An Efficient Algorithm for Vertex Enumeration of Arrangement

This paper presents a state-of-the-art algorithm for the vertex enumeration problem of arrangements, which is based on the proposed new pivot rule, called the Zero rule. The Zero rule possesses several desirable properties: i) It gets rid of the objective function; ii) Its terminal satisfies uniqueness; iii) We establish the if-and-only if condition between the Zero rule and its valid reverse, which is not enjoyed by earlier rules; iv) Applying the Zero rule recursively definitely terminates in $d$ steps, where $d$ is the dimension of input variables. Because of so, given an arbitrary arrangement with $v$ vertices of $n$ hyperplanes in $\mathbb{R}^d$, the algorithm's complexity is at most $\mathcal{O}(n^2d^2v)$ and can be as low as $\mathcal{O}(nd^4v)$ if it is a simple arrangement, while Moss' algorithm takes $\mathcal{O}(nd^2v^2)$, and Avis and Fukuda's algorithm goes into a loop or skips vertices because the if-and-only-if condition between the rule they chose and its valid reverse is not fulfilled. Systematic and comprehensive experiments confirm that the Zero rule not only does not fail but also is the most efficient

Constructing Sample-to-Class Graph for Few-Shot Class-Incremental Learning

Few-shot class-incremental learning (FSCIL) aims to build machine learning model that can continually learn new concepts from a few data samples, without forgetting knowledge of old classes. The challenges of FSCIL lies in the limited data of new classes, which not only lead to significant overfitting issues but also exacerbates the notorious catastrophic forgetting problems. As proved in early studies, building sample relationships is beneficial for learning from few-shot samples. In this paper, we promote the idea to the incremental scenario, and propose a Sample-to-Class (S2C) graph learning method for FSCIL. Specifically, we propose a Sample-level Graph Network (SGN) that focuses on analyzing sample relationships within a single session. This network helps aggregate similar samples, ultimately leading to the extraction of more refined class-level features. Then, we present a Class-level Graph Network (CGN) that establishes connections across class-level features of both new and old classes. This network plays a crucial role in linking the knowledge between different sessions and helps improve overall learning in the FSCIL scenario. Moreover, we design a multi-stage strategy for training S2C model, which mitigates the training challenges posed by limited data in the incremental process. The multi-stage training strategy is designed to build S2C graph from base to few-shot stages, and improve the capacity via an extra pseudo-incremental stage. Experiments on three popular benchmark datasets show that our method clearly outperforms the baselines and sets new state-of-the-art results in FSCIL