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In this paper, we investigate the sample size requirement for a general class of nuclear norm minimization methods for higher order tensor completion. We introduce a class of tensor norms by allowing for different levels of coherence, which…
We prove two identities that connect some natural tensor products in the category $\sf{LCS}$ of locally convex spaces with the tensor products in the category $\sf{Ste}$ of stereotype spaces. In particular, we give sufficient conditions…
We extend Deligne's notion of determinant functor to tensor triangulated categories. Specifically, to account for the multiexact structure of the tensor, we define a determinant functor on the 2-multicategory of triangulated categories and…
We describe the image of the canonical tensor functor from Deligne's interpolating category $Rep(GL_{m-n})$ to $Rep(GL(m|n))$ attached to the standard representation. This implies explicit tensor product decompositions between any two…
In this paper, we introduce a type of tensor neural network. For the first time, we propose its numerical integration scheme and prove the computational complexity to be the polynomial scale of the dimension. Based on the tensor product…
We study embeddings and norm estimates for tensor products of weighted reproducing kernel Hilbert spaces. These results lead to a transfer principle that is directly applicable to tractability studies of multivariate problems as integration…
We define a tensor product for permutative categories and prove a number of key properties. We show that this product makes the 2-category of permutative categories closed symmetric monoidal as a bicategory.
This work hopes to be an introduction to Deligne categories for someone familiar with classical representation theory and some category theory. In the first chapter, we motivate and define (symmetric) tensor categories, construct the…
The main purpose of this paper is to generalize and develop a few basic properties of the innerproduct space on a hypervector space. On this hypervector space we define the norm. Also we establish a important relation between normed…
We introduce the marked Brauer algebra and the marked Brauer category. These generalize the analogous constructions for the ordinary Brauer algebra to the setting of a homogeneous bilinear form on a $\mathbb{Z}_2$-graded vector space. We…
We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of bounded operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a…
This paper investigates the learning of 3rd-order tensors representing the semantics of transitive verbs. The meaning representations are part of a type-driven tensor-based semantic framework, from the newly emerging field of compositional…
The aim of this paper is two-fold. Firstly we provide necessary and sufficient criteria for the existence of a strict deformation quantization of algebraic tensor products of Poisson algebras, and secondly, we discuss the existence of…
This note documents the specification of normal forms in cubical type theory. The definition is already present in the proof of normalization for cubical type theory, but we present it in a more traditional style explicitly for reference.
Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the "synthetic" development of homotopy…
We obtain a complete characterization of the norm attainment set of a bounded linear functional on a normed space, in terms of a semi-inner-product defined on the space. Motivated by this result, we further apply the concept of…
This is the first of a pair of papers where we construct and investigate a closed monoidal structure on the category of generalized algebraic theories (in the sense of Cartmell). In the present text, as a starting point, we define the…
This is an expository paper on tensor products where the standard approaches for constructing concrete instances of algebraic tensor products of linear spaces, via quotient spaces or via linear maps of bilinear maps, are reviewed by…
We describe several equivalent models for the infinity-category of infinity-local systems of chain complexes over a space using the framework of quasi-categories. We prove that the given models are equivalent as infinity-categories by…
In recent years several classes of structured matrices are extended to classes of tensors in the context of tensor complementarity problem. The tensor complementarity problem is a class of nonlinear complementarity problem where the…