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Series: Job Candidate Talk

Series: Job Candidate Talk

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.

Series: Job Candidate Talk

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.

Series: Job Candidate Talk

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.

Series: Job Candidate Talk

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.

Series: Job Candidate Talk

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.

Series: Job Candidate Talk

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.

Series: Job Candidate Talk

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.

Series: Job Candidate Talk

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.

Series: Job Candidate Talk

TBA by Cheng Mao