An important challenge in Geometric Modeling is to classify polytopes with rational linear precision. Equivalently, in Algebraic Statistics one is interested in classifying scaled toric varieties, also known as discrete exponential families, for which the maximum likelihood estimator can be written in closed form as a rational function of the data (rational MLE). The toric fiber product (TFP) of statistical models is an operation to iteratively construct new models with rational MLE from lower dimensional ones. In this paper we introduce TFPs to the Geometric Modeling setting to construct polytopes with rational linear precision and give explicit formulae for their blending functions. A special case of the TFP is taking the Cartesian product of two polytopes and their blending functions. The Horn matrix of a statistical model with rational MLE is a key player in both Geometric Modeling and Algebraic Statistics; it proved to be fruitful providing a characterisation of those polytopes having the more restrictive property of strict linear precision. We give an explicit description of the Horn matrix of a TFP.
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Verifying the robustness of machine learning models against evasion attacks at test time is an important research problem. Unfortunately, prior work established that this problem is NP-hard for decision tree ensembles, hence bound to be intractable for specific inputs. In this paper, we identify a restricted class of decision tree ensembles, called large-spread ensembles, which admit a security verification algorithm running in polynomial time. We then propose a new approach called verifiable learning, which advocates the training of such restricted model classes which are amenable for efficient verification. We show the benefits of this idea by designing a new training algorithm that automatically learns a large-spread decision tree ensemble from labelled data, thus enabling its security verification in polynomial time. Experimental results on publicly available datasets confirm that large-spread ensembles trained using our algorithm can be verified in a matter of seconds, using standard commercial hardware. Moreover, large-spread ensembles are more robust than traditional ensembles against evasion attacks, while incurring in just a relatively small loss of accuracy in the non-adversarial setting.
Graph clustering is a longstanding research topic, and has achieved remarkable success with the deep learning methods in recent years. Nevertheless, we observe that several important issues largely remain open. On the one hand, graph clustering from the geometric perspective is appealing but has rarely been touched before, as it lacks a promising space for geometric clustering. On the other hand, contrastive learning boosts the deep graph clustering but usually struggles in either graph augmentation or hard sample mining. To bridge this gap, we rethink the problem of graph clustering from geometric perspective and, to the best of our knowledge, make the first attempt to introduce a heterogeneous curvature space to graph clustering problem. Correspondingly, we present a novel end-to-end contrastive graph clustering model named CONGREGATE, addressing geometric graph clustering with Ricci curvatures. To support geometric clustering, we construct a theoretically grounded Heterogeneous Curvature Space where deep representations are generated via the product of the proposed fully Riemannian graph convolutional nets. Thereafter, we train the graph clusters by an augmentation-free reweighted contrastive approach where we pay more attention to both hard negatives and hard positives in our curvature space. Empirical results on real-world graphs show that our model outperforms the state-of-the-art competitors.
Motivated by an application from geodesy, we introduce a novel clustering problem which is a $k$-center (or k-diameter) problem with a side constraint. For the side constraint, we are given an undirected connectivity graph $G$ on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in $G$. We call the resulting problems the connected $k$-center problem and the connected $k$-diameter problem. We prove several results on the complexity and approximability of these problems. Our main result is an $O(\log^2{k})$-approximation algorithm for the connected $k$-center and the connected $k$-diameter problem. For Euclidean metrics and metrics with constant doubling dimension, the approximation factor of this algorithm improves to $O(1)$. We also consider the special cases that the connectivity graph is a line or a tree. For the line we give optimal polynomial-time algorithms and for the case that the connectivity graph is a tree, we either give an optimal polynomial-time algorithm or a $2$-approximation algorithm for all variants of our model. We complement our upper bounds by several lower bounds.
We consider the online version of the piercing set problem, where geometric objects arrive one by one, and the online algorithm must maintain a valid piercing set for the already arrived objects by making irrevocable decisions. It is easy to observe that any deterministic online algorithm that solves this problem has a competitive ratio of at least $\Omega(n)$, which even holds when the objects are intervals. This paper considers the piercing set problem when objects are bounded scaled. We propose deterministic algorithms for bounded scaled fat objects. Piercing translated copies of an object is equivalent to the unit covering problem, which is well-studied in the online setup. Surprisingly, no upper bound of the competitive ratio was known for the unit covering problem when unit objects are anything other than balls and hypercubes. Our result gives an upper bound of the competitive ratio for the unit covering problem for various unit objects. For fixed-oriented hypercubes in $\mathbb{R}^d$ with the scaling factor in the range $[1,k]$, we propose an algorithm having a competitive ratio of at most~$3^d\log_2 k+2^d$. In the end, we show a lower bound of the competitive ratio for bounded scaled objects of various types like $\alpha$-fat objects in $\mathbb{R}^2$, axis-aligned hypercubes in $\mathbb{R}^d$, and balls in $\mathbb{R}^2$ and~$\mathbb{R}^3$.
Given a function $f$ on $\mathbb{F}_2^n$, we study the following problem. What is the largest affine subspace $\mathcal{U}$ such that when restricted to $\mathcal{U}$, all the non-trivial Fourier coefficients of $f$ are very small? For the natural class of bounded Fourier degree $d$ functions $f:\mathbb{F}_2^n \to [-1,1]$, we show that there exists an affine subspace of dimension at least $ \tilde\Omega(n^{1/d!}k^{-2})$, wherein all of $f$'s nontrivial Fourier coefficients become smaller than $ 2^{-k}$. To complement this result, we show the existence of degree $d$ functions with coefficients larger than $2^{-d\log n}$ when restricted to any affine subspace of dimension larger than $\Omega(dn^{1/(d-1)})$. In addition, we give explicit examples of functions with analogous but weaker properties. Along the way, we provide multiple characterizations of the Fourier coefficients of functions restricted to subspaces of $\mathbb{F}_2^n$ that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers.
A first-order formula is called primitive positive (pp) if it only admits the use of existential quantifiers and conjunction. Pp-formulas are a central concept in (fixed-template) constraint satisfaction since CSP($\Gamma$) can be viewed as the problem of deciding the primitive positive theory of $\Gamma$, and pp-definability captures gadget reductions between CSPs. An important class of tractable constraint languages $\Gamma$ is characterized by having few subpowers, that is, the number of $n$-ary relations pp-definable from $\Gamma$ is bounded by $2^{p(n)}$ for some polynomial $p(n)$. In this paper we study a restriction of this property, stating that every pp-definable relation is definable by a pp-formula of polynomial length. We conjecture that the existence of such short definitions is actually equivalent to $\Gamma$ having few subpowers, and verify this conjecture for a large subclass that, in particular, includes all constraint languages on three-element domains. We furthermore discuss how our conjecture imposes an upper complexity bound of co-NP on the subpower membership problem of algebras with few subpowers.
The first step when solving an infinite-dimensional eigenvalue problem is often to discretize it. We show that one must be extremely careful when discretizing nonlinear eigenvalue problems. Using examples, we show that discretization can: (1) introduce spurious eigenvalues, (2) entirely miss spectra, and (3) bring in severe ill-conditioning. While there are many eigensolvers for solving matrix nonlinear eigenvalue problems, we propose a solver for general holomorphic infinite-dimensional nonlinear eigenvalue problems that avoids discretization issues, which we prove is stable and converges. Moreover, we provide an algorithm that computes the problem's pseudospectra with explicit error control, allowing verification of computed spectra. The algorithm and numerical examples are publicly available in $\texttt{infNEP}$, which is a software package written in MATLAB.
Training machines to understand natural language and interact with humans is an elusive and essential task of artificial intelligence. A diversity of dialogue systems has been designed with the rapid development of deep learning techniques, especially the recent pre-trained language models (PrLMs). Among these studies, the fundamental yet challenging type of task is dialogue comprehension whose role is to teach the machines to read and comprehend the dialogue context before responding. In this paper, we review the previous methods from the technical perspective of dialogue modeling for the dialogue comprehension task. We summarize the characteristics and challenges of dialogue comprehension in contrast to plain-text reading comprehension. Then, we discuss three typical patterns of dialogue modeling. In addition, we categorize dialogue-related pre-training techniques which are employed to enhance PrLMs in dialogue scenarios. Finally, we highlight the technical advances in recent years and point out the lessons from the empirical analysis and the prospects towards a new frontier of researches.
The recent proliferation of knowledge graphs (KGs) coupled with incomplete or partial information, in the form of missing relations (links) between entities, has fueled a lot of research on knowledge base completion (also known as relation prediction). Several recent works suggest that convolutional neural network (CNN) based models generate richer and more expressive feature embeddings and hence also perform well on relation prediction. However, we observe that these KG embeddings treat triples independently and thus fail to cover the complex and hidden information that is inherently implicit in the local neighborhood surrounding a triple. To this effect, our paper proposes a novel attention based feature embedding that captures both entity and relation features in any given entity's neighborhood. Additionally, we also encapsulate relation clusters and multihop relations in our model. Our empirical study offers insights into the efficacy of our attention based model and we show marked performance gains in comparison to state of the art methods on all datasets.
Multimodal sentiment analysis is a very actively growing field of research. A promising area of opportunity in this field is to improve the multimodal fusion mechanism. We present a novel feature fusion strategy that proceeds in a hierarchical fashion, first fusing the modalities two in two and only then fusing all three modalities. On multimodal sentiment analysis of individual utterances, our strategy outperforms conventional concatenation of features by 1%, which amounts to 5% reduction in error rate. On utterance-level multimodal sentiment analysis of multi-utterance video clips, for which current state-of-the-art techniques incorporate contextual information from other utterances of the same clip, our hierarchical fusion gives up to 2.4% (almost 10% error rate reduction) over currently used concatenation. The implementation of our method is publicly available in the form of open-source code.
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