The goal of this paper is to investigate a family of optimization problems arising from list homomorphisms, and to understand what the best possible algorithms are if we restrict the problem to bounded-treewidth graphs. For a fixed $H$, the input of the optimization problem LHomVD($H$) is a graph $G$ with lists $L(v)$, and the task is to find a set $X$ of vertices having minimum size such that $(G-X,L)$ has a list homomorphism to $H$. We define analogously the edge-deletion variant LHomED($H$). This expressive family of problems includes members that are essentially equivalent to fundamental problems such as Vertex Cover, Max Cut, Odd Cycle Transversal, and Edge/Vertex Multiway Cut. For both variants, we first characterize those graphs $H$ that make the problem polynomial-time solvable and show that the problem is NP-hard for every other fixed $H$. Second, as our main result, we determine for every graph $H$ for which the problem is NP-hard, the smallest possible constant $c_H$ such that the problem can be solved in time $c^t_H\cdot n^{O(1)}$ if a tree decomposition of $G$ having width $t$ is given in the input.Let $i(H)$ be the maximum size of a set of vertices in $H$ that have pairwise incomparable neighborhoods. For the vertex-deletion variant LHomVD($H$), we show that the smallest possible constant is $i(H)+1$ for every $H$. The situation is more complex for the edge-deletion version. For every $H$, one can solve LHomED($H$) in time $i(H)^t\cdot n^{O(1)}$ if a tree decomposition of width $t$ is given. However, the existence of a specific type of decomposition of $H$ shows that there are graphs $H$ where LHomED($H$) can be solved significantly more efficiently and the best possible constant can be arbitrarily smaller than $i(H)$. Nevertheless, we determine this best possible constant and (assuming the SETH) prove tight bounds for every fixed $H$.
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We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via infinite-dimensional banded matrix factorizations and may be used to compute the modified Jacobi matrices all in linear complexity with respect to the truncation degree. A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified classical orthogonal polynomials.
This paper makes two contributions to the field of text-based patent similarity. First, it compares the performance of different kinds of patent-specific pretrained embedding models, namely static word embeddings (such as word2vec and doc2vec models) and contextual word embeddings (such as transformers based models), on the task of patent similarity calculation. Second, it compares specifically the performance of Sentence Transformers (SBERT) architectures with different training phases on the patent similarity task. To assess the models' performance, we use information about patent interferences, a phenomenon in which two or more patent claims belonging to different patent applications are proven to be overlapping by patent examiners. Therefore, we use these interferences cases as a proxy for maximum similarity between two patents, treating them as ground-truth to evaluate the performance of the different embedding models. Our results point out that, first, Patent SBERT-adapt-ub, the domain adaptation of the pretrained Sentence Transformer architecture proposed in this research, outperforms the current state-of-the-art in patent similarity. Second, they show that, in some cases, large static models performances are still comparable to contextual ones when trained on extensive data; thus, we believe that the superiority in the performance of contextual embeddings may not be related to the actual architecture but rather to the way the training phase is performed.
Deep generative models aim to learn the underlying distribution of data and generate new ones. Despite the diversity of generative models and their high-quality generation performance in practice, most of them lack rigorous theoretical convergence proofs. In this work, we aim to establish some convergence results for OT-Flow, one of the deep generative models. First, by reformulating the framework of OT-Flow model, we establish the $\Gamma$-convergence of the formulation of OT-flow to the corresponding optimal transport (OT) problem as the regularization term parameter $\alpha$ goes to infinity. Second, since the loss function will be approximated by Monte Carlo method in training, we established the convergence between the discrete loss function and the continuous one when the sample number $N$ goes to infinity as well. Meanwhile, the approximation capability of the neural network provides an upper bound for the discrete loss function of the minimizers. The proofs in both aspects provide convincing assurances for OT-Flow.
In the search for highly efficient decoders for short LDPC codes approaching maximum likelihood performance, a relayed decoding strategy, specifically activating the ordered statistics decoding process upon failure of a neural min-sum decoder, is enhanced by instilling three innovations. Firstly, soft information gathered at each step of the neural min-sum decoder is leveraged to forge a new reliability measure using a convolutional neural network. This measure aids in constructing the most reliable basis of ordered statistics decoding, bolstering the decoding process by excluding error-prone bits or concentrating them in a smaller area. Secondly, an adaptive ordered statistics decoding process is introduced, guided by a derived decoding path comprising prioritized blocks, each containing distinct test error patterns. The priority of these blocks is determined from the statistical data during the query phase. Furthermore, effective complexity management methods are devised by adjusting the decoding path's length or refining constraints on the involved blocks. Thirdly, a simple auxiliary criterion is introduced to reduce computational complexity by minimizing the number of candidate codewords before selecting the optimal estimate. Extensive experimental results and complexity analysis strongly support the proposed framework, demonstrating its advantages in terms of high throughput, low complexity, independence from noise variance, in addition to superior decoding performance.
The numerical approximation of dynamic poroelasticity, modeling flow in deformable porous media, by a family of continuous space-time finite element methods is investigated. Equal order approximation in space without any further stabilization is used for the displacement and pore pressure variable. Optimal order $L^\infty(L^2)$ error estimates are proved and numerically confirmed.
This paper introduces an innovative method for constructing copula models capable of describing arbitrary non-monotone dependence structures. The proposed method enables the creation of such copulas in parametric form, thus allowing the resulting models to adapt to diverse and intricate real-world data patterns. We apply this novel methodology to analyze the relationship between returns and trading volumes in financial markets, a domain where the existence of non-monotone dependencies is well-documented in the existing literature. Our approach exhibits superior adaptability compared to other models which have previously been proposed in the literature, enabling a deeper understanding of the dependence structure among the considered variables.
In this paper, we derive high-dimensional asymptotic properties of the Moore-Penrose inverse and the ridge-type inverse of the sample covariance matrix. In particular, the analytical expressions of the weighted sample trace moments are deduced for both generalized inverse matrices and are present by using the partial exponential Bell polynomials which can easily be computed in practice. The existent results are extended in several directions: (i) First, the population covariance matrix is not assumed to be a multiple of the identity matrix; (ii) Second, the assumption of normality is not used in the derivation; (iii) Third, the asymptotic results are derived under the high-dimensional asymptotic regime. Our findings are used to construct improved shrinkage estimators of the precision matrix, which asymptotically minimize the quadratic loss with probability one. Finally, the finite sample properties of the derived theoretical results are investigated via an extensive simulation study.
We propose a method for obtaining parsimonious decompositions of networks into higher order interactions which can take the form of arbitrary motifs.The method is based on a class of analytically solvable generative models, where vertices are connected via explicit copies of motifs, which in combination with non-parametric priors allow us to infer higher order interactions from dyadic graph data without any prior knowledge on the types or frequencies of such interactions. Crucially, we also consider 'degree--corrected' models that correctly reflect the degree distribution of the network and consequently prove to be a better fit for many real world--networks compared to non-degree corrected models. We test the presented approach on simulated data for which we recover the set of underlying higher order interactions to a high degree of accuracy. For empirical networks the method identifies concise sets of atomic subgraphs from within thousands of candidates that cover a large fraction of edges and include higher order interactions of known structural and functional significance. The method not only produces an explicit higher order representation of the network but also a fit of the network to analytically tractable models opening new avenues for the systematic study of higher order network structures.
Developing algorithms for accurate and comprehensive neural decoding of mental contents is one of the long-cherished goals in the field of neuroscience and brain-machine interfaces. Previous studies have demonstrated the feasibility of neural decoding by training machine learning models to map brain activity patterns into a semantic vector representation of stimuli. These vectors, hereafter referred as pretrained feature vectors, are usually derived from semantic spaces based solely on image and/or text features and therefore they might have a totally different characteristics than how visual stimuli is represented in the human brain, resulting in limiting the capability of brain decoders to learn this mapping. To address this issue, we propose a representation learning framework, termed brain-grounding of semantic vectors, which fine-tunes pretrained feature vectors to better align with the neural representation of visual stimuli in the human brain. We trained this model this model with functional magnetic resonance imaging (fMRI) of 150 different visual stimuli categories, and then performed zero-shot brain decoding and identification analyses on 1) fMRI and 2) magnetoencephalography (MEG). Interestingly, we observed that by using the brain-grounded vectors, the brain decoding and identification accuracy on brain data from different neuroimaging modalities increases. These findings underscore the potential of incorporating a richer array of brain-derived features to enhance performance of brain decoding algorithms.
Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.
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