Friday, January 18, 2019 - 11:15 , Location: Skiles 006 , Alex Blumenthal , Univ. of Maryland , Organizer: Rafael de la Llave
It is anticipated that chaotic regimes (characterized by, e.g., sensitivity with respect to initial conditions and loss of memory) arise in a wide variety of dynamical systems, including those arising from the study of ensembles of gas particles and fluid mechanics. However, in most cases the problem of rigorously verifying asymptotic chaotic regimes is notoriously difficult. For volume-preserving systems (e.g., incompressible fluid flow or Hamiltonian systems), these issues are exemplified by coexistence phenomena: even in quite simple models which should be chaotic, e.g. the Chirikov standard map, completely opposite dynamical regimes (elliptic islands vs. hyperbolic sets) can be tangled together in phase space in a convoluted way. Recent developments have indicated, however, that verifying chaos is tractable for systems subjected to a small amount of noise— from the perspective of modeling, this is not so unnatural, as the real world is inherently noisy. In this talk, I will discuss two recent results: (1) a large positive Lyapunov exponent for (extremely small) random perturbations of the Chirikov standard map, and (2) a positive Lyapunov exponent for the Lagrangian flow corresponding to various incompressible stochastic fluids models, including stochastic 2D Navier-Stokes and 3D hyperviscous Navier-Stokes on the periodic box. The work in this talk is joint with Jacob Bedrossian, Samuel Punshon-Smith, Jinxin Xue and Lai-Sang Young.
Thursday, January 17, 2019 - 11:00 , Location: Skiles 006 , Cheng Mao , Yale University , Organizer: Mayya Zhilova
A wide variety of applied tasks, such as ranking, clustering, graph matching and network reconstruction, can be formulated as a matrix estimation problem where the rows and columns of the matrix are shuffled by a latent permutation. The combinatorial nature of the unknown permutation and the non-convexity of the parameter space result in both statistical and algorithmic challenges. I will present recent developments of average-case models and efficient algorithms, primarily for the problems of ranking from comparisons and statistical seriation. On the statistical side, imposing shape constraints on the underlying matrix extends traditional parametric approaches, allowing for more robust and adaptive estimation. On the algorithmic front, I discuss efficient local algorithms with provable guarantees, one of which tightens a conjectured statistical-computational gap for a stochastically transitive ranking model.
Thursday, January 10, 2019 - 11:00 , Location: Skiles 006 , Subhabrata Sen , MIT , email@example.com , Organizer: Michael Damron
Network data arises frequently in modern scientific applications. These networks often have specific characteristics such as edge sparsity, heavy-tailed degree distribution etc. Some broad challenges arising in the analysis of such datasets include (i) developing flexible, interpretable models for network datasets, (ii) testing for goodness of fit, (iii) provably recovering latent structure from such data.In this talk, we will discuss recent progress in addressing very specific instantiations of these challenges. In particular, we will1. Interpret the Caron-Fox model using notions of graph sub-sampling, 2. Study model misspecification due to rare, highly “influential” nodes, 3. Discuss recovery of community structure, given additional covariates.
Tuesday, January 8, 2019 - 15:05 , Location: Skiles 006 , Pragya Sur , Statistics Department, Stanford University , Organizer: Christian Houdre
Logistic regression is arguably the most widely used and studied non-linear model in statistics. Classical maximum-likelihood theory based statistical inference is ubiquitous in this context. This theory hinges on well-known fundamental results---(1) the maximum-likelihood-estimate (MLE) is asymptotically unbiased and normally distributed, (2) its variability can be quantified via the inverse Fisher information, and (3) the likelihood-ratio-test (LRT) is asymptotically a Chi-Squared. In this talk, I will show that in the common modern setting where the number of features and the sample size are both large and comparable, classical results are far from accurate. In fact, (1) the MLE is biased, (2) its variability is far greater than classical results, and (3) the LRT is not distributed as a Chi-Square. Consequently, p-values obtained based on classical theory are completely invalid in high dimensions. In turn, I will propose a new theory that characterizes the asymptotic behavior of both the MLE and the LRT under some assumptions on the covariate distribution, in a high-dimensional setting. Empirical evidence demonstrates that this asymptotic theory provides accurate inference in finite samples. Practical implementation of these results necessitates the estimation of a single scalar, the overall signal strength, and I will propose a procedure for estimating this parameter precisely. This is based on joint work with Emmanuel Candes and Yuxin Chen.
Thursday, November 29, 2018 - 11:00 , Location: Skiles 006 , Dr. Veronica Ciocanel , Mathematical Biosciences Institute at The Ohio State University , firstname.lastname@example.org , Organizer: Howie Weiss
The cellular cytoskeleton ensures the dynamic transport, localization, and anchoring of various proteins and vesicles. In the development of egg cells into embryos, messenger RNA (mRNA) molecules bind and unbind to and from cellular roads called microtubules, switching between bidirectional transport, diffusion, and stationary states. Since models of intracellular transport can be analytically intractable, asymptotic methods are useful in understanding effective cargo transport properties as well as their dependence on model parameters.We consider these models in the framework of partial differential equations as well as stochastic processes and derive the effective velocity and diffusivity of cargo at large time for a general class of problems. Including the geometry of the microtubule filaments allows for better prediction of particle localization and for investigation of potential anchoring mechanisms. Our numerical studies incorporating model microtubule structures suggest that anchoring of mRNA-molecular motor complexes may be necessary in localization, to promote healthy development of oocytes into embryos. I will also briefly go over other ongoing projects and applications related to intracellular transport.
Wednesday, March 7, 2018 - 11:00 , Location: Skiles 005 , Adam Marcus , Princeton University , email@example.com , Organizer: Galyna Livshyts
I will discuss a recent line of research that uses properties of real rooted polynomials to get quantitative estimates in combinatorial linear algebra problems. I will start by discussing the main result that bridges the two areas (the "method of interlacing polynomials") and show some examples of where it has been used successfully (e.g. Ramanujan families and the Kadison Singer problem). I will then discuss some more recent work that attempts to make the method more accessible by providing generic tools and also attempts to explain the accuracy of the method by linking it to random matrix theory and (in particular) free probability. I will end by mentioning some current research initiatives as well as possible future directions.
Thursday, February 1, 2018 - 11:00 , Location: skiles 006 , Anderson Ye Zhang , Yale , Organizer: Heinrich Matzinger
The mean field variational inference is widely used in statistics and machine learning to approximate posterior distributions. Despite its popularity, there exist remarkably little fundamental theoretical justifications. The success of variational inference mainly lies in its iterative algorithm, which, to the best of our knowledge, has never been investigated for any high-dimensional or complex model. In this talk, we establish computational and statistical guarantees of mean field variational inference. Using community detection problem as a test case, we show that its iterative algorithm has a linear convergence to the optimal statistical accuracy within log n iterations. We are optimistic to go beyond community detection and to understand mean field under a general class of latent variable models. In addition, the technique we develop can be extended to analyzing Expectation-maximization and Gibbs sampler.
Tuesday, January 23, 2018 - 11:00 , Location: Skiles 005 , Konstantin Tikhomirov , Princeton University , firstname.lastname@example.org , Organizer: Galyna Livshyts
Convex-geometric methods, involving random projection operators and coverings, have been successfully used in the study of the largest and smallest singular values, delocalization of eigenvectors, and in establishing the limiting spectral distribution for certain random matrix models. Among further applications of those methods in computer science and statistics are restricted invertibility and dimension reduction, as well as approximation of covariance matrices of multidimensional distributions. Conversely, random linear operators play a very important role in geometric functional analysis. In this talk, I will discuss some recent results (by my collaborators and myself) within convex geometry and the theory of random matrices, focusing on invertibility of square non-Hermitian random matrices (with applications to numerical analysis and the study of the limiting spectral distribution of directed d-regular graphs), approximation of covariance matrices (in particular, a strengthening of the Bai–Yin theorem), as well as some applications of random operators in convex geometry.