Abstract:

This talk is organized into three parts.

1. Turing’s formula is introduced. Given an iid sample from an countable alphabet under a probability distribution, Turing’s formula (introduced by Good (1953), hence also known as the Good-Turing formula) is a mind-bending non-parametric estimator of total probability associated with letters of the alphabet that are NOT represented in the sample. Many of its statistical properties were not clearly known for a stretch of nearly sixty years until recently. Some of the newly established results, including various asymptotic normal laws, are described.

2. Turing’s perspective is described. Turing’s formula brought about a new perspective (or a new characterization) of probability distributions on general countable alphabets. The new perspective in turn provides a new way to do statistics on alphabets, where the usual statistical concepts associated with random variables (on the real line) no longer exist, for example, moments, tails, coefficients of correlation, characteristic functions don’t exist on alphabets (a major challenge of modern data sciences). The new perspective, in the form of entropic basis, is introduced.

3. Several applications are presented, including estimation of information entropy and diversity indices

Abstract: This work proposes new low rank approximation approaches with significant memory savings for large scale MR fingerprinting (MRF) problems.

We introduce a compressed MRF with randomized SVD method to significantly reduce the memory requirement for calculating a low rank approximation of large sized MRF dictionaries. We further relax this requirement by exploiting the structures of MRF dictionaries in the randomized SVD space and fitting them to low-degree polynomials to generate high resolution MRF parameter maps.

In vivo 1.5 and 3 Tesla brain scan data are used to validate the approaches. It is shown that T₁, T₂ and off-resonance maps are in good agreement with that of the standard MRF approach. Moreover, the memory savings is up to 1000 times for the MRF-FISP sequence and more than 15 times for the MRF-bSSFP sequence.

The proposed compressed MRF with randomized SVD and dictionary fitting methods are memory efficient low rank approximation methods, which can benefit the usage of MRF in clinical settings. They also have great potentials in large scale MRF problems, such as problems where multi-component chemical exchange effects are considered.