Structured Rectangular Tensors and Rectangular Tensor Complementarity Problems

In this paper, some properties of structured rectangular tensors are presented, and the relationship among these structured rectangular tensors is also given. It is shown that all the V-singular values of rectangular P-tensors are positive. Some necessary and/or sufficient conditions for a rectangular tensor to be a rectangular P-tensor are also obtained. A new subclass of rectangular tensors, which is called rectangular S-tensors, is introduced and it is proved that rectangular S-tensors can be defined by the feasible vectors of the corresponding rectangular tensor complementarity problem.

P-Tensor Product for Group C*-Algebras

In this paper, we introduce new tensor products ⊗ p ( 1 ≤ p ≤ + ∞ ) on C ℓ p * ( Γ ) ⊗ C ℓ p * ( Γ ) and ⊗ c 0 on C c 0 * ( Γ ) ⊗ C c 0 * ( Γ ) for any discrete group Γ . We obtain that for 1 ≤ p < + ∞ C ℓ p * ( Γ ) ⊗ m a x C ℓ p * ( Γ ) = C ℓ p * ( Γ ) ⊗ p C ℓ p * ( Γ ) if and only if Γ is amenable; C c 0 * ( Γ ) ⊗ m a x C c 0 * ( Γ ) = C c 0 * ( Γ ) ⊗ c 0 C c 0 * ( Γ ) if and only if Γ has Haagerup property. In particular, for the free group with two generators F 2 we show that C ℓ p * ( F 2 ) ⊗ p C ℓ p * ( F 2 ) ≇ C ℓ q * ( F 2 ) ⊗ q C ℓ q * ( F 2 ) for 2 ≤ q < p ≤ + ∞ .

Vanishing theorems for higher-order Killing and Codazzi

A Killing p-tensor (for an arbitrary natural number p ≥ 2) is a symmetric p-tensor with vanishing symmetrized covariant derivative. On the other hand, Codazzi p-tensor is a symmetric p-tensor with symmetric covariant derivative. Let M be a complete and simply connected Riemannian manifold of nonpositive (resp. non-negative) sectional curvature. In the first case we prove that an arbitrary symmetric traceless Killing p-tensor is parallel on M if its norm is a Lq -function for some q > 0. If in addition the volume of this manifold is infinite, then this tensor is equal to zero. In the second case we prove that an arbitrary traceless Codazzi p-tensor is equal to zero on a noncompact manifold M if its norm is a Lq -function for some q  1 .

Statistical Learning in Tree-Based Tensor Format

Tensor methods are widely used tools for the approximation of high dimensional functions. Such problems are encountered in uncertainty quantification and statistical learning, where the high dimensionality imposes to use specific techniques, such as rank-structured approximations [1]. In this work, we introduce a statistical learning algorithm for the approximation in tree-based tensor format, which are tensor networks whose graphs are dimension partition trees. This tensor format includes the Tucker format, the Tensor-Train format, as well as the more general Hierarchical tensor formats [4]. It can be interpreted as a deep neural network with a particular architecture [2]. The proposed algorithm uses random evaluations of a function to provide a tree-based tensor approximation, with adaptation of the tree-based rank by using a heuristic criterion based on the higher-order singular values to select the ranks to increase, and of the approximation spaces of the leaves of the tree. We then present a learning algorithm for the approximation under the form u(x) ≈ v(z_1,...,z_m) where v is a tensor in tree-based format and the z_i = g_i(x), 1 ≤ i ≤ m, are new variables. A strategy based on the projection pursuit regression [3] is proposed to compute the mappings g_i and increase the effective dimension m. The methods are illustrated on different examples to show their efficiency and adaptability as well as the power of representation of the tree-based tensor format, possibly combined with changes of variables.

Rapid acquisition of data dense solid-state CPMG NMR spectral sets using multi-dimensional statistical analysis

Tensor-rank decomposition methods have been applied to variable contact time 29Si{1H} CP/CPMG NMR data sets to extract NMR dynamics information and dramatically decrease conventional NMR acquisition times.