We define the k:th moment of a Banach space valued random variable as the expectation of its k:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals...

Deep neural networks are surprisingly efficient at solving practical tasks, but the theory behind this phenomenon is only starting to catch up with the practice. Numerous works show that depth is the key to this efficiency. A certain class of deep convolutional networks – namely those that correspond to the Hierarchical Tucker (HT) tensor decomposition – has been proven to have exponentially hi...

We continue the study of generalized tractability initiated in our previous paper “Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information”, J. Complexity, 23, 262-295 (2007). We study linear tensor product problems for which we can compute linear information which is given by arbitrary continuous linear functionals. We want to approxima...

We consider the Principal Component Analysis problem for large tensors of arbitrary order k under a single-spike (or rank-one plus noise) model. On the one hand, we use information theory, and recent results in probability theory, to establish necessary and sufficient conditions under which the principal component can be estimated using unbounded computational resources. It turns out that this ...

The {\em tensor power method} generalizes the matrix method to higher order arrays, or tensors. Like in case, fixed points of are eigenvectors tensor. While every real symmetric has an eigendecomposition, vectors generating a decomposition not always In this paper we show that whenever eigenvector is} generator tensor, then (if is sufficiently high) robust} , i.e., it attracting point method....

It is known that Horndeski theories can be transformed to a sub-class of GleyzesLanglois-Piazza-Vernizzi (GLPV) theories under the disformal transformation of the metric gμν → Ω(φ)gμν +Γ(φ,X)∇μφ∇νφ, where Ω is a function of a scalar field φ and Γ is another function depending on both φ and X = g∇μφ∇νφ. We show that, with the choice of unitary gauge, both curvature and tensor perturbations on th...

This article introduces a Tensor Network Kalman filter, which can estimate state vectors that are exponentially large without ever having to explicitly construct them. The Tensor Network Kalman filter also easily accommodates the case where several different state vectors need to be estimated simultaneously. The key lies in rewriting the standard Kalman equations as tensor equations and then im...

Deep neural networks are surprisingly efficient at solving practical tasks, but the theory behind this phenomenon is only starting to catch up with the practice. Numerous works show that depth is the key to this efficiency. A certain class of deep convolutional networks – namely those that correspond to the Hierarchical Tucker (HT) tensor decomposition – has been proven to have exponentially hi...

Let X a smooth quasi-projective algebraic surface, L a line bundle on X. Let X [n] the Hilbert scheme of n points on X and L the tautological bundle on X [n] naturally associated to the line bundle L on X. We explicitely compute the image Φ(L) of the tautological bundle L for the Bridgeland-King-Reid equivalence Φ : D(X ) → DbSn(X ), in terms of a complex C L of Snequivariant sheaves in DbSn(X ...

Evolution of primordial fluctuations in a Brans-Dicke type scalar-tensor gravity theory is comprehensively investigated. The harmonic attractor model, in which the scalar field has its harmonic effective potential in the Einstein conformal frame and the theory relaxes toward Einstein gravity with time, is considered. The evolution of adiabatic initial perturbations in flat SCDM models is examin...