Combining word semantics within complex Hilbert space for information retrieval

Complex numbers are a fundamental aspect of the mathematical formalism of quantum physics. Quantum-like models developed outside physics often overlooked the role of complex numbers. Specifically, previous models in Information Retrieval (IR) ignored complex numbers. We argue that to advance the use of quantum models of IR, one has to lift the constraint of real-valued representations of the information space, and package more information within the representation by means of complex numbers. As a first attempt, we propose a complex-valued representation for IR, which explicitly uses complex valued Hilbert spaces, and thus where terms, documents and queries are represented as complex-valued vectors. The proposal consists of integrating distributional semantics evidence within the real component of a term vector; whereas, ontological information is encoded in the imaginary component. Our proposal has the merit of lifting the role of complex numbers from a computational byproduct of the model to the very mathematical texture that unifies different levels of semantic information. An empirical instantiation of our proposal is tested in the TREC Medical Record task of retrieving cohorts for clinical studies.

Semantic oscillations : encoding context and structure in complex valued holographic vectors

In computational linguistics, information retrieval and applied cognition, words and concepts are often represented as vectors in high dimensional spaces computed from a corpus of text. These high dimensional spaces are often referred to as Semantic Spaces. We describe a novel and efficient approach to computing these semantic spaces via the use of complex valued vector representations. We report on the practical implementation of the proposed method and some associated experiments. We also briefly discuss how the proposed system relates to previous theoretical work in Information Retrieval and Quantum Mechanics and how the notions of probability, logic and geometry are integrated within a single Hilbert space representation. In this sense the proposed system has more general application and gives rise to a variety of opportunities for future research.

Multiview point cloud kernels for semisupervised learning [Lecture Notes]

In semisupervised learning (SSL), a predictive model is learn from a collection of labeled data and a typically much larger collection of unlabeled data. These paper presented a framework called multi-view point cloud regularization (MVPCR), which unifies and generalizes several semisupervised kernel methods that are based on data-dependent regularization in reproducing kernel Hilbert spaces (RKHSs). Special cases of MVPCR include coregularized least squares (CoRLS), manifold regularization (MR), and graph-based SSL. An accompanying theorem shows how to reduce any MVPCR problem to standard supervised learning with a new multi-view kernel.

Semantic spaces : measuring the distance between different subspaces

Semantic Space models, which provide a numerical representation of words’ meaning extracted from corpus of documents, have been formalized in terms of Hermitian operators over real valued Hilbert spaces by Bruza et al. [1]. The collapse of a word into a particular meaning has been investigated applying the notion of quantum collapse of superpositional states [2]. While the semantic association between words in a Semantic Space can be computed by means of the Minkowski distance [3] or the cosine of the angle between the vector representation of each pair of words, a new procedure is needed in order to establish relations between two or more Semantic Spaces. We address the question: how can the distance between different Semantic Spaces be computed? By representing each Semantic Space as a subspace of a more general Hilbert space, the relationship between Semantic Spaces can be computed by means of the subspace distance. Such distance needs to take into account the difference in the dimensions between subspaces. The availability of a distance for comparing different Semantic Subspaces would enable to achieve a deeper understanding about the geometry of Semantic Spaces which would possibly translate into better effectiveness in Information Retrieval tasks.

Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions

In this paper, a space fractional di®usion equation (SFDE) with non- homogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix raised to the same fractional power. Analytic solutions of the SFDE are derived. Finally, some numerical results are given to demonstrate that the MTT is a computationally e±cient and accurate method for solving SFDE.