Distribution of descents in matchings

Speaker: 

Gene Kim

Institution: 

USC

Time: 

Tuesday, February 20, 2018 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

The distribution of descents in certain conjugacy classes of S_n have been previously studied, and it is shown that its moments have interesting properties. This paper provides a bijective proof of the symmetry of the descents and major indices of matchings (also known as fixed point free involutions) and uses a generating function approach to prove an asymptotic normality theorem for the number of descents in matchings.

 

Efficient algorithms for phase retrieval in high dimensions

Speaker: 

Yan Shuo Tan

Institution: 

University of Michigan

Time: 

Thursday, February 8, 2018 - 11:00am to 12:00pm

Host: 

Location: 

RH 306P

Mathematical phase retrieval is the problem of solving systems of rank-1 quadratic equations. Over the last few years, there has been much interest in constructing algorithms with provable guarantees. Both theoretically and empirically, the most successful approaches have involved direct optimization of non-convex loss functions. In the first half of this talk, we will discuss how stochastic gradient descent for one of these loss functions provably results in (rapid) linear convergence with high probability. In the second half of the talk, we will discuss a semidefinite programming algorithm that simultaneously makes use of a sparsity prior on the solution vector, while overcoming possible model misspecification.

Bi-free probability and an approach to conjugate variables

Speaker: 

Ian Charlesworth

Institution: 

UCSD

Time: 

Tuesday, February 27, 2018 - 11:00am

Location: 

RH 306

Free entropy theory is an analogue of information theory in a non-commutative setting, which has had great applications to the examination of structural properties of von Neumann algebras. I will discuss some ongoing joint work with Paul Skoufranis to extend this approach to the setting of bi-free probability which attempts to study simultaneously ``left'' and ``right'' non-commutative variables. I will speak in particular of an approach to a bi-free Fisher information and bi-free conjugate variables -- analogues of Fisher's information measure and the score function of information theory. The focus will be on constructing these tools in the non-commutative setting, and time permitting, I will also mention some results such as bi-free Cramer-Rao and Stam inequalities, and some quirks of the bi-free setting which are not present in the free setting.

Phase transition in the spiked random tensors

Speaker: 

Wei-Kuo Chen

Institution: 

University of Minnesota

Time: 

Thursday, January 4, 2018 - 11:00am to 11:50am

Host: 

Location: 

RH 306

The problem of detecting a deformation in a symmetric Gaussian random tensor is concerned about whether there exists a statistical hypothesis test that can reliably distinguish a low-rank random spike from the noise. Recently Lesieur et al. (2017) proved that there exists a critical threshold so that when the signal-to-noise ratio exceeds this critical value, one can distinguish the spiked and unspiked tensors and weakly recover the spike via the minimal mean-square-error method. In this talk, we will show that in the case of the rank-one spike with Rademacher prior, this critical value strictly separates the distinguishability and indistinguishability of the two tensors under the total variation distance. Our approach is based on a subtle analysis of the high temperature behavior of the pure p-spin model, arising initially from the field of spin glasses. In particular, the signal-to-noise criticality is identified as the critical temperature, distinguishing the high and low temperature behavior, of the pure p-spin model.

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