Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

In natural language processing (NLP), the role of embeddings in representing linguistic semantics is crucial. Despite the prevalence of vector representations in embedding sets, they exhibit limitations in expressiveness and lack comprehensive set operations. To address this, we attempt to formulate and apply sets and their operations within pre-trained embedding spaces. Inspired by quantum logic, we propose to go beyond the conventional vector set representation with our novel subspace-based approach. This methodology constructs subspaces using pre-trained embedding sets, effectively preserving semantic nuances previously overlooked, and consequently consistently improving performance in downstream tasks.

In the decade since 2010, successes in artificial intelligence have been at the forefront of computer science and technology, and vector space models have solidified a position at the forefront of artificial intelligence. At the same time, quantum computers have become much more powerful, and announcements of major advances are frequently in the news. The mathematical techniques underlying both these areas have more in common than is sometimes realized. Vector spaces took a position at the axiomatic heart of quantum mechanics in the 1930s, and this adoption was a key motivation for the derivation of logic and probability from the linear geometry of vector spaces. Quantum interactions between particles are modelled using the tensor product, which is also used to express objects and operations in artificial neural networks. This paper describes some of these common mathematical areas, including examples of how they are used in artificial intelligence (AI), particularly in automated reasoning and natural language processing (NLP). Techniques discussed include vector spaces, scalar products, subspaces and implication, orthogonal projection and negation, dual vectors, density matrices, positive operators, and tensor products. Application areas include information retrieval, categorization and implication, modelling word-senses and disambiguation, inference in knowledge bases, and semantic composition. Some of these approaches can potentially be implemented on quantum hardware. Many of the practical steps in this implementation are in early stages, and some are already realized. Explaining some of the common mathematical tools can help researchers in both AI and quantum computing further exploit these overlaps, recognizing and exploring new directions along the way.

In the decade since 2010, successes in artificial intelligence have been at the forefront of computer science and technology, and vector space models have solidified a position at the forefront of artificial intelligence. At the same time, quantum computers have become much more powerful, and announcements of major advances are frequently in the news. The mathematical techniques underlying both these areas have more in common than is sometimes realized. Vector spaces took a position at the axiomatic heart of quantum mechanics in the 1930s, and this adoption was a key motivation for the derivation of logic and probability from the linear geometry of vector spaces. Quantum interactions between particles are modelled using the tensor product, which is also used to express objects and operations in artificial neural networks. This paper describes some of these common mathematical areas, including examples of how they are used in artificial intelligence (AI), particularly in automated reasoning and natural language processing (NLP). Techniques discussed include vector spaces, scalar products, subspaces and implication, orthogonal projection and negation, dual vectors, density matrices, positive operators, and tensor products. Application areas include information retrieval, categorization and implication, modelling word-senses and disambiguation, inference in knowledge bases, and semantic composition. Some of these approaches can potentially be implemented on quantum hardware. Many of the practical steps in this implementation are in early stages, and some are already realized. Explaining some of the common mathematical tools can help researchers in both AI and quantum computing further exploit these overlaps, recognizing and exploring new directions along the way.

The concepts developed by Chilean psychoanalyst Ignacio Matte Blanco in his books (Matte Blanco, 1980, 1988) represent the first attempt to formalize ideas that Sigmund Freud introduced in his book'The Interpretation of Dreams' (2011) and then elaborated on his later paper'The Unconscious'(1915). Matte Blanco's work presents the basis of a scientific discipline that takes a mathematical approach to psychoanalysis-Computational Psychoanalysis. Previous work on this topic has mainly dealt with the application of the p-adic model (Khrennikov, 2002), ultrametric topology (Murtagh, 2012; Lauro-Grotto, 2008), and the utilization of grupoid theory to Bi-logic concepts (Iurato, 2014). The author intends to present a new mathematical (logical) model that could formally describe phenomena that represent the fundamental characteristics of the unconscious psyche. In the first part of this paper, we give a short overview of the most important concepts and conclusions of Matte Blanco. The second part of the paper is designed to present the fundamental concepts of quantum logic and to explain the concept of Hilbert space which represents the'basis' for the quantum-logical system. Eventually, in the third part of the paper, we deal with the interpretation of the phenomena and concepts from Bi-logic in the context of quantum logic.

The most important requirements, for an operator to be viewed as a proposition, is that it must be hermitian and idempotent (which, in the Hilbert case corresponds to projectors). We interpret the above restrictions as follows. Hermitian operators have real eigenvalues. In particular, idempotent operators have eigenvalues 0 or 1, that is, they allow for asserting or negating in the classical way. When the operator is not hermitian, it is true that there is no way to interpret it directly as a logical proposition, because its eigenvalues are not real numbers, and the proposition cannot be asserted as usual.