ZiHao Zhang | Searcharxiv

arXiv2025

Fast Evaluation of Generalized Todd Polynomials: Applications to MacMahon's Partition Analysis and Integer Programming

The Todd polynomials, denoted as $td_k(b_1,b_2,\ldots,b_m)$, are characterised by their generating functions: $$\sum_{k\ge 0} td_k s^k = \prod_{i=1}^m \frac{b_i s}{e^{b_i s}-1}.$$ These polynomials serve as fundamental components in the Todd class of toric varieties, a concept of significant relevance in the study of lattice polytopes and number theory. We identify that generalised Todd polynomials emerge naturally within the framework of MacMahon's partition analysis, particularly in the context of computing Ehrhart series. We introduce an efficient method for the evaluation of generalised Todd polynomials for numerical values of $b_i$. This is achieved through the development of expedited operations in the quotient ring $\mathbb{Z}_p[[s]]$ modulo $s^{d}$, where $p$ is a large prime. The practical implications of our work are demonstrated through two applications: firstly, we facilitate a recalculated resolution of the Ehrhart series for magic squares of order 6, a problem initially addressed by the first author, reducing computation time from 70 days to approximately 1 day; secondly, we present a polynomial-time algorithm for Integer Linear Programming when the dimension is fixed, exhibiting a notable enhancement in computational efficiency.

math.CO↗

arXiv2022

Towards All-around Knowledge Transferring: Learning From Task-irrelevant Labels

Deep neural models have hitherto achieved significant performances on numerous classification tasks, but meanwhile require sufficient manually annotated data. Since it is extremely time-consuming and expensive to annotate adequate data for each classification task, learning an empirically effective model with generalization on small dataset has received increased attention. Existing efforts mainly focus on transferring task-relevant knowledge from other similar data to tackle the issue. These approaches have yielded remarkable improvements, yet neglecting the fact that the task-irrelevant features could bring out massive negative transfer effects. To date, no large-scale studies have been performed to investigate the impact of task-irrelevant features, let alone the utilization of this kind of features. In this paper, we firstly propose Task-Irrelevant Transfer Learning (TIRTL) to exploit task-irrelevant features, which mainly are extracted from task-irrelevant labels. Particularly, we suppress the expression of task-irrelevant information and facilitate the learning process of classification. We also provide a theoretical explanation of our method. In addition, TIRTL does not conflict with those that have previously exploited task-relevant knowledge and can be well combined to enable the simultaneous utilization of task-relevant and task-irrelevant features for the first time. In order to verify the effectiveness of our theory and method, we conduct extensive experiments on facial expression recognition and digit recognition tasks. Our source code will be also available in the future for reproducibility.

cs.LG↗