Tensor squeezed limits and the Higuchi bound

Low-rank Tensor Standard Forms and Limits∗

Abstract. Despite the growing use of tensor-based methods, tensors remain mysterious and misunderstood objects. In this work we develop a standard form for low-rank tensors that allows one to focus on the important defining parameters (internal angles and comparative sizes) while ignoring unimportant parameters (separable unitary transformations). This standard form eases the analysis of tensor...

Higuchi Dimension of Digital Images

There exist several methods for calculating the fractal dimension of objects represented as 2D digital images. For example, Box counting, Minkowski dilation or Fourier analysis can be employed. However, there appear to be some limitations. It is not possible to calculate only the fractal dimension of an irregular region of interest in an image or to perform the calculations in a particular dire...

Toshihiro Higuchi and Jun Saiki Hiroyuki Yamashiro ,

Statistical limits of spiked tensor models

We study the statistical limits of both detecting and estimating a rank-one deformation of a symmetric random Gaussian tensor. We establish upper and lower bounds on the critical signal-to-noise ratio, under a variety of priors for the planted vector: (i) a uniformly sampled unit vector, (ii) i.i.d. ±1 entries, and (iii) a sparse vector where a constant fraction ρ of entries are i.i.d. ±1 and t...

The bound r ≤ 16ǫ on the primordial tensor perturbation

I recall the well-known sufficient conditions for the bound r < 16ǫ on the spectrum of the primordial tensor perturbation. Two recent papers claim a violation of this bound, without stating explicitely any violation of the sufficient conditions. The following conditions are sufficient for the bound r < 16ǫ on the primordial tensor perturbation. • Inflation is of the slow-roll variety. • The cur...