This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to characterize the decision boundaries of a simple network of the form (Affine, ReLU, Affine). Our main finding is that the decision boundaries are a subset of a tropical hypersurface, which is intimately related to a polytope formed by the convex hull of two zonotopes. The generators of these zonotopes are functions of the network parameters. This geometric characterization provides new perspectives to three tasks. (i) We propose a new tropical perspective to the lottery ticket hypothesis, where we view the effect of different initializations on the tropical geometric representation of a network's decision boundaries. (ii) Moreover, we propose new tropical based optimization reformulations that directly influence the decision boundaries of the network for the task of network pruning. (iii) At last, we discuss the reformulation of the generation of adversarial attacks in a tropical sense. We demonstrate that one can construct adversaries in a new tropical setting by perturbing a specific set of decision boundaries by perturbing a set of parameters in the network.
We establish, for the first time, connections between feedforward neural networks with ReLU activation and tropical geometry --- we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden layer can be characterized by zonotopes, which serve as building blocks for deeper networks; we relate decision boundaries of such neural networks to tropical hypersurfaces, a major object of study in tropical geometry; and we prove that linear regions of such neural networks corresp…
Geometric Capacity of Transformers: A Tropical Geometry Perspective
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学人工智能学院）
School of Geosciences, China University of Petroleum（中国石油大学地球科学学院）；College of Computer Science and Technology, China University of Petroleum（中国石油大学计算机科学与技术学院）
Tropical cyclones (TC) generally carry large amounts of water vapor and can cause large-scale extreme rainfall. Passive microwave rainfall (PMR) estimation of TC with high spatial and temporal resolution is crucial for disaster warning of TC, but remains a challenging problem due to the low temporal resolution of microwave sensors. This study attempts to solve this problem by directly forecasting PMR from satellite infrared (IR) images of TC. We develop a generative adversarial network (GAN) to convert IR images into PMR, and establish the mapping relationship between TC cloud-top bright tempe…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；National Technical University of Athens（雅典国立技术大学）
In this paper we study an emerging class of neural networks based on the morphological operators of dilation and erosion. We explore these networks mathematically from a tropical geometry perspective as well as mathematical morphology. Our contributions are threefold. First, we examine the training of morphological networks via Difference-of-Convex programming methods and extend a binary morphological classifier to multiclass tasks. Second, we focus on the sparsity of dense morphological networks trained via gradient descent algorithms and compare their performance to their linear counterparts…
Deep learning models have been the subject of study from various perspectives, for example, their training process, interpretation, generalization error, robustness to adversarial attacks, etc. A trained model is defined by its decision boundaries, and therefore, many of the studies about deep learning models speculate about the decision boundaries, and sometimes make simplifying assumptions about them. So far, finding exact points on the decision boundaries of trained deep models has been considered an intractable problem. Here, we compute exact points on the decision boundaries of these mode…
Ulsan National Institute of Science and Technology（蔚山国立科学技术研究院）；Korea Advanced Institute of Science and Technology（韩国高等科学技术研究院）
Deep generative neural networks (DGNNs) have achieved realistic and high-quality data generation. In particular, the adversarial training scheme has been applied to many DGNNs and has exhibited powerful performance. Despite of recent advances in generative networks, identifying the image generation mechanism still remains challenging. In this paper, we present an explorative sampling algorithm to analyze generation mechanism of DGNNs. Our method efficiently obtains samples with identical attributes from a query image in a perspective of the trained model. We define generative boundaries which …
University of Maryland, College Park（马里兰大学帕克分校）；Rice University（莱斯大学）；New York University（纽约大学）
We discuss methods for visualizing neural network decision boundaries and decision regions. We use these visualizations to investigate issues related to reproducibility and generalization in neural network training. We observe that changes in model architecture (and its associate inductive bias) cause visible changes in decision boundaries, while multiple runs with the same architecture yield results with strong similarities, especially in the case of wide architectures. We also use decision boundary methods to visualize double descent phenomena. We see that decision boundary reproducibility d…
Important insights towards the explainability of neural networks reside in the characteristics of their decision boundaries. In this work, we borrow tools from the field of adversarial robustness, and propose a new perspective that relates dataset features to the distance of samples to the decision boundary. This enables us to carefully tweak the position of the training samples and measure the induced changes on the boundaries of CNNs trained on large-scale vision datasets. We use this framework to reveal some intriguing properties of CNNs. Specifically, we rigorously confirm that neural netw…
The Boundaries of Fair AI in Medical Image Prognosis: A Causal Perspective
Department of Computer Science and Engineering, The Ohio State University（计算机科学与工程系，俄亥俄州立大学）；Department of Biomedical Informatics, The Ohio State University（生物医学信息学系，俄亥俄州立大学）；Department of Ophthalmology and Visual Sciences, The Ohio State University（眼科学与视觉科学系，俄亥俄州立大学）
Mamba Guided Boundary Prior Matters: A New Perspective for Generalized Polyp Segmentation
University of Surrey（萨里大学）；Côte d’Azur University（蔚蓝海岸大学）；Malaviya National Institute of Technology Jaipur（贾伊普尔马拉维亚国立理工学院）；University of South Dakota（南达科他大学）
This paper provides a theoretical justification of the superior classification performance of deep rectifier networks over shallow rectifier networks from the geometrical perspective of piecewise linear (PWL) classifier boundaries. We show that, for a given threshold on the approximation error, the required number of boundary facets to approximate a general smooth boundary grows exponentially with the dimension of the data, and thus the number of boundary facets, referred to as boundary resolution, of a PWL classifier is an important quality measure that can be used to estimate a lower bound o…
Deep neural network classifiers partition input space into high confidence regions for each class. The geometry of these class manifolds (CMs) is widely studied and intimately related to model performance; for example, the margin depends on CM boundaries. We exploit the notions of Gaussian width and Gordon's escape theorem to tractably estimate the effective dimension of CMs and their boundaries through tomographic intersections with random affine subspaces of varying dimension. We show several connections between the dimension of CMs, generalization, and robustness. In particular we investiga…
We study the generalization of deep learning models in relation to the convex hull of their training sets. A trained image classifier basically partitions its domain via decision boundaries and assigns a class to each of those partitions. The location of decision boundaries inside the convex hull of training set can be investigated in relation to the training samples. However, our analysis shows that in standard image classification datasets, all testing images are considerably outside that convex hull, in the pixel space, in the wavelet space, and in the internal representations learned by de…
Understanding and Mitigating Overrefusal in LLMs from an Unveiling Perspective of Safety Decision Boundary
The State Key Laboratory of Blockchain and Data Security（区块链与数据安全国家重点实验室）；Hangzhou High-Tech Zone (Binjiang) Institute of Blockchain and Data Security（杭州高新技术区（滨江）区块链与数据安全研究院）；Language and Machine Intelligence Department, Ant Group（蚂蚁集团语言与机器智能部门）
