Term Vector Space Research Articles

Enumeration of anti-invariant subspaces and Touchard's formula for the entries of the q-Hermite Catalan matrix

We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the q-Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries.

On The Fuzzy Weak Complex Vector Spaces

The objective of this paper is to present for the first time the concept of fuzzy weak complex vector spaces depending on the ring of fuzzy weak complex numbers. Also, we examine some elementary algebraic properties of fuzzy weak complex vector spaces in terms of theorems. On the other hand, we illustrate many related examples to explain the novelty of these spaces.

Infinite-dimensional linear algebra and solvability of partial differential equations

We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called in the article \emph{regular}, acting on the algebraic dual $\mathcal D^*(\Omega)$ of the space of test-functions $\mathcal D(\Omega)$. The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within $\mathcal D^*(\Omega)$. We discuss our result in contrast to and comparison with similar results about the restrictions of the regular operators on the space of Schwartz distribution $\mathcal D^\prime(\Omega)$, where these operators are often non-surjective.

On a Characterization of Finite-Dimensional Vector Spaces

We provide a characterization of the finite dimensionality of vector spaces in terms of the right-sided invertibility of linear operators on them.

Predictive article recommendation using natural language processing and machine learning to support evidence updates in domain-specific knowledge graphs

ObjectivesDescribe an augmented intelligence approach to facilitate the update of evidence for associations in knowledge graphs.MethodsNew publications are filtered through multiple machine learning study classifiers, and filtered publications are combined with articles already included as evidence in the knowledge graph. The corpus is then subjected to named entity recognition, semantic dictionary mapping, term vector space modeling, pairwise similarity, and focal entity match to identify highly related publications. Subject matter experts review recommended articles to assess inclusion in the knowledge graph; discrepancies are resolved by consensus.ResultsStudy classifiers achieved F-scores from 0.88 to 0.94, and similarity thresholds for each study type were determined by experimentation. Our approach reduces human literature review load by 99%, and over the past 12 months, 41% of recommendations were accepted to update the knowledge graph.ConclusionIntegrated search and recommendation exploiting current evidence in a knowledge graph is useful for reducing human cognition load.

Distributional semantic modeling: a revised technique to train term/word vector space models applying the ontology-related approach

We design a new technique for the distributional semantic modeling with a neural network-based approach to learn distributed term representations (or term embeddings) – term vector space models as a result, inspired by the recent ontology-related approach (using different types of contextual knowledge such as syntactic knowledge, terminological knowledge, semantic knowledge, etc.) to the identification of terms (term extraction) and relations between them (relation extraction) called semantic pre-processing technology – SPT. Our method relies on automatic term extraction from the natural language texts and subsequent formation of the problem-oriented or application-oriented (also deeply annotated) text corpora where the fundamental entity is the term (includes non-compositional and compositional terms). This gives us an opportunity to changeover from distributed word representations (or word embeddings) to distributed term representations (or term embeddings). The main practical result of our work is the development kit (set of toolkits represented as web service APIs and web application), which provides all necessary routines for the basic linguistic pre-processing and the semantic pre-processing of the natural language texts in Ukrainian for future training of term vector space models.

A novel filter feature selection method using rough set for short text data

High dimensionality problem is an important concern for short text classification due to its effect on computational cost and accuracy of classifiers. Also, short text data, besides being high dimensional, has an incomplete, inconsistent and sparse structure. Selection of important features that provide a better representation is a solution for high dimensionality problem. In this study, we developed a novel filter feature selection method, Proportional Rough Feature Selector (PRFS), which uses the rough set for a regional distinction according to the value set of term to identify documents that exactly belong to a class or that is possibly belong to a class. Documents possible to belong to a class are penalized by multiplying with a coefficient named α. Additionally, the effect of sparsity in the term vector space is calculated with the help of rough set. The PRFS is compared with state-of-the-art filter feature selection methods such as Gini index, information gain, distinguishing feature selector, recently proposed max–min ratio, and normalized difference measure methods. The comparison is carried out using various feature sizes on four different short text datasets with a Macro-F1 success measure. Experimental results demonstrated that the PRFS offers either better or competitive performance with respect to other feature selection methods in terms of Macro-F1. This study may be a pioneering study in this research field as it proposes a novel feature selection method for short text classification using a rough set theory.

Induced L-bornological vector spaces and L-Mackey convergence1

Motivated by the concept of lattice-bornological vector spaces of J. Paseka, S. Solovyov and M. Stehlík, which extends bornological vector spaces to the fuzzy setting over a complete lattice, this paper continues to study the theory of L-bornological vector spaces. The specific description of L-bornological vector spaces is presented, some properties of Lowen functors between the category of bornological vector spaces and the category of L-bornological vector spaces are discussed. In addition, the notions and some properties of L-Mackey convergence and separation in L-bornological vector spaces are showed. The equivalent characterization of separation in L-bornological vector spaces in terms of L-Mackey convergence is obtained in particular.

Combining Fact Extraction and Verification with Neural Semantic Matching Networks

The increasing concern with misinformation has stimulated research efforts on automatic fact checking. The recentlyreleased FEVER dataset introduced a benchmark factverification task in which a system is asked to verify a claim using evidential sentences from Wikipedia documents. In this paper, we present a connected system consisting of three homogeneous neural semantic matching models that conduct document retrieval, sentence selection, and claim verification jointly for fact extraction and verification. For evidence retrieval (document retrieval and sentence selection), unlike traditional vector space IR models in which queries and sources are matched in some pre-designed term vector space, we develop neural models to perform deep semantic matching from raw textual input, assuming no intermediate term representation and no access to structured external knowledge bases. We also show that Pageview frequency can also help improve the performance of evidence retrieval results, that later can be matched by using our neural semantic matching network. For claim verification, unlike previous approaches that simply feed upstream retrieved evidence and the claim to a natural language inference (NLI) model, we further enhance the NLI model by providing it with internal semantic relatedness scores (hence integrating it with the evidence retrieval modules) and ontological WordNet features. Experiments on the FEVER dataset indicate that (1) our neural semantic matching method outperforms popular TF-IDF and encoder models, by significant margins on all evidence retrieval metrics, (2) the additional relatedness score and WordNet features improve the NLI model via better semantic awareness, and (3) by formalizing all three subtasks as a similar semantic matching problem and improving on all three stages, the complete model is able to achieve the state-of-the-art results on the FEVER test set (two times greater than baseline results).1

Twisted Hilbert spaces of 3d supersymmetric gauge theories

We study aspects of 3d mathcal{N}=2 supersymmetric gauge theories on the product of a line and a Riemann surface. Performing a topological twist along the Riemann surface leads to an effective supersymmetric quantum mechanics on the line. We propose a construction of the space of supersymmetric ground states as a graded vector space in terms of a certain cohomology of generalized vortex moduli spaces on the Riemann surface. This exhibits a rich dependence on deformation parameters compatible with the topological twist, including superpotentials, real mass parameters, and background vector bundles associated to flavour symmetries. By matching spaces of supersymmetric ground states, we perform new checks of 3d abelian mirror symmetry.

Salience Estimation via Variational Auto-Encoders for Multi-Document Summarization

We propose a new unsupervised sentence salience framework for Multi-Document Summarization (MDS), which can be divided into two components: latent semantic modeling and salience estimation. For latent semantic modeling, a neural generative model called Variational Auto-Encoders (VAEs) is employed to describe the observed sentences and the corresponding latent semantic representations. Neural variational inference is used for the posterior inference of the latent variables. For salience estimation, we propose an unsupervised data reconstruction framework, which jointly considers the reconstruction for latent semantic space and observed term vector space. Therefore, we can capture the salience of sentences from these two different and complementary vector spaces. Thereafter, the VAEs-based latent semantic model is integrated into the sentence salience estimation component in a unified fashion, and the whole framework can be trained jointly by back-propagation via multi-task learning. Experimental results on the benchmark datasets DUC and TAC show that our framework achieves better performance than the state-of-the-art models.

The inner product on exterior powers of a complex vector space

We give a formula for the inner product of forms on a Hermitian vector space in terms of linear combinations of iterates of the adjoint of the Lefschetz operator. As an application, we reprove the Kobayashi–Lübke inequality for Hermite–Einstein bundles.

Rapid polynomial approximation in $\boldsymbol{L_2}$-spaces with Freud weights on the real line

The weights $W_{\alpha}(x)=\mathrm{exp}(-|x|^{\alpha})$ $(\alpha>1)$ form a subclass of Freud weights on the real line. Primarily from a functional analytic angle, we investigate the subspace of $L_{2}(\mathbb{R}, W_{\alpha}^{2} dx)$ consisting of those elements that can be rapidly approximated by polynomials. This subspace has a natural Fr\'echet topology, in which it is isomorphic to the space of rapidly decreasing sequences. We show that it consists of smooth functions and obtain concrete results on its topology. For $\alpha=2$, there is a complete and elementary description of this topological vector space in terms of the Schwartz functions.

Efficient PET-CT image retrieval using graphs embedded into a vector space.

Combined positron emission tomography and computed tomography (PET-CT) produces functional data (from PET) in relation to anatomical context (from CT) and it has made a major contribution to improved cancer diagnosis, tumour localisation, and staging. The ability to retrieve PET-CT images from large archives has potential applications in diagnosis, education, and research. PET-CT image retrieval requires the consideration of modality-specific 3D image features and spatial contextual relationships between features in both modalities. Graph-based retrieval methods have recently been applied to represent contextual relationships during PET-CT image retrieval. However, accurate methods are computationally complex, often requiring offline processing, and are unable to retrieve images at interactive rates. In this paper, we propose a method for PET-CT image retrieval using a vector space embedding of graph descriptors. Our method defines the vector space in terms of the distance between a graph representing a PET-CT image and a set of fixed-sized prototype graphs; each vector component measures the dissimilarity of the graph and a prototype. Our evaluation shows that our method is significantly faster (≈800× speedup, p <; 0.05) than retrieval using the graph-edit distance while maintaining comparable precision (5% difference, p > 0.05).

A characterisation of inner product spaces by the maximal circumradius of spheres

We give a new characterisation of inner product spaces amongst normed vector spaces in terms of the maximal circumradius of spheres. It turns out that a normed vector space is an inner product space if and only if all spheres are not degenerate, i.e., the maximal circumradius of points on the sphere equals the radius of the sphere.

Evaluation of Co-occurring Terms in Clinical Documents Using Latent Semantic Indexing

ObjectivesMeasurement of similarities between documents is typically influenced by the sparseness of the term-document matrix employed. Latent semantic indexing (LSI) may improve the results of this type of analysis.MethodsIn this study, LSI was utilized in an attempt to reduce the term vector space of clinical documents and newspaper editorials.ResultsAfter applying LSI, document similarities were revealed more clearly in clinical documents than editorials. Clinical documents which can be characterized with co-occurring medical terms, various expressions for the same concepts, abbreviations, and typographical errors showed increased improvement with regards to a correlation between co-occurring terms and document similarities.ConclusionsOur results showed that LSI can be used effectively to measure similarities in clinical documents. In addition, correlation between the co-occurrence of terms and similarities realized in this study is an important positive feature associated with LSI.

Ring of normal cones

Normal categories were introduced by Nambooripad [4], who proved that the set of normal cones in such a category forms a regular semigroup. In this paper, we specialize normal categories to what we call RR-categories, and prove that the set of normal cones in such a category forms a regular ring. As an example, we analyze the regular ring of finite-rank operators on an infinite-dimensional vector space in terms of RR-categories. This example can be realised as the characterisation of a regular ring in terms of its left ideals.

Monomial bases for broken circuit complexes

Let F be a field and let G be a finite graph with a total ordering on its edge set. Richard Stanley noted that the Stanley–Reisner ring F ( G ) of the broken circuit complex of G is Cohen–Macaulay. Jason Brown gave an explicit description of a homogeneous system of parameters for F ( G ) in terms of fundamental cocircuits in G . So F ( G ) modulo this hsop is a finite dimensional vector space. We conjecture an explicit monomial basis for this vector space in terms of the circuits of G and prove that the conjecture is true for two infinite families of graphs. We also explore an application of these ideas to bounding the number of acyclic orientations of G from above.

Spectral decomposition of real symmetric quadratic $\lambda $-matrices and its applications

Spectral decomposition provides a canonical representation of an operator over a vector space in terms of its eigenvalues and eigenfunctions. The canonical form often facilitates discussions which, otherwise, would be complicated and involved. Spectral decomposition is of fundamental importance in many applications. The well-known GLR theory generalizes the classical result of eigendecomposition to matrix polynomials of higher degrees, but its development is based on complex numbers. This paper modifies the GLR theory for the special application to real symmetric quadratic matrix polynomials, Q( A) = MX2 + CX + K, M nonsingular, subject to the specific restriction that all matrices in the representation be realvalued. It is shown that the existence of the real spectral decomposition can be characterized through the notion of real standard pair which, in turn, can be constructed from the spectral data. Applications to a variety of challenging inverse problems are discussed.

Variational form of the large deviation functional

Under exponential tightness hypothesis, we show that the large deviation functional Λ ¯ ( · ) has the same variational form as when large deviations hold. As an application, we identify the affine regularization of a rate function on any real Hausdorff topological vector space in terms of Λ ¯ ( · ) . Some classical results involving the generalized log-moment generating function are improved.