Past

Plateau's problem asks whether every boundary curve in 3-space is spanned by an area minimizing surface. Various interpretations of this problem have been solved using eg. geometric measure theory. Froehlich and Struwe proposed a PDE approach, in which the desired surface is produced using smooth sections of a twisted line bundle over the complement of the boundary curve. The idea is to consider sections of this bundle which minimize an analogue of the Allen--Cahn functional (a classical model for phase transition phenomena) and show that these concentrate energy on a solution of Plateau's problem. After some background on the link between phase transition models and minimal surfaces, I will describe new work with Marco Guaraco in which we produce smooth solutions of Plateau's problem using this approach. 

In 2019 Masser and Zannier proved that "most" abelian varieties over the algebraic numbers are not isogenous to the jacobian of any curve (where "most" refers to an ordering by some suitable height function). We will see how this result fits in the general Zilber-Pink Conjecture picture and we discuss some (rather concrete) analogous problems in a power of the modular curve Y(1).

Given two mathematical objects, the most basic question is whether they are the same. We will discuss this question for triangulations of three-manifolds. In practice there is fast software to answer this question and theoretically the problem is known to be decidable. However, our understanding is limited and known theoretical algorithms could have extremely long run-times. I will describe a programme to show that the 3-manifold homeomorphism problem is in the complexity class NP, and discuss the important sub-case of Seifert fibered spaces. 

Neural SDEs are continuous-time generative models for sequential data. State-of-the-art performance for irregular time series generation has been previously obtained by training these models adversarially as GANs. However, as typical for GAN architectures, training is notoriously unstable, often suffers from mode collapse, and requires specialised techniques such as weight clipping and gradient penalty to mitigate these issues. In this talk, I will introduce a novel class of scoring rules on path space based on signature kernels and use them as an objective for training Neural SDEs non-adversarially. The strict properness of such kernel scores and the consistency of the corresponding estimators, provide existence and uniqueness guarantees for the minimiser. With this formulation, evaluating the generator-discriminator pair amounts to solving a system of linear path-dependent PDEs which allows for memory-efficient adjoint-based backpropagation. Moreover, because the proposed kernel scores are well-defined for paths with values in infinite-dimensional spaces of functions, this framework can be easily extended to generate spatiotemporal data. This procedure permits conditioning on a rich variety of market conditions and significantly outperforms alternative ways of training Neural SDEs on a variety of tasks including the simulation of rough volatility models, the conditional probabilistic forecasts of real-world forex pairs where the conditioning variable is an observed past trajectory, and the mesh-free generation of limit order book dynamics.

We construct vertex algebras associated to divisors $S$ in toric Calabi-Yau threefolds $Y$, satisfying conjectures of Gaiotto-Rapcak and Feigin-Gukov, and in particular such that the characters of these algebras are given by a local analogue of the Vafa-Witten partition function of the underlying reduced subvariety $S^{red}$. These results are part of a broader program to establish a dictionary between the enumerative geometry of coherent sheaves on surfaces and Calabi-Yau threefolds, and the representation theory of vertex algebras and affine Yangian-type quantum groups.

How do you make the most of graduate supervisions?  Whether you are a first year graduate wanting to learn about how to manage meetings with your supervisor, or a later year DPhil student, postdoc or faculty member willing to share their experiences and give advice, please come along to this informal discussion led by DPhil students for the first Fridays@4 session of the term.  You can also continue the conversation and learn more about graduate student life at Oxford at Happy Hour afterwards.

Various constructions relevant to practical problems such as clustering and graph classification give rise to multiparameter persistence modules (MPPM), that is, linear representations of non-totally ordered sets. Much of the mathematical interest in multiparameter persistence comes from the fact that there exists no tractable classification of MPPM up to isomorphism, meaning that there is a lot of room for devising invariants of MPPM that strike a good balance between discriminating power and complexity of their computation. However, there is no consensus on what type of information we want these invariants to provide us with, and, in particular, there seems to be no good notion of “global” or “high persistence” features of MPPM.

With the goal of substantiating these claims, as well as making them more precise, I will start with an overview of some of the known invariants of MPPM, including joint works with Bauer and Oudot. I will then describe recent work of Bjerkevik, which contains relevant open questions and which will help us make sense of the notion of global feature in multiparameter persistence.

Many natural and social phenomena involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, cells in a developing tissue, or locusts in a swarm. Here I will focus on two examples of such emergent behavior in biology, specifically cell interactions during pattern formation in zebrafish skin and gametophyte development in ferns. Different modeling approaches provide complementary insights into these systems and face different challenges. For example, vertex-based models describe cell shape, while more efficient agent-based models treat cells as particles. Continuum models, which track the evolution of cell densities, are more amenable to analysis, but it is often difficult to relate their few parameters to specific cell interactions. In this talk, I will overview our models of cell behavior in biological patterns and discuss our ongoing work on quantitatively relating different types of models using topological data analysis and data-driven techniques.

We will kick off the start of the academic year and the Junior Algebra and Representation Theory seminar (JART) with a fun social event in the common room. Come catch up with your fellow students about what happened over the summer, meet the new students and play some board games. We'll go for lunch together afterwards.

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L-functions over number fields. There is an extremely similar function field analogue, worked out by Andrade-Keating. I will explain that one can relate this problem to understanding the homology of the braid group with symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove an improved range for homological stability with these coefficients. Together, these results imply the conjectured asymptotics for all moments in the function field case, for all sufficiently large (but fixed) q.

A Path Shadowing Monte-Carlo method provides prediction of future paths given any generative model.

At a given date, it averages future quantities over generated price paths whose past history matches, or “shadows”, the actual (observed) history.

We test our approach using paths generated from a maximum entropy model of financial prices,

based on the recently introduced “Scattering Spectra” which are multi-scale analogues of the standard skewness and kurtosis.

This model promotes diversity of generated paths while reproducing the main statistical properties of financial prices, including stylized facts on volatility roughness.

Our method yields state-of-the-art predictions for future realized volatility. It also allows one to determine conditional option smiles for the S&P500.

These smiles depend only on the distribution of the price process, and are shown to outperform both the current version of the Path Dependent Volatility model and the option market itself.

This work considers weighted and preconditioned GMRES. The objective is to provide a way of choosing the preconditioner and the inner product, also called weight, that ensure fast convergence. The main focus of the article is on Hermitian preconditioning (even for non-Hermitian problems).

It is indeed proposed to choose a Hermitian preconditioner H, and to apply GMRES in the inner product induced by H. If moreover, the problem matrix A is positive definite, then a new convergence bound is proved that depends only on how well H preconditions the Hermitian part of A, and on a measure of how non-Hermitian A is. In particular, if a scalable preconditioner is known for the Hermitian part of A, then the proposed method is also scalable. I will also illustrate this result numerically.

Over the years, I've often taught a first course in asymptotics and perturbation methods, even though I don't know much about the subject. In this talk, I'll discuss a textbook example of a singularly perturbed nonlinear boundary-value problem that has revealed delightful new surprises, every time I teach it. These include a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the solutions' initial conditions that can be calculated by elementary means.

If some structure, mathematical or otherwise, is giving you grief, then often the first thing to do is to attempt to break the offending object down into (finitely many) simpler pieces.

In group theory, when we speak of questions of *accessibility* we are referring to the ability to achieve precisely this. The idea of an 'accessible group' was first coined by Terry Wall in the 70s, and since then has left quite a mark on our field (and others). In this talk I will introduce the toolbox required to study accessibility, and walk you and your groups through some reasons to be accessible.

I will discuss various definitions of quantum or noncommutative graphs that have appeared in the literature, along with motivating examples.  One definition is due to Weaver, where examples arise from quantum channels and the study of quantum zero-error communication.  This definition works for any von Neumann algebra, and is "spatial": an operator system satisfying a certain operator bimodule condition.  Another definition, first due to Musto, Reutter, and Verdon, involves a generalisation of the concept of an adjacency matrix, coming from the study of (simple, undirected) graphs.  Here we study finite-dimensional C*-algebras with a given faithful state; examples are perhaps less obvious.  I will discuss generalisations of the latter framework when the state is not tracial, and discuss various notions of a "morphism" of the resulting objects

In a recent work with Sam Mellick and Amanda Wilkens, we proved that higher rank semisimple Lie groups satisfy a generalization of Gaboriau fixed price property (originally defined for countable groups) to the setting of locally compact second countable groups. As one of the corollaries, under mild conditions, we can prove that the rank (minimal number of generators) or the first mod-p Betti number of a higher rank lattice grow sublinearly in the covolume.  The proof relies on surprising geometric properties of Poisson-Voronoi tessellations in higher-rank symmetric spaces, which could be of independent interest. 

Complex networks have become the main paradigm for modeling the dynamics of interacting systems. However, networks are intrinsically limited to describing pairwise interactions, whereas real-world systems are often characterized by interactions involving groups of three or more units. In this talk, I will consider social systems as a natural testing ground for higher-order network approaches (hypergraphs and simplicial complexes). I will briefly introduce models of social contagion and norm evolution on hypergraphs to show how the inclusion of higher-order mechanisms can lead to the emergence of novel phenomena such as discontinuous transitions and critical mass effects. I will then present some recent results on the role that structural features play on the emergent dynamics, and introduce a measure of hyper-coreness to characterize the centrality of nodes and inform seeding strategies. Finally, I will delve into the microscopic dynamics of empirical higher-order structures. I will study the mechanisms governing their temporal dynamics both at the node and group level, characterizing how individuals navigate groups and how groups form and dismantle. I will conclude by proposing a dynamical hypergraph model that closely reproduces the empirical observations.
 

Let $G$ be a $d$-regular graph of growing degree on $n$ vertices, and form a random subgraph $G_p$ of $G$ by retaining edge of $G$ independently with probability $p=p(d)$. Which conditions on $G$ suffice to observe a phase transition at $p=1/d$, similar to that in the binomial random graph $G(n,p)$, or, say, in a random subgraph of the binary hypercube $Q^d$?

We argue that in the supercritical regime $p=(1+\epsilon)/d$, $\epsilon>0$ being a small constant, postulating that every vertex subset $S$ of $G$ of at most $n/2$ vertices has its edge boundary at least $C|S|$, for some large enough constant $C=C(\epsilon)>0$, suffices to guarantee the likely appearance of the giant component in $G_p$. Moreover, its asymptotic order is equal to that in the random graph $G(n,(1+\epsilon)/n)$, and all other components are typically much smaller.

We further give examples demonstrating the tightness of this result in several key senses.

A joint work with Sahar Diskin, Joshua Erde and Mihyun Kang.

We introduce a sparse and very high order hp-finite element method for the weak form of the Helmholtz equation.  The domain may be a disk, an annulus, or a cylinder. The cells of the mesh are an innermost disk (omitted if the domain is an annulus) and concentric annuli.

We demonstrate the effectiveness of this method on PDEs with radial direction discontinuities in the coefficients and data. The discretization matrix is always symmetric and positive-definite in the positive-definite Helmholtz regime. Moreover, the Fourier modes decouple, reducing a two-dimensional PDE solve to a series of one-dimensional solves that may be computed in parallel, scaling with linear complexity. In the positive-definite case, we utilize the ADI method of Fortunato and Townsend to apply the method to a 3D cylinder with a quasi-optimal complexity solve.

Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. In this talk, we study the growth rate of these functions adapted to finite characteristic quotients. One potential application of this result is towards linearity of the mapping class group.

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time.

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time. 

Students presenting are:

Adrian Martini, supervisor Alison Ethridge

Fang Rui Lim, supervisor Rama Cont

Thomas Groves, supervisor Dmitry Beylaev

Sarah-Jean Meyer, supervisor Massimiliano Gubinelli

We establish exponentially-fast mixing for passive scalars driven by two well-known examples of random divergence-free vector fields. The first one is the alternating shear flow model proposed by Pierrehumbert, in which case we set up a dynamics-based framework to construct such space-time smooth universal exponential mixers. The second example is the statistically stationary, homogeneous, isotropic Kraichnan model of fluid turbulence. In this case, the proof follows a new explicit identity for the evolution of negative Sobolev norms of the scalar. This is based on joint works with Alex Blumenthal (Georgia Tech) and Michele Coti Zelati (ICL), and Michele Coti Zelati and Theodore Drivas (Stony Brook), respectively.

The twin prime conjecture asserts that there are infinitely many primes p for which p+2 is also prime. This conjecture appears far out of reach of current mathematical techniques. However, in 2013 Zhang achieved a breakthrough, showing that there exists some positive integer h for which p and p+h are both prime infinitely often. Equidistribution estimates for primes in arithmetic progressions to smooth moduli were a key ingredient of his work. In this talk, I will sketch what role these estimates play in proofs of bounded gaps between primes. I will also show how a refinement of the q-van der Corput method can be used to improve on equidistribution estimates of the Polymath project for primes in APs to smooth moduli.

In a classical work, Bowen and Margulis proved the equidistribution of
closed geodesics in any hyperbolic manifold. Together with Jeremy Kahn
and Vladimir Marković, we asked ourselves what happens in a
three-manifold if we replace curves by surfaces. The natural analog of a
closed geodesic is then a minimal surface, as totally geodesic surfaces
exist only very rarely. Nevertheless, it still makes sense (for various
reasons, in particular to ensure uniqueness of the minimal
representative) to restrict our attention to surfaces that are almost
totally geodesic.

The statistics of these surfaces then depend very strongly on how we
order them: by genus, or by area. If we focus on surfaces whose *area*
tends to infinity, we conjecture that they do indeed equidistribute; we
proved a partial result in this direction. If, however, we focus on
surfaces whose *genus* tends to infinity, the situation is completely
opposite: we proved that they then accumulate onto the totally geodesic
surfaces of the manifold (if there are any).

We describe the compact scaling limits of uniformly random quadrangulations with boundaries on a surface of arbitrary fixed genus. These limits, called Brownian surfaces, are homeomorphic to the surface of the given genus with or without boundaries depending on the scaling regime of the boundary perimeters of the quadrangulation. They are constructed by appropriate gluings of pieces derived from Brownian geometrical objects (the Brownian plane and half-plane). In this talk, I will review their definition and discuss possible alternative constructions. This is based on joint work with Jérémie Bettinelli.

Classically, topological vector bundles are classified by homotopy classes of maps into infinite Grassmannians. This allows us to study topological vector bundles using obstruction theory: we can detect whether a vector bundle has a trivial subbundle by means of cohomological invariants. In the context of algebraic geometry, one can ask whether algebraic vector bundles over smooth affine varieties can be classified in a similar way. Recent advances in motivic homotopy theory give a positive answer, at least over an algebraically closed base field. Moreover, the behaviour of vector bundles over general base fields has surprising connections with the theory of quadratic forms.

Transfer learning is an emerging and popular paradigm for utilizing existing knowledge from  previous learning tasks to improve the performance of new ones. In this talk, we will first present transfer learning in the early diagnosis of eye diseases: diabetic retinopathy and retinopathy of prematurity.  

We will discuss how this empirical  study leads to the mathematical analysis of the feasibility and transferability  issues in transfer learning. We show how a mathematical framework for the general procedure of transfer learning helps establish  the feasibility of transfer learning as well as  the analysis of the associated transfer risk, with applications to financial time series data.

Come and meet the Mathematrix committee and find out more about what the society is for and what events we are putting on this term. There will be free pizza! 

https://www.maths.ox.ac.uk/study-here/postgraduate-study/mathematrix

Interpreting data using mechanistic mathematical models provides a foundation for discovery and decision-making in all areas of science and engineering. Key steps in using mechanistic mathematical models to interpret data include: (i) identifiability analysis; (ii) parameter estimation; and (iii) model prediction. Here we present a systematic, computationally efficient likelihood-based workflow that addresses all three steps in a unified way. Recently developed methods for constructing profile-wise prediction intervals enable this workflow and provide the central linkage between different workflow components. These methods propagate profile-likelihood-based confidence sets for model parameters to predictions in a way that isolates how different parameter combinations affect model predictions. We show how to extend these profile-wise prediction intervals to two-dimensional interest parameters, and then combine profile-wise prediction confidence sets to give an overall prediction confidence set that approximates the full likelihood-based prediction confidence set well.  We apply our methods to a range of synthetic data and real-world ecological data describing re-growth of coral reefs on the Great Barrier Reef after some external disturbance, such as a tropical cyclone or coral bleaching event.
 

Interacting particle systems provide flexible and powerful models that are useful in many application areas such as sociology (agents), molecular dynamics (proteins) etc. However, particle systems with large numbers of particles are very complex and difficult to handle, both analytically and computationally. Therefore, a common strategy is to derive effective equations that describe the time evolution of the empirical particle density, the so-called Dean-Kawasaki equation.

Our aim is to derive and study continuum models for the mesoscopic behavior of particle systems. In particular, we are interested in finite size effects. We will introduce nonlinear and non-Gaussian models that approximate the Dean-Kawasaki equation, in the special case of non-interacting particles. We want to study the well-posedness of these nonlinear SPDE models and to control the weak error of the SPDE approximation.  This is the joint work with H. Kremp (TU Wien) and N. Perkowski (FU Berlin).

We discuss $C^1$-regularity and developability of isometric immersions of flat domains into $\mathbb{R}^3$ enjoying a local fractional Sobolev $W^{1+s;2/s}$-regularity for $2/3 \leq s < 1$, generalising the known results on Sobolev (by Pakzad) and H\"{o}lder (by De Lellis--Pakzad) regimes. Ingredients of the proof include analysis of the weak Codazzi equations of isometric immersions, the study of $W^{1+s;2/s}$-gradient deformations with symmetric derivative and vanishing distributional Jacobian determinant, and the theory of compensated compactness. Joint work with M. Reza Pakzad and Armin Schikorra.

Floer theory, which mimics an infinite dimensional Morse approach to the study of critical points of smooth functions, appeared as an attempt to prove Arnold conjecture. The theory is more or less well understood in some compact cases.

Non-compact symplectic manifolds can sometimes be compactified as singular symplectic manifolds where the symplectic form "blows up" along a hypersurface in a controlled way (b^m-symplectic manifolds). In natural examples in Celestial mechanics such as the 3-body problem, these compactifications are given by regularization transformations à la Moser/Mc Gehee etc.

I will use the theory of b^m-symplectic/b^m-contact manifolds (introduced by Scott, Guillemin-Miranda Weitsman, and Miranda-Oms) as a guinea pig to propose ways to extend the study of Hamiltonian/Reeb Dynamics to singular symplectic/contact manifolds. This, in particular, yields new results for non-compact symplectic manifolds and for special (but, yet, meaningful) classes of Poisson manifolds.

Inspiration comes from several results extending the Weinstein conjecture to the context of b^m-contact manifolds and its connection to the study of escape orbits in Celestial mechanics and Fluid Dynamics. Those examples motivate a model for (singular) Floer homology.

I'll describe the motivating examples/results and some ideas to attack the general questions.

The Hardy Lectureship was founded in 1967 in memory of G.H. Hardy (LMS President 1926-1928 & 1939-1941). The Hardy Lectureship is a lecture tour of the UK by a mathematician with a high reputation in research.

Schauder estimates are a classical tool in linear and nonlinear elliptic and parabolic PDEs. They describe how regularity of coefficients reflects in regularity of solutions. They basically have a perturbative nature. This means that they can be obtained by perturbing the estimates available for problems without coefficients. This paradigm works as long as one deals with uniformly elliptic equations. The nonuniformly elliptic case is a different story and Schauder's theory turns out to be not perturbative any longer, as shown by counterexamples. In my talk, I will present a method allowing to bypass the perburbative schemes and leading to Schauder estimates in the nonuniformly elliptic regime. For this I will concentrate on the case of nonuniformly elliptic functionals with nearly linear growth, also covering a borderline case of so-called double phase energies. From recent, joint work with Cristiana De Filippis (Parma). 

Counting incompressible surfaces in hyperbolic 3-manifolds.

1:30pm

Marc Lackenby (Oxford)

Incompressible embedded surfaces play a central role in 3-manifold theory. It is a natural and interesting question to ask how many such surfaces are contained in a given 3-manifold M, as a function of their genus g. I will present a new theorem that provides a surprisingly small upper bound. For any given g, there a polynomial p_g with the following property. The number of closed incompressible surfaces of genus g in a hyperbolic 3-manifold M is at most p_g(vol(M)). This is joint work with Anastasiia Tsvietkova.

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The rates of growth in hyperbolic groups.

2:45pm

Koji Fujiwara (Kyoto)

For a finitely generated group of exponential growth, we study the set of exponential growth rates for all possible finite generating sets. Let G be a hyperbolic group. It turns out that the set of growth rates is well-ordered.  Also, given a number, there are only finitely many generating sets that have this number as the growth rate. I also plan to discuss the set of growth for a family of groups.

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Character varieties of random groups.

4:00pm

Emmanuel Breuillard (Oxford)

In joint work with P. Varju and O. Becker we study the representation and character varieties of random finitely presented groups with values in a complex semisimple Lie group.  We compute the dimension and number of irreducible components of the character variety of a random group. In particular we show that random one-relator groups have many rigid Zariski-dense representations. The proofs use a fair amount of number theory and are conditional on GRH. Key to them is the use of expander graphs for finite simple groups of Lie type as well as a new spectral gap result for random walks on linear groups.

I will explain a formalism for anticyclotomic Euler systems for a large class of Galois representations and explain how to prove analogs of Kolyvagins' celebrated "rank one" results. A novelty of this approach lies in the use of primes that split in the CM field. This is joint work with Dimitar Jetchev and Jan Nekovar. I will also describe some higher-dimensional examples of such Euler systems.

1:30 Milena Vuletic

Simulation of Arbitrage-Free Implied Volatility Surfaces

We present a computationally tractable method for simulating arbitrage-free implied volatility surfaces. We illustrate how our method may be combined with a factor model based on historical SPX implied volatility data to generate dynamic scenarios for arbitrage-free implied volatility surfaces. Our approach conciliates static arbitrage constraints with a realistic representation of statistical properties of implied volatility co-movements.

2:00 Nicola Muca Cirone

Neural Signature Kernels

Motivated by the paradigm of reservoir computing, we consider randomly initialized controlled ResNets defined as Euler-discretizations of neural controlled differential equations (Neural CDEs), a unified architecture which enconpasses both RNNs and ResNets. We show that in the infinite-width-depth limit and under proper scaling, these architectures converge weakly to Gaussian processes indexed on some spaces of continuous paths and with kernels satisfying certain partial differential equations (PDEs) varying according to the choice of activation function, extending the results of Hayou (2022); Hayou & Yang (2023) to the controlled and homogeneous case. In the special, homogeneous, case where the activation is the identity, we show that the equation reduces to a linear PDE and the limiting kernel agrees with the signature kernel of Salvi et al. (2021a). We name this new family of limiting kernels neural signature kernels. Finally, we show that in the infinite-depth regime, finite-width controlled ResNets converge in distribution to Neural CDEs with random vector fields which, depending on whether the weights are shared across layers, are either time-independent and Gaussian or behave like a matrix-valued Brownian motion.

2:30 Break

2:50-3:50 Renyuan Xu, Assistant Professor, University of Southern California

Reversible and Irreversible Decisions under Costly Information Acquisition 

Many real-world analytics problems involve two significant challenges: estimation and optimization. Due to the typically complex nature of each challenge, the standard paradigm is estimate-then-optimize. By and large, machine learning or human learning tools are intended to minimize estimation error and do not account for how the estimations will be used in the downstream optimization problem (such as decision-making problems). In contrast, there is a line of literature in economics focusing on exploring the optimal way to acquire information and learn dynamically to facilitate decision-making. However, most of the decision-making problems considered in this line of work are static (i.e., one-shot) problems which over-simplify the structures of many real-world problems that require dynamic or sequential decisions.

As a preliminary attempt to introduce more complex downstream decision-making problems after learning and to investigate how downstream tasks affect the learning behavior, we consider a simple example where a decision maker (DM) chooses between two products, an established product A with known return and a newly introduced product B with an unknown return. The DM will make an initial choice between A and B after learning about product B for some time. Importantly, our framework allows the DM to switch to Product A later on at a cost if Product B is selected as the initial choice. We establish the general theory and investigate the analytical structure of the problem through the lens of the Hamilton—Jacobi—Bellman equation and viscosity solutions. We then discuss how model parameters and the opportunity to reverse affect the learning behavior of the DM.

This is based on joint work with Thaleia Zariphopoulou and Luhao Zhang from UT Austin.
 

Principal Component Analysis (PCA) is a fundamental and ubiquitous tool in statistics and data analysis.
The bare-bones idea is this. Given a data set of n points y_1, ..., y_n, form their sample covariance S. Eigenvectors corresponding to large eigenvalues--namely directions along which the variation within the data set is large--are usually thought of as "important"  or "signal-bearing"; in contrast, weak directions are often interpreted as "noise", and discarded along the proceeding steps of the data analysis pipeline. Principal component (PC) selection is an important methodological question: how large should an eigenvalue be so as to be considered "informative"?
Our main deliverable is ScreeNOT: a novel, mathematically-grounded procedure for PC selection. It is intended as a fully algorithmic replacement for the heuristic and somewhat vaguely-defined procedures that practitioners often use--for example the popular "scree test".
Towards tackling PC selection systematically, we model the data matrix as a low-rank signal plus noise matrix Y = X + Z; accordingly, PC selection is cast as an estimation problem for the unknown low-rank signal matrix X, with the class of permissible estimators being singular value thresholding rules. We consider a formulation of the problem under the spiked model. This asymptotic setting captures some important qualitative features observed across numerous real-world data sets: most of the singular values of Y are arranged neatly in a "bulk", with very few large outlying singular values exceeding the bulk edge. We propose an adaptive algorithm that, given a data matrix, finds the optimal truncation threshold in a data-driven manner under essentially arbitrary noise conditions: we only require that Z has a compactly supported limiting spectral distribution--which may be a priori unknown. Under the spiked model, our algorithm is shown to have rather strong oracle optimality properties: not only does it attain the best error asymptotically, but it also achieves (w.h.p.) the best error--compared to all alternative thresholds--at finite n.

This is joint work with Matan Gavish (Hebrew University of Jerusalem) and David Donoho (Stanford).