# Schedule for: 22w5116 - New interfaces of Stochastic Analysis and Rough Paths

Beginning on Sunday, September 4 and ending Friday September 9, 2022

All times in Banff, Alberta time, MDT (UTC-6).

Sunday, September 4 | |
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16:00 - 17:30 | Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |

Monday, September 5 | |
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07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

08:45 - 09:00 |
Introduction and Welcome by BIRS Staff ↓ A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions. (TCPL 201) |

08:55 - 09:20 |
Weijun Xu: Periodic homogenisation for phi 4 2 ↓ We consider a periodic homogenisation problem for \phi^4_2, a toy model that combines both renormalisation in singular SPDEs and homogenisation. In this case, we show that two different orders of the two limiting procedures yield the same limit, for both the dynamics and the invariant measure. Joint work in progress with Yilin Chen. (Online) |

09:20 - 09:45 |
Ismael Bailleul: The Anderson operator ↓ The continuous Anderson operator H is a perturbation of the Laplace-Beltrami operator by a random space white noise potential. We consider this 'singular' operator on a two-dimensional closed Riemannian manifold. One can use functional analysis arguments to construct the operator as an unbounded operator on L2. We prove a sharp Gaussian small-time asymptotic for the heat kernel of H that leads amongst others to strong norm estimates for quasimodes. We introduce a new random field, called Anderson Gaussian free field, and prove that the law of its random partition function characterizes the law of the spectrum of H (Online) |

09:45 - 10:10 |
Remi Catellier: Regularization by noise for rough differential equations driven by Gaussian rough paths ↓ We consider the rough differential equation with drift driven by a Gaussian geometric rough path. Under natural conditions on the rough path, namely non-determinism, and uniform ellipticity conditions on the diffusion coefficient, we prove path-by-path well-posedness of the equation for poorly regular drifts. In the case of the fractional Brownian motion B H for H > 1 4 , we prove that the drift may be taken to be κ > 0 Hölder continuous and bounded for κ > 3 2 − 1 2H. A flow transform of the equation and Malliavin calculus for Gaussian rough paths are used to achieve such a result. (Online) |

10:10 - 10:11 | Virtual Group Photo (Online) |

10:10 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 10:55 |
Xue-Mei Li: Non-markovian multi-scale stochastic systems ↓ Multi-time scale is prevalent in mathematical modelling, so is auto-correlated noise. I shall discuss Multi-scale stochastic differential equations drive by fractional Brownian motions, to which the recent progress in rough paths and stochastic analysis allowing to make progress for the first time. This talk with be for non-expert, accessible to anybody interested in time evolution of random variables (Online) |

10:55 - 11:20 |
Tomoyuki Ichiba: A pathwise approach to directed chain stochastic differential equations ↓ We study pathwise properties of the system of diffusions on infinite directed chain graph described by stochastic differential equations with mean-field interaction and the interactions from the neighborhood nodes under Lipschitz assumptions on coefficients. We discuss the continuity of the law of the solution in the weak topology of measures with respect to the law of inputs and apply to the heterogenous particle approximation problems. Then we construct rough pathwise solution to the directed chain equations. (Online) |

11:20 - 11:45 |
Alberto Ohashi: Rough paths and symmetric-Stratonovich integrals driven by singular covariance Gaussian processes ↓ In this talk, we will present the relation between a stochastic version of the rough path integral with the symmetric-Stratonovich integral in the sense of regularization. Under mild regularity conditions in the sense of Malliavin calculus, we establish equality between stochastic rough path and symmetric-Stratonovich integrals driven by a class of Gaussian processes. As a by-product, we show that solutions of multi-dimensional rough differential equations driven by a large class of Gaussian rough paths they are actually solutions to Stratonovich stochastic differential equations. We obtain almost sure convergence rates of the first-order Stratonovich scheme to rough paths integrals in the sense of Gubinelli. In case the time-increment of the Malliavin derivative of the integrands is regular enough, the rates are essentially sharp. The framework applies to a large class of Gaussian processes whose the second-order derivative of the covariance function is a sigma-finite non-positive measure on $\mathbb{R}^2_+$ off diagonal. (Online) |

11:45 - 12:10 |
William Salkeld: An introduction to rough mean-field equations ↓ I will explain some of the problems with mean-field limits for rough differential equations and how probabilistic rough paths, random controlled rough paths and coupled hopf algebras provide solutions. (Online) |

12:10 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

14:00 - 14:20 |
Group Photo ↓ Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo! (TCPL Foyer) |

14:20 - 14:50 |
George Wynne: Kernel Stein Discrepancy for Measures on Hilbert Spaces ↓ Kernel Stein discrepancy (KSD) is a measure of discrepancy between a target distribution and a candidate distribution which does not need samples from the target distribution to estimate. KSD has found wide application in computational statistics and machine learning, for example in goodness-of-fit tests and sample quality assessment. So far it has only been applied for finite dimensional data. This talk will cover the extension of KSD to data lying in a Hilbert space, for example trajectories of stochastic differential equations and will demonstrate the use of KSD for the aforementioned tasks with this data. (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 16:00 |
Rosa Preiß: Smooth rough paths, their geometry and algebraic renormalization ↓ Joined work with Carlo Bellingeri, Peter Friz and Sylvie Paycha
We introduce the class of “smooth rough paths” and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer–Cartan perspective is the key to a purely algebraic form of Lyons’ extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of “sum of rough paths”. (TCPL 201) |

16:00 - 16:30 |
Samy Tindel: Hyperbolic Anderson model in the Skorohod and rough settings ↓ In this talk I will first give a brief overview of some standard results concerning the wave equation. Then I will describe some recent advances aiming at a proper definition of noisy wave equations, when specialized to a bilinear setting (called hyperbolic Anderson model). First I will focus on the so-called Skorohod setting, where an explicit chaos decomposition of the solution is available. A good control of the chaos expansion is then achieved thanks to an exponentiation trick. Next I will turn to a pathwise approach, which is based on a novel Strichartz type estimate for the wave operator. If time allows it, I will show the main steps of this analytic estimate. (TCPL 201) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |

Tuesday, September 6 | |
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07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

08:30 - 08:55 |
Chong Liu: Higher Rank Signatures and Filtrations ↓ Filtration is an abstract and important notion that appears naturally in stochastic analysis,
which models the information flow generated by underlying stochastic processes. However, many well–known statistical methods cannot detect filtrations as they are based on weak topology, and consequently they may lead to significant errors for those circumstances where the evolution of information plays a crucial role. In this talk we will introduce a new methodology based on the signature kernel learning approach developed by Terry Lyons which can be used for giving a precise description of filtrations hidden behind observed signals. We will then illustrate that this method provides a feasible statistical tool for lots of filtration–sensitive cases; in particular, it allows to reduce highly non–linear path-and-filtration dependent functionals (e.g. the pricing of American option) to a linear regression problem, which reveals an interesting combination of (Hopf) algebra and kernel learning. (Online) |

08:55 - 09:20 |
Peter Friz: Weak Rates For Rough Vol ↓ joint work with T. Wagenhofer and W. Salkeld (Online) |

09:20 - 09:45 |
Bruno Dupire: Signatures and Expansions of Functionals ↓ European option payoffs can be generated by combinations of hockeystick payoffs or of monomials. Interestingly, path dependent options can be generated by combinations of signatures, which are the building blocks of path dependence.
We focus on the case of 1 asset together with time, typically the evolution of the price x as a function of the time t. The signature of a path for a given word with letters in the alphabet {t,x} (sometimes called augmented signature of dimension 1) is an iterated Stratonovich integral with respect to the letters of the word and it plays the role of a monomial in a Taylor expansion.
For a given time horizon T the signature elements associated to short words are contained in the linear space generated by the signature elements associated to longer words and we construct an incremental basis of signature elements. It allows writing a smooth path dependent payoff as a converging series of signature elements, a result stronger than the density property of signature elements from the Stone-Weierstrass theorem.
We recall the main concepts of the Functional Itô Calculus, a natural framework to model path dependence and draw links between two approximation results, the Taylor expansion and the Wiener chaos decomposition. The Taylor expansion is obtained by iterating the functional Stratonovich formula whilst the Wiener chaos decomposition is obtained by iterating the functional Itô formula applied to a conditional expectation. (Online) |

09:45 - 10:10 |
Christa Cuchiero: Signature methods in stochastic portfolio theory ↓ In the context of stochastic portfolio theory we introduce a novel class of portfolios which we call linear path-functional portfolios. These are portfolios which are determined by certain transformations of linear functions of a collections of feature maps that are non-anticipative path functionals of an underlying semimartingale. As main example for such feature maps we consider (random) signature of the (ranked) market weights. We prove that these portfolios are universal in the sense that every continuous (possibly path-dependent) portfolio function of the market weights can be uniformly approximated by signature portfolios. We also show that signature portfolios can approximate the log-optimal portfolio in several classes of non-Markovian models arbitrarily well and illustrate numerically that the trained signature portfolios are remarkably close to the theoretical log-optimal portfolios.
Besides these universality features, the main numerical advantage lies in the fact that several optimization tasks like maximizing expected logarithmic utility or mean-variance optimization within the class of linear path-functional portfolios reduces to a convex quadratic optimization problem, thus making it computationally highly tractable.
We apply our method to real market data and show generic out-performance on out-of-sample data even under transaction costs.
The talk is based on joint work with Janka Möller. (Online) |

10:10 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 10:55 |
David Prömel: Model-free portfolio theory: a rough path approach ↓ Classical approaches to optimal portfolio selection problems are based on probabilistic models for the asset returns or prices. However, by now it is well observed that the performance of optimal portfolios are highly sensitive to model misspecifications. To account for various type of model risk, robust and model-free approaches have gained more and more importance in portfolio theory.Based on a rough path foundation, we develop a model-free approach to stochastic portfolio theory and Cover's universal portfolio. The use of rough path theory allows treating significantly more general portfolios in a model-free setting, compared to previous model-free approaches. Without the assumption of any underlying probabilistic model, we present pathwise Master formulae analogously to the classical ones in stochastic portfolio theory, describing the growth of wealth processes generated by pathwise portfolios relative to the wealth process of the market portfolio, and we show that the appropriately scaled asymptotic growth rate of Cover's universal portfolio is equal to the one of the best retrospectively chosen portfolio. The talk is based on a joint work with Andrew Allan, Christa Cuchiero and Chong Liu. (Online) |

10:55 - 11:20 |
Uzu Lim: Tangent Space and Dimension Estimation with the Wasserstein Distance ↓ Consider a finite sample drawn near a smooth compact submanifold M of a Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate tangent spaces and the dimension of M with high confidence. The algorithm for estimation is a local version of the principal component analysis. Our results accommodate for: (1) non-uniformity of the data distribution, (2) non-uniform noise that may vary across the manifold, and (3) explicitly computes all of the constants appearing in the theorem. The proof uses a matrix concentration inequality to estimate covariance matrices and quantifies nonlinearity of manifold using the Wasserstein distance. (Online) |

11:20 - 11:45 |
Jiajie Tao: Optimization on signature manifold ↓ The signature space is not Euclidean while the logsignature space is. We present an optimization algorithm on signature space by transforming the problem into an optimization problem on the logsignature space. We provide a natural representation of the tensor exponential map in order to derive the gradient from its dual element, the differential. (Online) |

11:45 - 12:10 |
Fabian Harang: Non-linear Young equations in the plane and pathwise regularization by noise ↓ Regularization by noise for stochastic differential equations has been a long studied topic in the field of stochastic analysis. After the work of Gubinelli and Catellier in 2016 on a pathwise analogue to the probabilistic analysis of regularization by noise based on what they called averaged fields in combination with the concept of non-linear Young equations, this area of study has recently received much attention. In this talk we will discuss an extension of this approach to pathwise differential equations in the plane. These are essentially hyperbolic non-linear PDEs with additive noise. We extend the so-called local time formulation of the regularization by noise approach to these equations, and extend the concept of non-linear Young equations to rectangular domains in order to prove wellposedness of these equations, even when the nonlinear coefficient is a generalized function (e.g. distribution). We illustrate the application of this construction by proving wellposedness of a wave equation with a noisy boundary constructed from two independent fractional Brownian motions.
In the end we will discuss further potential applications and extensions of this framework and present several open challenges.
This talk is based on a joint collaboration with Florian Bechtold (Bielefeld University) and Nimit Rana (Bielefeld University), see arXiv:2206.05360 (Online) |

12:10 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

13:30 - 14:00 |
Pierre Nyquist: Large deviations in data science ↓ In addition to having become a cornerstone of modern probability, the theory of large deviations has proven very useful in the analysis and design of efficient stochastic numerical methods. Examples include methods for rare-event sampling and general-purpose Markov chain Monte Carlo methods. Large deviations are also a natural way to analyse gradient flows, which are now being used extensively within data science. Despite the successful use of tools from large deviation theory, and their connections to stochastic control, in these areas, they are largely unexplored in the machine learning setting. In this talk we will discuss some initial attempts at bringing this theory into the data science context. As examples we will consider some recent algorithms proposed for finding mixed equilibria in zero-sum games and stochastic approximation methods. (TCPL 201) |

14:00 - 14:30 |
Anastasia Papavasiliou: The inverse problem for CDEs ↓ I will present a novel algorithm for constructing a control driving a controlled differential equation from discrete observations of the response. By using signatures, our approach is pathwise, resulting to a uniform convergence with respect to sampling rate, in $p$-variation. I will also discuss applications of the algorithm, most notably on the construction of the likelihood for discretely observed random rough differential equations. (TCPL 201) |

14:30 - 15:00 |
Paul Gassiat: Reflected differential equations and rough paths ↓ The solution to a reflected SDE is a stochastic process which is constrained to take values in a spatial domain D. When the underlying signal is a semimartingale, the well-posedness of these equations has been known since the 80s. A natural question is whether rough path methods can be applied to this type of equations, for instance to extend the class of driving noises beyond the semimartingale framework. In this talk, I will present what is currently known on this topic (focusing in particular on uniqueness of solutions, which turns out to depend on the regularity of the noise) and discuss some open questions. (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |

Wednesday, September 7 | |
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07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

08:30 - 08:55 |
Sebastian Riedel: Random dynamical systems and rough paths ↓ The theory of Random dynamical systems (RDS) is a powerful tool that allows to study the long-time behaviour of solutions to stochastic evolution equations. Since the theory presumes a pathwise solution concept, it is often not clear whether an evolution equation on an infinite dimensional space generates an RDS or not. We use rough paths theory to show that a certain class of stochastic delay differential equations induce an RDS on random Banach spaces. We prove that this structure is indeed useful by proving the existence of local random invariant manifolds. (Online) |

08:55 - 09:20 |
Antoine Lejay: Rough Invariant Imbedding ↓ We extend the invariant embedding principle in the context of rough path to solve a differential equation with an affine 2-points boundary values problem. We consider first the question of existence and uniqueness in short time, as well as existence for any time using the theory of degree. The use of rough paths allows one to deal with a broad family of stochastic procesess and to avoid the technical difficulties related to the use of anticipative stochastic calculus. This is a joint work with Renaud Marty (IECL). (Online) |

09:20 - 09:45 |
Khoa Le: Numerical solutions for singular stochastic differential equations ↓ We consider a generic and explicit tamed Euler--Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusion coefficient is uniformly elliptic, Hölder continuous and weakly differentiable in the spatial variables while the drift satisfies the Ladyzhenskaya--Prodi--Serrin condition, as considered by Krylov and Röckner (2005). In the discrete scheme, the drift is tamed by replacing it by an approximation. A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and be fine-tuned to achieve the standard 1/2-strong convergence rate with a logarithmic factor. (Online) |

09:45 - 10:10 |
Ilya Chevyrev: Feature Engineering with Regularity Structures ↓ In many machine learning tasks, it is crucial to extract low-dimensional and descriptive features from a data set. In this talk, I present a method to extract features from multi-dimensional space-time signals which is motivated by the success of signatures in machine learning together with the success of regularity structures in the analysis of SPDEs. I will present a flexible definition of a model feature vector along with numerical experiments in which we combine these features with basic supervised linear regression to predict solutions to parabolic and dispersive PDEs with a given forcing and boundary conditions. Interestingly, in the dispersive case, the prediction power relies heavily on whether the boundary conditions are appropriately included in the model. The talk is based on the following joint work with Andris Gerasimovics and Hendrik Weber. (Online) |

10:10 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 10:55 |
Antoine Hocquet: Quasilinear rough evolution equations ↓ I will present recent results obtained with Alexandra Neamtu on quasilinear evolution equations driven by a rough input, based on functional analysis techniques and a generalization of the sewing lemma.
Applications in the context of stochastic PDEs and stochastic flows will be given. (Online) |

10:55 - 11:20 |
Paul Hager: Optimal Stopping and Control with Signatures ↓ We present a new method for solving optimal stopping problems based on a representation of the stopping rule by linear and non-linear functionals (deep n.n.) of the rough path signature and prove that maximizing over these classes of signature stopping times, in fact, solves the stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a deterministic optimization problem depending only on the (truncated) expected signature. The theory encompasses processes such as fractional Brownian motion, which fail to be either semi-martingales or Markov processes. This is from a joint work with C. Bayer, S. Riedel and J. Schoenmakers.
In the talk we further examine other signature based methods in optimal control. (Online) |

11:20 - 11:45 |
Leonard Schmitz: Multiparameter quasisymmetric functions ↓ Quasisymmetric functions have recently been used in time series analysis as polynomial features that are invariant under, so-called, dynamic time warping.
We extend this notion to data indexed by multiple parameters and thus provide dynamic warping invariants for tensors of arbitrary shape, including time series, images, 3D images, or videos. We show that multiparameter quasisymmetric functions are complete in a certain sense, and provide a quasi-shuffle identity by equipping the underlying Hopf algebra with a multidimensional quasi-shuffle. The compatible coproduct is based on diagonal concatenation of the input data, leading to a (weak) form of Chen’s identity. (Online) |

11:45 - 12:10 |
Alexander Schell: Nonlinear and Robust Independent Component Analysis for Stochastic Processes ↓ Blind Source Separation (BSS) aims to recover a signal S from its mixture X = f(S) under the condition that the effecting transformation f is invertible but unknown. This being a basic problem with numerous real-world applications, a fundamental issue is to understand how the solutions to this problem behave when their supporting statistical assumptions are violated. In this talk, we present a method to achieve Blind Source Separation for f nonlinear, and (time permitting) also introduce a general framework to analyse violations of common identifiability assumptions and quantify the effect these have on the blind recovery of S from X (Online) |

12:10 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

13:30 - 17:30 | Free Afternoon (Banff National Park) |

17:30 - 19:30 |
Dinner ↓ |

Thursday, September 8 | |
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07:00 - 08:45 |
Breakfast ↓ |

08:30 - 08:55 |
Josef Teichmann: A representation theoretic view on signature transforms ↓ We construct by means of probabilistic methods dynamical systems
driven by rough paths, which contain precisely the same information as
signature itself. (Online) |

08:55 - 09:20 |
Nikolas Nüsken: Estimating hidden parameters in stochastic multiscale systems using McKean-Vlasov dynamics and rough paths ↓ Motivated by the challenge of incorporating data into misspecified and multiscale dynamical mod-els, we study a McKean-Vlasov equation that contains the data stream as a common driving rough path. This setting allows us to prove well-posedness as well as continuity with respect to the driver in an appropriate rough-path topology. The latter property is key in our subsequent development of a robust data assimilation methodology: We establish propagation of chaos for the associated interacting particle system, which in turn is suggestive of a numerical scheme that can be viewed as an extension of the ensemble Kalman filter to a rough-path framework.
Finally, we discuss a data-driven method based on subsampling to construct suitable rough path lifts and demonstrate the robustness of our scheme in a number of numerical experiments related to parameter estimation problems in multiscale contexts. (Online) |

09:20 - 09:45 |
Joel Dyer: Path signatures in simulation-based inference ↓ Computer simulations are used widely across scientific disciplines, often taking the form of stochastic black-box models consuming fixed parameters and generating a random output. In general for such models, no likelihood function is available, often due to the complexity of the simulators. Consequently, it is often convenient to adopt so-called likelihood-free or simulation-based inference methods that mimic conventional likelihood-based procedures using data simulated at different parameter values. While many such approaches exist for iid data, adapting these techniques to simulators that generate sequential data can be challenging. In this talk, we will discuss our recent work on simulation-based parameter inference for dynamic, stochastic simulators with the use of path signatures. We will argue that signatures and their recent kernelisation naturally and flexibly enable both approximate Bayesian and frequentist inference with time-series simulators of different kinds, with competitive empirical performance in a variety of benchmark experiments. (Online) |

09:45 - 10:10 |
James Foster: Markov Chain Cubature for Bayesian Inference ↓ Markov Chain Monte Carlo (MCMC) is widely regarded as the “go-to” approach for computing integrals with respect to posterior distributions in Bayesian inference. Whilst there are a large variety of MCMC methods, many prominent algorithms can be viewed as approximations of stochastic differential equations (SDEs). For example, the unadjusted Langevin algorithm (ULA) is obtained as an Euler discretization of the Langevin diffusion and has seen particular interest due to its scalability and connections to the optimization literature.
On the other hand, “Cubature on Wiener Space” (Lyons and Victoir, 2004) provides a powerful alternative to Monte Carlo for simulating SDEs. In the cubature paradigm, SDE solutions are represented as a cloud of particles and propagated via deterministic cubature formulae. However, such formulae can dramatically increase the number of particles, and thus SDE cubature requires “distribution compression” to be practical.
In this talk, we will show that by applying cubature to ULA and resampling particles in a spatially balanced manner, we obtain a gradient-based particle method for Bayesian inference. We shall discuss the theory underpinning SDE cubature and the key properties of the Langevin diffusion that enable numerical errors to be controlled over long time horizons. Finally, we compare our cubature algorithm to ULA and Stein Variational Gradient Descent on Gaussian mixture and Bayesian logistic regression models. (Online) |

10:10 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 10:55 |
Csaba Toth: Random Signature Fourier Features ↓ We revisit the signature kernel and provide a randomized construction that 1) converges to the true kernel in probability 2) comes with simple algorithms that scales linearly with sequence length while simultatenously enjoying the benefits of lifting the path to an infinite-dimensional state space 3) shows little loss of performance or even improvements compared to other variants while exhibiting much better computational complexity. The key is the use of random Fourier features and structured random projections for tensors together with concentration inequalities that allow obtaining good bounds. Joint work with Harald Oberhauser and Zoltan Szabo (Online) |

10:55 - 11:20 |
Zoltan Szabo: Kernel Machines with Shape Constraints ↓ Shape constraints enable one to incorporate prior knowledge into predictive models in a principled way with several successful applications. Including this side information in a hard fashion (for instance, at each point of an interval) for rich function classes however is a quite challenging task. I am going to present a convex optimization framework to encode hard affine constraints on function values and derivatives in the flexible family of kernel machines. The efficiency of the approach will be demonstrated in joint quantile regression (analysis of aircraft departures), convoy localization and safety-critical control (piloting an underwater vehicle while avoiding obstacles). [This is joint work with Pierre-Cyril Aubin-Frankowski. Preprint: https://arxiv.org/abs/2101.01519] (Online) |

11:20 - 11:45 |
Emilio Ferrucci: Branched rough paths on manifolds ↓ This talk is based on the recent preprint https://arxiv.org/abs/2205.00582. A branched rough path X consists of a rough integral calculus for X:[0,T]→R^d which may fail to satisfy integration by parts. Using Kelly's bracket extension [Kel12], we define a notion of pushforward of branched rough paths through smooth maps, which leads naturally to a definition of branched rough path on a smooth manifold. Once a covariant derivative is fixed, we are able to give a canonical, coordinate-free definition of integral against such rough paths. After characterising quasi-geometric rough paths in terms of their bracket extension, we use the same framework to define manifold-valued rough differential equations (RDEs) driven by quasi-geometric rough paths. These results extend previous work on 3>p-rough paths [ABCF22], itself a generalisation of the Ito calculus on manifolds developed by Meyer and Emery [Mey81, E89, E90], to the setting of non-geometric rough calculus of arbitrarily low regularity. (Online) |

11:45 - 12:10 |
Horatio Boedihardjo: A non-vanishing property for the signature of a bounded variation path ↓ Given a bounded variation path, what can we say about it's signature? It is classical that the signature is a group-like element and that the n-th term of the signature decay at the speed of n!. In this talk, we will show a third property, that the sequence of signature cannot contain infinitely many zeros. Together with the result of Chang, Lyons and Ni, this means the signature of reduced bounded variations paths have an exact decay rate n!. This work gives rise to many interesting questions, including what would be the complex version of the uniqueness theorem for signature and the analogous non-vanishing results for general geometric rough path (even though the nonvanishing property itself is not true for general rough paths). Joint work with Xi Geng. (Online) |

12:10 - 13:00 |
Lunch ↓ |

13:30 - 14:00 |
Joscha Diehl: Graph counting signatures and bicommutative Hopf algebras ↓ When counting subgraphs, well-known algebraic identities arise, which can be addressed by the theory of combinatorial Hopf algebras. For the many
notions of subgraphs, e.g., restricted, connected, or induced, there appear several definitions of graph counting operations in the literature. In this talk I will show how certain bialgebras help in translating between them. (Joint work with D. Caudillo, K. Ebrahimi-Fard, E. Verri) (TCPL 201) |

14:00 - 14:30 |
Christian Bayer: Stability of Deep Neural Networks via discrete rough paths ↓ Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as functions of the layer, we propose to derive stability bounds in terms of the total p-variation of trained weights for any p∈[1,3]. Unlike the C1-theory underlying the neural ODE literature, our estimates remain bounded even in the limiting case of weights behaving like Brownian motions, as suggested in [arXiv:2105.12245]. Mathematically, we interpret residual neural network as solutions to (rough) difference equations, and analyse them based on recent results of discrete time signatures and rough path theory. (TCPL 201) |

14:30 - 15:00 |
Patric Bonnier: Kernelized Cumulants: Beyond Kernel Mean Embeddings ↓ We study cumulants on reproducing kernel Hilbert spaces (RKHS) and show that they can be computed using the kernel trick. This provides a new set of all-purpose statistics associated with kernel embeddings that generalise the kernel mean embedding (KME), and these statistics characterise a distribution even when the underlying kernel is non-characteristic. Joint work with Zolt{\'a}n Szab{\'o} and Harald Oberhauser. (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

17:30 - 19:30 |
Dinner ↓ |

Friday, September 9 | |
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07:00 - 08:45 |
Breakfast ↓ |

08:30 - 08:55 |
Tong Xin: Sampling with constraints ↓ Sampling-based inference and learning techniques, especially Bayesian inference, provide an essential approach to handling uncertainty in machine learning (ML). As these techniques are increasingly used in daily life, it becomes essential to safeguard the ML systems with various trustworthy-related constraints, such as fairness, safety, interpretability. We propose a family of constrained sampling algorithms which generalize Langevin Dynamics (LD) and Stein Variational Gradient Descent (SVGD) to incorporate a moment constraint or a level set specified by a general nonlinear function. By exploiting the gradient flow structure of LD and SVGD, we derive algorithms for handling constraints, including a primal-dual gradient approach and the constraint controlled gradient descent approach. We investigate the continuous-time mean-field limit of these algorithms and show that they have $O(1/t)$ convergence under mild conditions. (Online) |

08:55 - 09:20 |
Nikolas Tapia: Generalized iterated-sums signatures ↓ We explore the algebraic properties of a generalised version of the iterated-sums signature, inspired by previous work of F.~Király and H.~Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties. (Online) |

09:20 - 09:45 |
Hang Lou: Path Development Network with Finite dimensional Lie Group ↓ Path signature, a mathematically principled and universal feature of sequential data, boosts performance of deep learning-based models in various sequential data tasks as a complimentary feature. However, it suffers from the curse of dimensionality when the path dimension is high. To tackle this problem, we propose a novel, trainable path development layer, which exploits representations of sequential data with the help of finite-dimensional matrix Lie groups. Our proposed layer, analogous to recurrent neural networks (RNN) but possessing an explicit, simple recurrent unit, can alleviate the gradient issues of RNNs with suitably chosen Lie groups. As a continuous time series model, our layer proves to be advantageous for irregular time series modelling. Empirical results on a range of datasets show that the development layer consistently and significantly outperforms, in terms of accuracy and dimensionality, signature features. Moreover, the compact hybrid model (obtained by stacking one-layer LSTM with the development layer) achieves the state-of-the-art against various RNN and continuous time series models. In addition, our layer enhances the performance of modelling dynamics constrained to Lie groups. (Online) |

09:45 - 10:10 |
Andrew Allan: Càdlàg rough differential equations with reflecting barriers ↓ We investigate rough differential equations (RDEs) with a time-dependent reflecting lower barrier, where both the driving (rough) path and the barrier itself may have jumps. Such reflected RDEs are interesting, particularly due to the nonuniqueness of multidimensional solutions. Assuming the driving signals allow for Young integration, we provide existence, uniqueness and stability results. When the driving signal is a càdlàg p-rough path for p < 3, we establish existence to general reflected rough differential equations, as well as uniqueness in the one-dimensional case. (Online) |

10:10 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:00 |
Checkout by 11AM ↓ 5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM. (Front Desk - Professional Development Centre) |

10:30 - 10:55 |
Danyu Yang: A remainder estimate for branched rough differential equations ↓ Based on isomorphisms of Hopf algebras, we obtain a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths. (Online) |

10:55 - 11:25 |
James-Michael Leahy: The Rough Euler-Poincaré equations and their calibration ↓ We recall our framework for variational principles for fluid dynamics on rough paths, in which advection is constrained to be the sum of a smooth and geometric rough-in-time vector field. The corresponding rough Euler-Poincaré equations satisfy a Kelvin circulation theorem. These equations conserve circulation for incompressible fluids and, otherwise, nontrivial circulation dynamics are generated by gradients of advected quantities arising from broken relabeling symmetry of inhomogeneous initial conditions. By parameterizing the fine-scale fluid motion with a rough vector field, we establish a flexible framework for parsimonious non-Markovian parameterizations of subgrid-scale dynamics.
We will then discuss the calibration problem of these models and a new result on the estimation of the noise coefficient of fractional Brownian motion-driven equations using a type of scaled quadratic variation.
This talk is based on joint work with Dan Crisan and Darryl Holm at Imperial College London, UK, and Torstein Nilssen at the University of Agder, Norway. (TCPL 201) |

11:25 - 11:55 |
Darrick Lee: Mapping Space Signatures ↓ We introduce the mapping space signature, a generalization of the path signature for maps from higher dimensional cubical domains, which is motivated by the topological perspective of iterated integrals by K. T. Chen. We show that the mapping space signature shares many of the analytic and algebraic properties of the path signature; in particular it is universal and characteristic with respect to Jacobian equivalence classes of cubical maps. This is joint work with Chad Giusti, Vidit Nanda, and Harald Oberhauser. (TCPL 201) |

12:00 - 13:30 | Lunch from 11:30 to 13:30 (Vistas Dining Room) |