搜索结果：Complementary

In this article, we characterize the class of complementary edge ideals which satisfy the licci property in terms of the underlying graph. Using this characterization, we associate the licci property of a complementary edge ideal to its other algebraic properties. Finally, we provide two different probability regimes for which the complementary edge ideals of random graphs are licci with high probability and not licci with high probability respectively.

The present research presents potentials and complementary potentials used in the one-dimensional nonlocal integral formulations. The pure stress and the pure strain nonlocal formulations were considered. While the potential used in the strain driven formulation is well known, the complementary potential has not yet been presented in the literature. The same applies to the stress driven formulation. The equivalent formulations are obtained by resorting to the Legendre transformation, and their equivalence is proved. It is also shown that these results can be used to postulate a novel potential, i.e. a kind of mixed stress-strain potential, which is, however, as ill-conditioned as the pure strain-driven formulation. Finally, an example is given that practically confirms that the stress-driven formulations resulting from the potential and the complementary potential are equivalent.

Artificial intelligence explanations can make complex predictive models more comprehensible. To be effective, however, they should anticipate and mitigate possible misinterpretations, e.g., arising when users infer incorrect information that is not explicitly conveyed. To this end, we propose complementary explanations -- a novel method that pairs explanations to compensate for their respective limitations. A complementary explanation adds insights that clarify potential misconceptions stemming from the primary explanation while ensuring their coherency and avoiding redundancy. We introduce a framework for designing and evaluating complementary explanation pairs based on pertinent qualitative properties and quantitative metrics. Our approach allows to construct complementary explanations that minimise the chance of their misinterpretation.

Knowledge distillation (KD)transfers the dark knowledge from a complex teacher to a compact student. However, heterogeneous architecture distillation, such as Vision Transformer (ViT) to ResNet18, faces challenges due to differences in spatial feature representations.Traditional KD methods are mostly designed for homogeneous architectures and hence struggle to effectively address the disparity. Although heterogeneous KD approaches have been developed recently to solve these issues, they often incur high computational costs and complex designs, or overly rely on logit alignment, which limits their ability to leverage the complementary features. To overcome these limitations, we propose Heterogeneous Complementary Distillation (HCD),a simple yet effective framework that integrates complementary teacher and student features to align representations in shared logits.These logits are decomposed and constrained to facilitate diverse knowledge transfer to the student. Specifically, HCD processes the student's intermediate features through convolutional projector and adaptive pooling, concatenates them with teacher's feature from the penultimate layer and then maps them via the Complementa

Quantum signal processing is a framework for implementing polynomial functions on quantum computers. To implement a given polynomial $P$, one must first construct a corresponding complementary polynomial $Q$. Existing approaches to this problem employ numerical methods that are not amenable to explicit error analysis. We present a new approach to complementary polynomials using complex analysis. Our main mathematical result is a contour integral representation for a canonical complementary polynomial. On the unit circle, this representation has a particularly simple and efficacious Fourier analytic interpretation, which we use to develop a Fast Fourier Transform-based algorithm for the efficient calculation of $Q$ in the monomial basis with explicit error guarantees. Numerical evidence that our algorithm outperforms the state-of-the-art optimization-based method for computing complementary polynomials is provided.

While precise data observation is essential for the learning processes of predictive models, it can be challenging owing to factors such as insufficient observation accuracy, high collection costs, and privacy constraints. In this paper, we examines cases where some qualitative features are unavailable as precise information indicating "what it is," but rather as complementary information indicating "what it is not." We refer to features defined by precise information as ordinary features (OFs) and those defined by complementary information as complementary features (CFs). We then formulate a new learning scenario termed Complementary Feature Learning (CFL), where predictive models are constructed using instances consisting of OFs and CFs. The simplest formalization of CFL applies conventional supervised learning directly using the observed values of CFs. However, this approach does not resolve the ambiguity associated with CFs, making learning challenging and complicating the interpretation of the predictive model's specific predictions. Therefore, we derive an objective function from an information-theoretic perspective to estimate the OF values corresponding to CFs and to predic

We revisit the topic of power-free morphisms, focusing on the properties of the class of complementary morphisms. Such morphisms are defined over a $2$-letter alphabet, and map the letters 0 and 1 to complementary words. We prove that every prefix of the famous Thue-Morse word $\mathbf{t}$ gives a complementary morphism that is $3^+$-free and hence $α$-free for every real number $α>3$. We also describe, using a 4-state binary finite automaton, the lengths of all prefixes of $\mathbf{t}$ that give cubefree complementary morphisms. Next, we show that $3$-free (cubefree) complementary morphisms of length $k$ exist for all $k ot\in \{3,6\}$. Moreover, if $k$ is not of the form $3\cdot2^n$, then the images of letters can be chosen to be factors of $\mathbf{t}$. Finally, we observe that each cubefree complementary morphism is also $α$-free for some $α<3$; in contrast, no binary morphism that maps each letter to a word of length 3 (resp., a word of length 6) is $α$-free for any $α<3$. In addition to more traditional techniques of combinatorics on words, we also rely on the Walnut theorem-prover. Its use and limitations are discussed.

We call a linear code $C$ with length $n$ over a field $F$, a linear complementary equi-dual code, when there exists a linear code $D$ over $F$ such that $D$ is permutation equivalent to $C^\perp$ and $(C,D)$ is a linear complementary pair of codes, that is, $C+ D=F^n$ and $C\cap D=0$. We first state a necessary condition on a code $C$ to be linear complementary equi-dual. Then, we conjecture that this necessary condition is also sufficient and present several statements which support this conjecture.

In this paper, we introduce the concept of complementary edge ideals of graphs and study their algebraic properties and invariants.

A cyclic complementary extension of a finite group $A$ is a finite group $G$ which contains $A$ and a cyclic subgroup $C$ such that $A\cap C=\{1_G\}$ and $G=AC$. For any fixed generator $c$ of the cyclic factor $C=\langle c\rangle$ of order $n$ in a cyclic complementary extension $G=AC$, the equations $cx=\varphi(x)c^{Π(x)}$, $x\in A$, determine a permutation $\varphi:A\to A$ and a function $Π:A\to\mathbb{Z}_n$ on $A$ characterized by the properties: (a) $\varphi(1_A)=1_A$ and $Π(1_A)\equiv1\pmod{n}$; (b) $\varphi(xy)=\varphi(x)\varphi^{Π(x)}(y)$ and $Π(xy)\equiv\sum_{i=1}^{Π(x)}Π(\varphi^{i-1}(y))\pmod{n}$, for all $x,y\in A$. The permutation $\varphi$ is called a skew-morphism of $A$ and has already been extensively studied. One of the main contributions of the present paper is the recognition of the importance of the function $Π$, which we call the extended power function associated with $\varphi$. We show that {\em every} cyclic complementary extension of $A$ is determined and can be constructed from a skew-morphism $\varphi$ of $A$ and an extended power function $Π$ associated with $\varphi$. As an application, we present a classification of cyclic complementary extensions of

In recent years, linear complementary pairs (LCP) of codes and linear complementary dual (LCD) codes have gained significant attention due to their applications in coding theory and cryptography. In this work, we construct explicit LCPs of codes and LCD codes from function fields of genus $g \geq 1$. To accomplish this, we present pairs of suitable divisors giving rise to non-special divisors of degree $g-1$ in the function field. The results are applied in constructing LCPs of algebraic geometry codes and LCD algebraic geometry (AG) codes in Kummer extensions, hyperelliptic function fields, and elliptic curves.

A weakly-supervised learning framework named as complementary-label learning has been proposed recently, where each sample is equipped with a single complementary label that denotes one of the classes the sample does not belong to. However, the existing complementary-label learning methods cannot learn from the easily accessible unlabeled samples and samples with multiple complementary labels, which are more informative. In this paper, to remove these limitations, we propose the novel multi-complementary and unlabeled learning framework that allows unbiased estimation of classification risk from samples with any number of complementary labels and unlabeled samples, for arbitrary loss functions and models. We first give an unbiased estimator of the classification risk from samples with multiple complementary labels, and then further improve the estimator by incorporating unlabeled samples into the risk formulation. The estimation error bounds show that the proposed methods are in the optimal parametric convergence rate. Finally, the experiments on both linear and deep models show the effectiveness of our methods.

Traditional approaches for complementary product recommendations rely on behavioral and non-visual data such as customer co-views or co-buys. However, certain domains such as fashion are primarily visual. We propose a framework that harnesses visual cues in an unsupervised manner to learn the distribution of co-occurring complementary items in real world images. Our model learns a non-linear transformation between the two manifolds of source and target complementary item categories (e.g., tops and bottoms in outfits). Given a large dataset of images containing instances of co-occurring object categories, we train a generative transformer network directly on the feature representation space by casting it as an adversarial optimization problem. Such a conditional generative model can produce multiple novel samples of complementary items (in the feature space) for a given query item. The final recommendations are selected from the closest real world examples to the synthesized complementary features. We apply our framework to the task of recommending complementary tops for a given bottom clothing item. The recommendations made by our system are diverse, and are favored by human expert

In this paper, we study the classification problem in which we have access to easily obtainable surrogate for true labels, namely complementary labels, which specify classes that observations do \textbf{not} belong to. Let $Y$ and $\bar{Y}$ be the true and complementary labels, respectively. We first model the annotation of complementary labels via transition probabilities $P(\bar{Y}=i|Y=j), i eq j\in\{1,\cdots,c\}$, where $c$ is the number of classes. Previous methods implicitly assume that $P(\bar{Y}=i|Y=j), \forall i eq j$, are identical, which is not true in practice because humans are biased toward their own experience. For example, as shown in Figure 1, if an annotator is more familiar with monkeys than prairie dogs when providing complementary labels for meerkats, she is more likely to employ "monkey" as a complementary label. We therefore reason that the transition probabilities will be different. In this paper, we propose a framework that contributes three main innovations to learning with \textbf{biased} complementary labels: (1) It estimates transition probabilities with no bias. (2) It provides a general method to modify traditional loss functions and extends standard d

The pair $(Q, \mathscr{K})$ is a {\it knowledge space} if $\bigcup\mathscr{K}=Q$ and $\mathscr{K}$ is closed under union, where $Q$ is a nonempty set and $\mathscr{K}$ is a family of subsets of $Q$. A knowledge space $(Q, \mathscr{K})$ is called {\it complementary} if there exists a non-discrete knowledge space $(Q, \mathscr{L})$ such that the following (i) and (ii) satisfy: (i) for any $q\in Q$, there are finitely many $K_{1}, \cdots, K_{n}\in \mathscr{K}$ and $L_{1}, \cdots, L_{m}\in \mathscr{L}$ such that $$(\bigcap_{i=1}^{n}K_{i})\cap (\bigcap_{j=1}^{m}L_{j})=\{q\};$$ (ii) $\mathscr{K}\cap \mathscr{L}=\{\emptyset, Q\}$. In this paper, the existence of a complementary knowledge space for each knowledge space is proved, and a method of the construction of complementary finite knowledge spaces is given.

Given a graph $G$ with vertices $\{v_1,\ldots,v_n\}$, we define $\mathcal{S}(G)$ to be the set of symmetric matrices $A=[a_{i,j}]$ such that for $i e j$ we have $a_{i,j} e 0$ if and only if $v_iv_j\in E(G)$. Motivated by the Graph Complement Conjecture, we say that a graph $G$ is complementary vanishing if there exist matrices $A \in \mathcal{S}(G)$ and $B \in \mathcal{S}(\overline{G})$ such that $AB=O$. We provide combinatorial conditions for when a graph is or is not complementary vanishing, and we characterize which graphs are complementary vanishing in terms of certain minimal complementary vanishing graphs. In addition to this, we determine which graphs on at most $8$ vertices are complementary vanishing.

Given a positive, non-increasing sequence $a$ with finite sum equal to $1$, we consider the family of all closed subsets of $[0,1]$ whose complementary open intervals have lengths given by a rearrangement of the sequence $a$. We study the full range of possible $θ$-intermediate dimensions of these sets and, under suitable assumptions on the sequence, we show that this range forms a closed interval, whose endpoints we compute explicitly. This paper fills a gap in the literature concerning the dimensional properties of complementary sets.

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study binary linear complementary dual $[n,k]$ codes with the largest minimum weight among all binary linear complementary dual $[n,k]$ codes. We characterize binary linear complementary dual codes with the largest minimum weight for small dimensions. A complete classification of binary linear complementary dual $[n,k]$ codes with the largest minimum weight is also given for $1 \le k \le n \le 16$.

In this paper we express the difference of two complementary Beatty sequences, as the sum of two Beatty sequences closely related to them. In the process we introduce a new Algorithm that generalizes the well known Minimum Excluded algorithm and provides a method to generate combinatorially any pair of complementary Beatty sequences.