into the sensitivity to the rotation, the translation, the scale, and local 3D structures. Besides, we also propose metrics to evaluate the spatial smoothness of encoding 3D structures, and the representation complexity of the DNN. Based on such analysis, experiments expose representation problems with classic DNNs, and explain the utility of the adversarial training.

Interpretable Compositional Convolutional Neural Networks

The reasonable definition of semantic interpretability presents the core challenge in explainable AI. This paper proposes a method to modify a traditional convolutional neural network (CNN) into an interpretable compositional CNN, in order to learn filters that encode meaningful visual patterns in intermediate convolutional layers. In a compositional CNN, each filter is supposed to consistently represent a specific compositional object part or image region with a clear meaning. The compositional CNN learns from image labels for classification without any annotations of parts or regions for supervision. Our method can be broadly applied to different types of CNNs. Experiments have demonstrated the effectiveness of our method.

Automated Segmentation of Brain Gray Matter Nuclei on Quantitative Susceptibility Mapping Using Deep Convolutional Neural Network

Abnormal iron accumulation in the brain subcortical nuclei has been reported to be correlated to various neurodegenerative diseases, which can be measured through the magnetic susceptibility from the quantitative susceptibility mapping (QSM). To quantitively measure the magnetic susceptibility, the nuclei should be accurately segmented, which is a tedious task for clinicians. In this paper, we proposed a double-branch residual-structured U-Net (DB-ResUNet) based on 3D convolutional neural network (CNN) to automatically segment such brain gray matter nuclei. To better tradeoff between segmentation accuracy and the memory efficiency, the proposed DB-ResUNet fed image patches with high resolution and the patches with low resolution but larger field of view into the local and global branches, respectively. Experimental results revealed that by jointly using QSM and T$_\text{1}$ weighted imaging (T$_\text{1}$WI) as inputs, the proposed method was able to achieve better segmentation accuracy over its single-branch counterpart, as well as the conventional atlas-based method and the classical 3D-UNet structure. The susceptibility values and the volumes were also measured, which indicated that the measurements from the proposed DB-ResUNet are able to present high correlation with values from the manually annotated regions of interest.