while streaming pca (also known as oja’s algorithm) was proposed about four decades ago and has roots going back to 1949, theoretical resolution in terms of obtaining optimal convergence rates has been obtained only in the last decade. however, we are not aware of any available distributional guarantees, which can help provide confidence intervals on the quality of the solution. in this talk, i will present the problem of quantifying uncertainty for the estimation error of the leading eigenvector using oja's algorithm for streaming pca, where the data are generated iid from some unknown distribution. combining classical tools from the u-statistics literature with recent results on high-dimensional central limit theorems for quadratic forms of random vectors and concentration of matrix products, we establish a distributional approximation result for the error between the population eigenvector and the output of oja's algorithm. we also propose an online multiplier bootstrap algorithm and establish conditions under which the bootstrap distribution is close to the corresponding sampling distribution with high probability. while there are optimal rates for the streaming pca problem, they typically apply to the iid setting, whereas in many applications like distributed optimization, the data is generated from a markov chain and the goal is to infer parameters of the limiting stationary distribution. if time permits, i will also present our near-optimal finite sample guarantees which remove the logarithmic dependence on the sample size in previous work, where markovian data is downsampled to get a nearly independent data stream. 

note: the speaker will also give an overview of her latest research.

we overview the cost of a group action, which measures how much information is needed to generate its induced orbit equivalence relation, and the ideal poisson-voronoi tessellation (ipvt), a new random limit with interesting geometric features. in recent work, we use the ipvt to prove all measure preserving and free actions of a higher rank semisimple lie group on a standard probability space have cost 1, answering gaboriau's fixed price question for this class of groups. we sketch a proof, which relies on some simple dynamics of the group action and the definition of a poisson point process. no prior knowledge on cost, ipvts, or lie groups will be assumed. this is joint work with mikolaj fraczyk and sam mellick.

in this talk, i will sketch a surprising proof due to gyorgy elekes on a non-trivial lower bound for the sums-versus-product problem in combinatorial number theory.   

we consider first-order methods with constant step size for minimizing locally lipschitz coercive functions that are tame in an o-minimal structure on the real field. we prove that if the method is approximated by subgradient trajectories, then the iterates eventually remain in a neighborhood of a connected component of the set of critical points. under suitable method-dependent regularity assumptions, this result applies to the subgradient method with momentum, the stochastic subgradient method with random reshuffling and momentum, and the random-permutations cyclic coordinate descent method.

this talk is based on ongoing work of the speaker. we will discuss the stochastic embeddability of snowflakes of finite nagata-dimensional spaces into ultrametric spaces and the induced stochastic embeddings of their hyperbolic fillings into trees. several results follow as applications, for example:
(1) for any uniformly concave gauge $\omega$, the wasserstein 1-metric over $([0,1]^n,\omega(\|\cdot\|))$ bilipschitzly embeds into $\ell^1$.
(2) the wasserstein 1-metric over any finitely generated gromov hyperbolic group bilipschitzly embeds into $\ell^1$.

arithmeticity of quaternionic modular forms on g_2 abstract: quaternionic modular forms (qmfs) on the split exceptional group g_2 are a special class of automorphic functions on this group, whose origin goes back to work of gross-wallach and gan-gross-savin. while the group g_2 does not possess any holomorphic modular forms, the quaternionic modular forms seem to be able to be a good substitute.  in particular, qmfs on g_2 possess a semi-classical fourier expansion and fourier coefficients, just like holomorphic modular forms on shimura varieties.  i will explain the proof that the cuspidal qmfs of even weight at least 6 admit an arithmetic structure: there is a basis of the space of all such cusp forms, for which every fourier coefficient of every element of this basis lies in the cyclotomic extension of q.

in the 90s kollár proved non-rationality of many degrees and dimensions of complex fano hypersurfaces by considering their degenerations modulo p and taking advantage of the surprising existence and positivity of differential forms on their reductions. in a series of papers with chen, church, and ji, we have shown that these differential forms in characteristic p can give insight into many other aspects of the birational geometry of complex hypersurfaces: the degree of irrationality, rational endomorphisms, the birational automorphism group, and rational fibrations in low genus curves.

(based on joint work with ruofan jiang) the hilbert scheme of points on a variety  parametrizes -dimensional quotients of the structure sheaf. when  is a planar singular curve, its enumerative invariants have close relation to mathematical physics, knot theory and combinatorics. in this talk, we investigate two analogous moduli spaces, one being a direct generalization of the hilbert scheme. our results reveal their surprising relations to hall polynomials, matrix equations, modular forms, etc.