Models of a Non-Associative Composition

International audienceWe characterise the polarised evaluation order through a categorical structure where the hypothesis that composition is associative is relaxed. Duploid is the name of the structure, as a reference to Jean-Louis Loday's duplicial algebras. The main result is a reflection Adj→Dupl where Dupl is a category of duploids and duploid functors, and Adj is the category of adjunctions and pseudo maps of adjunctions. The result suggests that the various biases in denotational semantics: indirect, call-by-value, call-by-name... are a way of hiding the fact that composition is not always associative.Nous caractérisons l'ordre d'évaluation polarisé à travers une structure catégorielle dont l'hypothèse que la composition est associative est relâchée. Duploïde est le nom de la structure, par référence aux algèbres dupliciales de Loday. Le résultat principal est une réflection Adj→Dupl où Dupl est une catégorie des duploïdes et des foncteurs de duploïdes, et Adj est la catégorie des adjonctions et des pseudo-morphismes d'adjonctions. Le résultat suggère que les biais des sémantiques dénotationnelles: indirectes, en appel par valeur, en appel par nom... sont des façons de cacher le fait que la composition n'est pas toujours associative

Deformation theory of representations of prop(erad)s

We study the deformation theory of morphisms of properads and props thereby extending to a non-linear framework Quillen's deformation theory for commutative rings. The associated chain complex is endowed with a Lie algebra up to homotopy structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with a canonical Lie algebra up to homotopy structure in general and a Lie algebra structure only in the Koszul case. In particular, we explicit the deformation complex of morphisms from the properad of associative bialgebras. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of a Lie algebra up to homotopy structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.Comment: Version 4 : Statement about the properad of (non-commutative) Frobenius bialgebras fixed in Section 4. [82 pages

Non-Associative Geometry and the Spectral Action Principle

Chamseddine and Connes have argued that the action for Einstein gravity, coupled to the SU(3)\times SU(2)\times U(1) standard model of particle physics, may be elegantly recast as the "spectral action" on a certain "non-commutative geometry." In this paper, we show how this formalism may be extended to "non-associative geometries," and explain the motivations for doing so. As a guiding illustration, we present the simplest non-associative geometry (based on the octonions) and evaluate its spectral action: it describes Einstein gravity coupled to a G_2 gauge theory, with 8 Dirac fermions (which transform as a singlet and a septuplet under G_2). This is just the simplest example: in a forthcoming paper we show how to construct more realistic models that include Higgs fields, spontaneous symmetry breaking and fermion masses.Comment: 24 pages, no figures, matches JHEP versio

Learning to Rank Question Answer Pairs with Holographic Dual LSTM Architecture

We describe a new deep learning architecture for learning to rank question answer pairs. Our approach extends the long short-term memory (LSTM) network with holographic composition to model the relationship between question and answer representations. As opposed to the neural tensor layer that has been adopted recently, the holographic composition provides the benefits of scalable and rich representational learning approach without incurring huge parameter costs. Overall, we present Holographic Dual LSTM (HD-LSTM), a unified architecture for both deep sentence modeling and semantic matching. Essentially, our model is trained end-to-end whereby the parameters of the LSTM are optimized in a way that best explains the correlation between question and answer representations. In addition, our proposed deep learning architecture requires no extensive feature engineering. Via extensive experiments, we show that HD-LSTM outperforms many other neural architectures on two popular benchmark QA datasets. Empirical studies confirm the effectiveness of holographic composition over the neural tensor layer.Comment: SIGIR 2017 Full Pape