**Upcoming MAMS Colloquium Series**

### Spring 2024

**5/3/2024, Friday. 3:15-4:15 pm in Wickenden 321**

**Speaker**: Dr. Dootika Vats (Indian Institute of Technology Kanpur)

**Title**: Efficient Multivariate Initial Sequence Estimator in MCMC

**Abstract**: Assessing the quality of a Markov chain Monte Carlo sampler is critical to reliable inference in any Bayesian problem. Due to the complex inter-dependence of modern Bayesian posteriors, it is essential to estimate the limiting covariance matrix in a Markov chain CLT. There are many estimators that have been developed in the literature, however, almost all exhibit significant under-estimation, leading to inaccurate effective sample size estimates and premature terminations. An estimator that exhibits superior finite sample properties is the multivariate initial sequence estimator. However, this estimator can be highly inefficient and thereby not practical. We propose an efficient multivariate generalization of the initial sequence estimator, that retains the finite sample and asymptotic properties, but is orders of magnitude faster. We demonstrate the utility of our proposed estimator over some practical Bayesian posteriors.

**4/19/2024, Friday. 3:15-4:15 pm in Wickenden 321**

**Speaker**: Dr. Bharath Sriperumbudur (Penn State University)

**Title**: Spectral Regularized Kernel Hypothesis Tests

**Abstract**: Over the last decade, an approach that has gained a lot of popularity in tackling non-parametric testing problems on general (i.e., non-Euclidean) domains is based on the notion of reproducing kernel Hilbert space (RKHS) embedding of probability distributions. The main goal of our work is to understand the optimality of two-sample and goodness-of-fit tests constructed based on this approach. First, we show that the popular MMD (maximum mean discrepancy) based hypothesis tests are not optimal in terms of the separation boundary measured in Hellinger distance. Second, we propose a modification to the MMD test based on spectral regularization by taking into account the covariance information (which is not captured by the MMD test) and prove the proposed test to be minimax optimal with a smaller separation boundary than that achieved by the MMD test. Third, we propose an adaptive version of the above test, which involves a data-driven strategy to choose the regularization parameter and show the adaptive test to be almost minimax optimal up to a logarithmic factor. Moreover, our results hold for the permutation variant of the test where the test threshold is chosen elegantly through the permutation of the samples. Through numerical experiments on synthetic and real-world data, we demonstrate the superior performance of the proposed test in comparison to many popular tests.

(Based on joint work with Omar Hagrass (PSU) and Bing Li (PSU))

**4/12/2024, Friday. 3:15-4:15 pm in Wickenden 321**

**Speaker**: Dr. Lixin Shen (Math Department, Syracus University)

**Title**: Computing Proximity Operators of Scale and Signed Permutation Invariant Functions

**Abstract**: This presentation focuses on computing proximity operators for scale and signed permutation invariant functions. A scale-invariant function remains unchanged under uniform scaling, while a signed permutation invariant function retains its structure despite permutations and sign changes applied to its input variables. Noteworthy examples include the $\ell_0$ function and the ratios of $\ell_1/\ell_2$ and its square, with their proximity operators being particularly crucial in sparse signal recovery. We delve into the properties of scale and signed permutation invariant functions, delineating the computation of their proximity operators into three sequential steps: the $\vw$-step, $r$-step, and $d$-step. These steps collectively form a procedure termed as WRD, with the $\vw$-step being of utmost importance and requiring careful treatment. Leveraging this procedure, we present a method for explicitly computing the proximity operator of $(\ell_1/\ell_2)^2$ and introduce an efficient algorithm for the proximity operator of $\ell_1/\ell_2$. This presentation is accessible to senior undergraduate and graduate students.

**4/5/2024, Friday. 12:45-1:45 pm in Clapp Hall 108**

**Speaker**: Dr. Persi Diaconis (Stanford University)

**Title**: The Mathematics of Solitaire

**Abstract**: Millions of people play solitaire (usual Klondike) every day. It is an embarrassment that mathematicians can’t answer the questions ‘what are the odds of winning?’, ‘What’s a good way of playing?’, In Vegas, you can ‘buy a deck’ for $52 and get $5 for each card played up. Is this fair? (HAH). I’ll report what we know; surely you say one of the fancy computer programs (alpha zero, mu zero) can do it (NOPE). There is a ‘simple solitaire’ where we can ‘do the math’. Surprisingly, this has links to some of the deepest corners of modern probability–random matrix theory–and the work of Elizabeth Meckes. I’ll try to explain all this in English, for a general audience.

**4/5/2024, Friday. 3:15-4:15 pm in Wickenden 321**

**Speaker**: Dr. Persi Diaconis (Stanford University)

**Title**: Adding numbers and shuffling cards

**Abstract**: When numbers are added in the usual way, ‘carries’ occur. For typical numbers, how do the carries go? How many are typical and, if you just had a carry, is it more or less likely that there will be a carry in the next column? It turns out that carries form a Markov chain with an ‘AMAZING’ transition matrix (are any matrices amazing?). This same matrix occurs in the analysis of the usual method of shuffling cards (riffle shuffling). I’ll explain the ‘seven shuffles theorem’ and the connection with carries. The same matrix occurs in taking sections of generating functions for the Veronese embedding and as the ‘Foulkes characters’ of the symmetric group. And then, well, carries are cocycles and the story goes on. I’ll try to explain it in ‘mathematical English’. This is joint work with Jason Fulman.

**3/29/2024, Friday. 3:15-4:15 pm in Wickenden 321**

**Speaker**: Dr. Michael Pokojovy (Old Dominion University)

**Title**: Univariate Fast Initial Response Statistical Process Control with Taut Strings

**Abstract**: We present a novel real-time univariate monitoring scheme for detecting a sustained departure of a process mean from some given standard assuming a constant variance. Our proposed stopping rule is based on the total variation of a nonparametric taut string estimator of the process mean and is designed to provide a desired average run length for an in-control situation. Compared to the more prominent CUSUM fast initial response (FIR) methodology and allowing for a restart following a false alarm, the proposed two-sided taut string (TS) scheme produces a significant reduction in average run length for a wide range of changes in the mean that occur at or immediately after process monitoring begins. A decision rule for when to choose our proposed TS chart compared to the CUSUM FIR chart that takes into account both false alarm rate and average run length to detect a shift in the mean is proposed and implemented. This is joint work with J. Marcus Jobe (Miami University, Oxford, OH).

**2/9/2024, Friday. 3:15-4:15 pm in Wickenden 321**

**Speaker**: Dr. Mona Merling (University of Pennsylvania)

**Title**: Higher scissors congruence invariants for manifolds

**Abstract**: The classical scissors congruence problem asks whether given two polyhedra with the same volume, one can cut one into a finite number of smaller polyhedra and reassemble these to form the other. There is an analogous definition of an SK (German “schneiden und kleben,” cut and paste) relation for manifolds and classically defined scissors congruence (SK) groups for manifolds. Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic K-theory, and applying these tools to studying the Grothendieck ring of varieties. I will talk about a new application of this framework: we will construct a K-theory spectrum of manifolds, which lifts the classical SK group, and a derived version of the Euler characteristic. I will discuss what this higher homotopical lift of the Euler characteristic sees on the level of pi_1.