The problem of determining whether a graph $G$ contains another graph $H$ as a minor, referred to as the minor containment problem, is a fundamental problem in the field of graph algorithms. While it is NP-complete when $G$ and $H$ are general graphs, it is sometimes tractable on more restricted graph classes. This study focuses on the case where both $G$ and $H$ are trees, known as the tree minor containment problem. Even in this case, the problem is known to be NP-complete. In contrast, polynomial-time algorithms are known for the case when both trees are caterpillars or when the maximum degree of $H$ is a constant. Our research aims to clarify the boundary of tractability and intractability for the tree minor containment problem. Specifically, we provide dichotomies for the computational complexities of the problem based on three structural parameters: the diameter, pathwidth, and path eccentricity.
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The issue of generative pretraining for vision models has persisted as a long-standing conundrum. At present, the text-to-image (T2I) diffusion model demonstrates remarkable proficiency in generating high-definition images matching textual inputs, a feat made possible through its pre-training on large-scale image-text pairs. This leads to a natural inquiry: can diffusion models be utilized to tackle visual perception tasks? In this paper, we propose a simple yet effective scheme to harness a diffusion model for visual perception tasks. Our key insight is to introduce learnable embeddings (meta prompts) to the pre-trained diffusion models to extract proper features for perception. The effect of meta prompts are two-fold. First, as a direct replacement of the text embeddings in the T2I models, it can activate task-relevant features during feature extraction. Second, it will be used to re-arrange the extracted features to ensures that the model focuses on the most pertinent features for the task on hand. Additionally, we design a recurrent refinement training strategy that fully leverages the property of diffusion models, thereby yielding stronger visual features. Extensive experiments across various benchmarks validate the effectiveness of our approach. Our approach achieves new performance records in depth estimation tasks on NYU depth V2 and KITTI, and in semantic segmentation task on CityScapes. Concurrently, the proposed method attains results comparable to the current state-of-the-art in semantic segmentation on ADE20K and pose estimation on COCO datasets, further exemplifying its robustness and versatility.
A Gr\"obner basis computation for the Weyl algebra with respect to a tropical term order and by using a homogenization-dehomogenization technique is sufficiently sluggish. A significant number of reductions to zero occur. To improve the computation, a tropical F5 algorithm is developed for this context. As a member of the family of signature-based algorithms, this algorithm keeps track of where Weyl algebra elements come from to anticipate reductions to zero. The total order for ordering module monomials or signatures in this paper is designed as close as possible to the definition of the tropical term order. As in Vaccon et al. (2021), this total order is not compatible with the tropical term order.
The quantum communication cost of computing a classical sum of distributed sources is studied over a quantum erasure multiple access channel (QEMAC). $K$ classical messages are distributed across $S$ servers, who also share quantum entanglement in advance. Each server $s\in[S]$ manipulates and sends its quantum subsystem $\mathcal{Q}_s$ to the receiver who computes the sum of the messages. The download cost from Server $s\in [S]$ is the logarithm of the dimension of $\mathcal{Q}_s$. The rate $R$ is defined as the number of instances of the sum computed at the receiver, divided by the total download cost from all the servers. In the symmetric setting with $K= {S \choose \alpha} $ messages where each message is replicated among a unique subset of $\alpha$ servers, and the answers from any $\beta$ servers may be erased, we show that the capacity (maximal rate) is $C= \max\left\{ \min \left\{ \frac{2(\alpha-\beta)}{S}, \frac{S-2\beta}{S} \right\}, \frac{\alpha-\beta}{S} \right\}$.
We explore the concept of separating systems of vertex sets of graphs. A separating system of a set $X$ is a collection of subsets of $X$ such that for any pair of distinct elements in $X$, there exists a set in the separating system that contains exactly one of the two elements. A separating system of the vertex set of a graph $G$ is called a vertex-separating path (tree) system of $G$ if the elements of the separating system are paths (trees) in the graph $G$. In this paper, we focus on the size of the smallest vertex-separating path (tree) system for different types of graphs, including trees, grids, and maximal outerplanar graphs.
We study the problem of contextual feature selection, where the goal is to learn a predictive function while identifying subsets of informative features conditioned on specific contexts. Towards this goal, we generalize the recently proposed stochastic gates (STG) Yamada et al. [2020] by modeling the probabilistic gates as conditional Bernoulli variables whose parameters are predicted based on the contextual variables. Our new scheme, termed conditional-STG (c-STG), comprises two networks: a hypernetwork that establishes the mapping between contextual variables and probabilistic feature selection parameters and a prediction network that maps the selected feature to the response variable. Training the two networks simultaneously ensures the comprehensive incorporation of context and feature selection within a unified model. We provide a theoretical analysis to examine several properties of the proposed framework. Importantly, our model leads to improved flexibility and adaptability of feature selection and, therefore, can better capture the nuances and variations in the data. We apply c-STG to simulated and real-world datasets, including healthcare, housing, and neuroscience, and demonstrate that it effectively selects contextually meaningful features, thereby enhancing predictive performance and interpretability.
Translational distance-based knowledge graph embedding has shown progressive improvements on the link prediction task, from TransE to the latest state-of-the-art RotatE. However, N-1, 1-N and N-N predictions still remain challenging. In this work, we propose a novel translational distance-based approach for knowledge graph link prediction. The proposed method includes two-folds, first we extend the RotatE from 2D complex domain to high dimension space with orthogonal transforms to model relations for better modeling capacity. Second, the graph context is explicitly modeled via two directed context representations. These context representations are used as part of the distance scoring function to measure the plausibility of the triples during training and inference. The proposed approach effectively improves prediction accuracy on the difficult N-1, 1-N and N-N cases for knowledge graph link prediction task. The experimental results show that it achieves better performance on two benchmark data sets compared to the baseline RotatE, especially on data set (FB15k-237) with many high in-degree connection nodes.
Knowledge graph completion aims to predict missing relations between entities in a knowledge graph. While many different methods have been proposed, there is a lack of a unifying framework that would lead to state-of-the-art results. Here we develop PathCon, a knowledge graph completion method that harnesses four novel insights to outperform existing methods. PathCon predicts relations between a pair of entities by: (1) Considering the Relational Context of each entity by capturing the relation types adjacent to the entity and modeled through a novel edge-based message passing scheme; (2) Considering the Relational Paths capturing all paths between the two entities; And, (3) adaptively integrating the Relational Context and Relational Path through a learnable attention mechanism. Importantly, (4) in contrast to conventional node-based representations, PathCon represents context and path only using the relation types, which makes it applicable in an inductive setting. Experimental results on knowledge graph benchmarks as well as our newly proposed dataset show that PathCon outperforms state-of-the-art knowledge graph completion methods by a large margin. Finally, PathCon is able to provide interpretable explanations by identifying relations that provide the context and paths that are important for a given predicted relation.
Incompleteness is a common problem for existing knowledge graphs (KGs), and the completion of KG which aims to predict links between entities is challenging. Most existing KG completion methods only consider the direct relation between nodes and ignore the relation paths which contain useful information for link prediction. Recently, a few methods take relation paths into consideration but pay less attention to the order of relations in paths which is important for reasoning. In addition, these path-based models always ignore nonlinear contributions of path features for link prediction. To solve these problems, we propose a novel KG completion method named OPTransE. Instead of embedding both entities of a relation into the same latent space as in previous methods, we project the head entity and the tail entity of each relation into different spaces to guarantee the order of relations in the path. Meanwhile, we adopt a pooling strategy to extract nonlinear and complex features of different paths to further improve the performance of link prediction. Experimental results on two benchmark datasets show that the proposed model OPTransE performs better than state-of-the-art methods.
It is important to detect anomalous inputs when deploying machine learning systems. The use of larger and more complex inputs in deep learning magnifies the difficulty of distinguishing between anomalous and in-distribution examples. At the same time, diverse image and text data are available in enormous quantities. We propose leveraging these data to improve deep anomaly detection by training anomaly detectors against an auxiliary dataset of outliers, an approach we call Outlier Exposure (OE). This enables anomaly detectors to generalize and detect unseen anomalies. In extensive experiments on natural language processing and small- and large-scale vision tasks, we find that Outlier Exposure significantly improves detection performance. We also observe that cutting-edge generative models trained on CIFAR-10 may assign higher likelihoods to SVHN images than to CIFAR-10 images; we use OE to mitigate this issue. We also analyze the flexibility and robustness of Outlier Exposure, and identify characteristics of the auxiliary dataset that improve performance.
Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.
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