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Expressive large-scale neural networks enable training powerful models for prediction tasks. However, in many engineering and science domains, such models are intended to be used not just for prediction, but for design -- e.g., creating new proteins that serve as effective therapeutics, or creating new materials or chemicals that maximize a downstream performance measure. Thus, researchers have recently grown an interest in building deep learning methods that solve offline \emph{model-based optimization} (MBO) problems, in which design candidates are optimized with respect to surrogate models learned from offline data. However, straightforward application of predictive models that are effective at predicting in-distribution properties of a design are not necessarily the best suited for use in creating new designs. Thus, the most successful algorithms that tackle MBO draw inspiration from reinforcement learning and generative modeling to meet the in-distribution constraints. Meanwhile, recent theoretical works have observed that exploiting the structure of the target black-box function is an effective strategy for solving MBO from offline data. Unfortunately, discovering such structure remains an open problem. In this paper, following first principles, we develop a model that learns the structure of an MBO task and empirically leads to improved designs. To this end, we introduce \emph{Cliqueformer} -- a scalable transformer-based architecture that learns the black-box function's structure in the form of its \emph{functional graphical model} (FGM), thus bypassing the problem of distribution shift, previously tackled by conservative approaches. We evaluate Cliqueformer on various tasks, ranging from high-dimensional black-box functions from MBO literature to real-world tasks of chemical and genetic design, consistently demonstrating its state-of-the-art performance.

Learning from multi-step off-policy data collected by a set of policies is a core problem of reinforcement learning (RL). Approaches based on importance sampling (IS) often suffer from large variances due to products of IS ratios. Typical IS-free methods, such as $n$-step Q-learning, look ahead for $n$ time steps along the trajectory of actions (where $n$ is called the lookahead depth) and utilize off-policy data directly without any additional adjustment. They work well for proper choices of $n$. We show, however, that such IS-free methods underestimate the optimal value function (VF), especially for large $n$, restricting their capacity to efficiently utilize information from distant future time steps. To overcome this problem, we introduce a novel, IS-free, multi-step off-policy method that avoids the underestimation issue and converges to the optimal VF. At its core lies a simple but non-trivial \emph{highway gate}, which controls the information flow from the distant future by comparing it to a threshold. The highway gate guarantees convergence to the optimal VF for arbitrary $n$ and arbitrary behavioral policies. It gives rise to a novel family of off-policy RL algorithms that safely learn even when $n$ is very large, facilitating rapid credit assignment from the far future to the past. On tasks with greatly delayed rewards, including video games where the reward is given only at the end of the game, our new methods outperform many existing multi-step off-policy algorithms.

While machine learning models are typically trained to solve prediction problems, we might often want to use them for optimization problems. For example, given a dataset of proteins and their corresponding fluorescence levels, we might want to optimize for a new protein with the highest possible fluorescence. This kind of data-driven optimization (DDO) presents a range of challenges beyond those in standard prediction problems, since we need models that successfully predict the performance of new designs that are better than the best designs seen in the training set. It is not clear theoretically when existing approaches can even perform better than the naive approach that simply selects the best design in the dataset. In this paper, we study how structure can enable sample-efficient data-driven optimization. To formalize the notion of structure, we introduce functional graphical models (FGMs) and show theoretically how they can provide for principled data-driven optimization by decomposing the original high-dimensional optimization problem into smaller sub-problems. This allows us to derive much more practical regret bounds for DDO, and the result implies that DDO with FGMs can achieve nearly optimal designs in situations where naive approaches fail due to insufficient coverage of the offline data. We further present a data-driven optimization algorithm that inferes the FGM structure itself, either over the original input variables or a latent variable representation of the inputs.

Federated Learning is a collaborative training framework that leverages heterogeneous data distributed across a vast number of clients. Since it is practically infeasible to request and process all clients during the aggregation step, partial participation must be supported. In this setting, the communication between the server and clients poses a major bottleneck. To reduce communication loads, there are two main approaches: compression and local steps. Recent work by Mishchenko et al. [2022] introduced the new ProxSkip method, which achieves an accelerated rate using the local steps technique. Follow-up works successfully combined local steps acceleration with partial participation [Grudzie\'n et al., 2023, Condat et al. 2023] and gradient compression [Condat et al. [2022]. In this paper, we finally present a complete method for Federated Learning that incorporates all necessary ingredients: Local Training, Compression, and Partial Participation. We obtain state-of-the-art convergence guarantees in the considered setting. Moreover, we analyze the general sampling framework for partial participation and derive an importance sampling scheme, which leads to even better performance. We experimentally demonstrate the advantages of the proposed method in practice., Comment: 33 pages, 3 algorithms, 1 figure

Effective offline RL methods require properly handling out-of-distribution actions. Implicit Q-learning (IQL) addresses this by training a Q-function using only dataset actions through a modified Bellman backup. However, it is unclear which policy actually attains the values represented by this implicitly trained Q-function. In this paper, we reinterpret IQL as an actor-critic method by generalizing the critic objective and connecting it to a behavior-regularized implicit actor. This generalization shows how the induced actor balances reward maximization and divergence from the behavior policy, with the specific loss choice determining the nature of this tradeoff. Notably, this actor can exhibit complex and multimodal characteristics, suggesting issues with the conditional Gaussian actor fit with advantage weighted regression (AWR) used in prior methods. Instead, we propose using samples from a diffusion parameterized behavior policy and weights computed from the critic to then importance sampled our intended policy. We introduce Implicit Diffusion Q-learning (IDQL), combining our general IQL critic with the policy extraction method. IDQL maintains the ease of implementation of IQL while outperforming prior offline RL methods and demonstrating robustness to hyperparameters. Code is available at https://github.com/philippe-eecs/IDQL., Comment: 9 Pages, 4 Figures, 3 Tables

The necessity for cooperation among intelligent machines has popularised cooperative multi-agent reinforcement learning (MARL) in AI research. However, many research endeavours heavily rely on parameter sharing among agents, which confines them to only homogeneous-agent setting and leads to training instability and lack of convergence guarantees. To achieve effective cooperation in the general heterogeneous-agent setting, we propose Heterogeneous-Agent Reinforcement Learning (HARL) algorithms that resolve the aforementioned issues. Central to our findings are the multi-agent advantage decomposition lemma and the sequential update scheme. Based on these, we develop the provably correct Heterogeneous-Agent Trust Region Learning (HATRL), and derive HATRPO and HAPPO by tractable approximations. Furthermore, we discover a novel framework named Heterogeneous-Agent Mirror Learning (HAML), which strengthens theoretical guarantees for HATRPO and HAPPO and provides a general template for cooperative MARL algorithmic designs. We prove that all algorithms derived from HAML inherently enjoy monotonic improvement of joint return and convergence to Nash Equilibrium. As its natural outcome, HAML validates more novel algorithms in addition to HATRPO and HAPPO, including HAA2C, HADDPG, and HATD3, which generally outperform their existing MA-counterparts. We comprehensively test HARL algorithms on six challenging benchmarks and demonstrate their superior effectiveness and stability for coordinating heterogeneous agents compared to strong baselines such as MAPPO and QMIX.

The celebrated FedAvg algorithm of McMahan et al. (2017) is based on three components: client sampling (CS), data sampling (DS) and local training (LT). While the first two are reasonably well understood, the third component, whose role is to reduce the number of communication rounds needed to train the model, resisted all attempts at a satisfactory theoretical explanation. Malinovsky et al. (2022) identified four distinct generations of LT methods based on the quality of the provided theoretical communication complexity guarantees. Despite a lot of progress in this area, none of the existing works were able to show that it is theoretically better to employ multiple local gradient-type steps (i.e., to engage in LT) than to rely on a single local gradient-type step only in the important heterogeneous data regime. In a recent breakthrough embodied in their ProxSkip method and its theoretical analysis, Mishchenko et al. (2022) showed that LT indeed leads to provable communication acceleration for arbitrarily heterogeneous data, thus jump-starting the $5^{\rm th}$ generation of LT methods. However, while these latest generation LT methods are compatible with DS, none of them support CS. We resolve this open problem in the affirmative. In order to do so, we had to base our algorithmic development on new algorithmic and theoretical foundations., Comment: 38 pages, 2 figures, 5 algorithms, 2 tables

Tremendous progress has been made in reinforcement learning (RL) over the past decade. Most of these advancements came through the continual development of new algorithms, which were designed using a combination of mathematical derivations, intuitions, and experimentation. Such an approach of creating algorithms manually is limited by human understanding and ingenuity. In contrast, meta-learning provides a toolkit for automatic machine learning method optimisation, potentially addressing this flaw. However, black-box approaches which attempt to discover RL algorithms with minimal prior structure have thus far not outperformed existing hand-crafted algorithms. Mirror Learning, which includes RL algorithms, such as PPO, offers a potential middle-ground starting point: while every method in this framework comes with theoretical guarantees, components that differentiate them are subject to design. In this paper we explore the Mirror Learning space by meta-learning a "drift" function. We refer to the immediate result as Learnt Policy Optimisation (LPO). By analysing LPO we gain original insights into policy optimisation which we use to formulate a novel, closed-form RL algorithm, Discovered Policy Optimisation (DPO). Our experiments in Brax environments confirm state-of-the-art performance of LPO and DPO, as well as their transfer to unseen settings., Comment: NeurIPS 2022

The necessity for cooperation among intelligent machines has popularised cooperative multi-agent reinforcement learning (MARL) in the artificial intelligence (AI) research community. However, many research endeavors have been focused on developing practical MARL algorithms whose effectiveness has been studied only empirically, thereby lacking theoretical guarantees. As recent studies have revealed, MARL methods often achieve performance that is unstable in terms of reward monotonicity or suboptimal at convergence. To resolve these issues, in this paper, we introduce a novel framework named Heterogeneous-Agent Mirror Learning (HAML) that provides a general template for MARL algorithmic designs. We prove that algorithms derived from the HAML template satisfy the desired properties of the monotonic improvement of the joint reward and the convergence to Nash equilibrium. We verify the practicality of HAML by proving that the current state-of-the-art cooperative MARL algorithms, HATRPO and HAPPO, are in fact HAML instances. Next, as a natural outcome of our theory, we propose HAML extensions of two well-known RL algorithms, HAA2C (for A2C) and HADDPG (for DDPG), and demonstrate their effectiveness against strong baselines on StarCraftII and Multi-Agent MuJoCo tasks.

Large sequence model (SM) such as GPT series and BERT has displayed outstanding performance and generalization capabilities on vision, language, and recently reinforcement learning tasks. A natural follow-up question is how to abstract multi-agent decision making into an SM problem and benefit from the prosperous development of SMs. In this paper, we introduce a novel architecture named Multi-Agent Transformer (MAT) that effectively casts cooperative multi-agent reinforcement learning (MARL) into SM problems wherein the task is to map agents' observation sequence to agents' optimal action sequence. Our goal is to build the bridge between MARL and SMs so that the modeling power of modern sequence models can be unleashed for MARL. Central to our MAT is an encoder-decoder architecture which leverages the multi-agent advantage decomposition theorem to transform the joint policy search problem into a sequential decision making process; this renders only linear time complexity for multi-agent problems and, most importantly, endows MAT with monotonic performance improvement guarantee. Unlike prior arts such as Decision Transformer fit only pre-collected offline data, MAT is trained by online trials and errors from the environment in an on-policy fashion. To validate MAT, we conduct extensive experiments on StarCraftII, Multi-Agent MuJoCo, Dexterous Hands Manipulation, and Google Research Football benchmarks. Results demonstrate that MAT achieves superior performance and data efficiency compared to strong baselines including MAPPO and HAPPO. Furthermore, we demonstrate that MAT is an excellent few-short learner on unseen tasks regardless of changes in the number of agents. See our project page at https://sites.google.com/view/multi-agent-transformer.

Value function decomposition is becoming a popular rule of thumb for scaling up multi-agent reinforcement learning (MARL) in cooperative games. For such a decomposition rule to hold, the assumption of the individual-global max (IGM) principle must be made; that is, the local maxima on the decomposed value function per every agent must amount to the global maximum on the joint value function. This principle, however, does not have to hold in general. As a result, the applicability of value decomposition algorithms is concealed and their corresponding convergence properties remain unknown. In this paper, we make the first effort to answer these questions. Specifically, we introduce the set of cooperative games in which the value decomposition methods find their validity, which is referred as decomposable games. In decomposable games, we theoretically prove that applying the multi-agent fitted Q-Iteration algorithm (MA-FQI) will lead to an optimal Q-function. In non-decomposable games, the estimated Q-function by MA-FQI can still converge to the optimum under the circumstance that the Q-function needs projecting into the decomposable function space at each iteration. In both settings, we consider value function representations by practical deep neural networks and derive their corresponding convergence rates. To summarize, our results, for the first time, offer theoretical insights for MARL practitioners in terms of when value decomposition algorithms converge and why they perform well., Comment: 37 pages

Modern deep reinforcement learning (RL) algorithms are motivated by either the generalised policy iteration (GPI) or trust-region learning (TRL) frameworks. However, algorithms that strictly respect these theoretical frameworks have proven unscalable. Surprisingly, the only known scalable algorithms violate the GPI/TRL assumptions, e.g. due to required regularisation or other heuristics. The current explanation of their empirical success is essentially "by analogy": they are deemed approximate adaptations of theoretically sound methods. Unfortunately, studies have shown that in practice these algorithms differ greatly from their conceptual ancestors. In contrast, in this paper we introduce a novel theoretical framework, named Mirror Learning, which provides theoretical guarantees to a large class of algorithms, including TRPO and PPO. While the latter two exploit the flexibility of our framework, GPI and TRL fit in merely as pathologically restrictive corner cases thereof. This suggests that the empirical performance of state-of-the-art methods is a direct consequence of their theoretical properties, rather than of aforementioned approximate analogies. Mirror learning sets us free to boldly explore novel, theoretically sound RL algorithms, a thus far uncharted wonderland.

In vivo assessment of cancer and precise location of altered tissues at initial stages of molecular disorders are important diagnostic challenges. Positronium is copiously formed in the free molecular spaces in the patient's body during positron emission tomography (PET). The positronium properties vary according to the size of inter- and intramolecular voids and the concentration of molecules in them such as, e.g., molecular oxygen, O2; therefore, positronium imaging may provide information about disease progression during the initial stages of molecular alterations. Current PET systems do not allow acquisition of positronium images. This study presents a new method that enables positronium imaging by simultaneous registration of annihilation photons and deexcitation photons from pharmaceuticals labeled with radionuclides. The first positronium imaging of a phantom built from cardiac myxoma and adipose tissue is demonstrated. It is anticipated that positronium imaging will substantially enhance the specificity of PET diagnostics., Comment: 10 pages, 5 figures

This tutorial provides a broad introduction to Bayesian data assimilation that will be useful to practitioners, in interpreting algorithms and results, and for theoretical studies developing novel schemes with an understanding of the rich history of geophysical data assimilation and its current directions. The simple case of data assimilation in a 'perfect' model is primarily discussed for pedagogical purposes. Some mathematical results are derived at a high-level in order to illustrate key ideas about different estimators. However, the focus of this work is on the intuition behind these methods, where more formal and detailed treatments of the data assimilation problem can be found in the various references. In surveying a variety of widely used data assimilation schemes, the key message of this tutorial is how the Bayesian analysis provides a consistent framework for the estimation problem and how this allows one to formulate its solution in a variety of ways to exploit the operational challenges in the geosciences.

Developing reinforcement learning algorithms that satisfy safety constraints is becoming increasingly important in real-world applications. In multi-agent reinforcement learning (MARL) settings, policy optimisation with safety awareness is particularly challenging because each individual agent has to not only meet its own safety constraints, but also consider those of others so that their joint behaviour can be guaranteed safe. Despite its importance, the problem of safe multi-agent learning has not been rigorously studied; very few solutions have been proposed, nor a sharable testing environment or benchmarks. To fill these gaps, in this work, we formulate the safe MARL problem as a constrained Markov game and solve it with policy optimisation methods. Our solutions -- Multi-Agent Constrained Policy Optimisation (MACPO) and MAPPO-Lagrangian -- leverage the theories from both constrained policy optimisation and multi-agent trust region learning. Crucially, our methods enjoy theoretical guarantees of both monotonic improvement in reward and satisfaction of safety constraints at every iteration. To examine the effectiveness of our methods, we develop the benchmark suite of Safe Multi-Agent MuJoCo that involves a variety of MARL baselines. Experimental results justify that MACPO/MAPPO-Lagrangian can consistently satisfy safety constraints, meanwhile achieving comparable performance to strong baselines.

Trust region methods rigorously enabled reinforcement learning (RL) agents to learn monotonically improving policies, leading to superior performance on a variety of tasks. Unfortunately, when it comes to multi-agent reinforcement learning (MARL), the property of monotonic improvement may not simply apply; this is because agents, even in cooperative games, could have conflicting directions of policy updates. As a result, achieving a guaranteed improvement on the joint policy where each agent acts individually remains an open challenge. In this paper, we extend the theory of trust region learning to MARL. Central to our findings are the multi-agent advantage decomposition lemma and the sequential policy update scheme. Based on these, we develop Heterogeneous-Agent Trust Region Policy Optimisation (HATPRO) and Heterogeneous-Agent Proximal Policy Optimisation (HAPPO) algorithms. Unlike many existing MARL algorithms, HATRPO/HAPPO do not need agents to share parameters, nor do they need any restrictive assumptions on decomposibility of the joint value function. Most importantly, we justify in theory the monotonic improvement property of HATRPO/HAPPO. We evaluate the proposed methods on a series of Multi-Agent MuJoCo and StarCraftII tasks. Results show that HATRPO and HAPPO significantly outperform strong baselines such as IPPO, MAPPO and MADDPG on all tested tasks, therefore establishing a new state of the art.

Policy gradient (PG) methods are popular reinforcement learning (RL) methods where a baseline is often applied to reduce the variance of gradient estimates. In multi-agent RL (MARL), although the PG theorem can be naturally extended, the effectiveness of multi-agent PG (MAPG) methods degrades as the variance of gradient estimates increases rapidly with the number of agents. In this paper, we offer a rigorous analysis of MAPG methods by, firstly, quantifying the contributions of the number of agents and agents' explorations to the variance of MAPG estimators. Based on this analysis, we derive the optimal baseline (OB) that achieves the minimal variance. In comparison to the OB, we measure the excess variance of existing MARL algorithms such as vanilla MAPG and COMA. Considering using deep neural networks, we also propose a surrogate version of OB, which can be seamlessly plugged into any existing PG methods in MARL. On benchmarks of Multi-Agent MuJoCo and StarCraft challenges, our OB technique effectively stabilises training and improves the performance of multi-agent PPO and COMA algorithms by a significant margin.

Chaos is ubiquitous in physical systems. The associated sensitivity to initial conditions is a significant obstacle in forecasting the weather and other geophysical fluid flows. Data assimilation is the process whereby the uncertainty in initial conditions is reduced by the astute combination of model predictions and real-time data. This chapter reviews recent findings from investigations on the impact of chaos on data assimilation methods: for the Kalman filter and smoother in linear systems, analytic results are derived; for their ensemble-based versions and nonlinear dynamics, numerical results provide insights. The focus is on characterising the asymptotic statistics of the Bayesian posterior in terms of the dynamical instabilities, differentiating between deterministic and stochastic dynamics. We also present two novel results. Firstly, we study the functioning of the ensemble Kalman filter in the context of a chaotic, coupled, atmosphere-ocean model with a quasi-degenerate spectrum of Lyapunov exponents, showing the importance of having sufficient ensemble members to track all of the near-null modes. Secondly, for the fully non-Gaussian method of the particle filter, numerical experiments are conducted to test whether the curse of dimensionality can be mitigated by discarding observations in the directions of little dynamical growth of uncertainty. The results refute this option, most likely because the particles already embody this information on the chaotic system. The results also suggest that it is the rank of the unstable-neutral subspace of the dynamics, and not that of the observation operator, that determines the required number of particles. We finally discuss how knowledge of the random attractor can play a role in the development of future data assimilation schemes for chaotic multiscale systems with large scale separation.

Email responses often contain items-such as a file or a hyperlink to an external document-that are attached to or included inline in the body of the message. Analysis of an enterprise email corpus reveals that 35% of the time when users include these items as part of their response, the attachable item is already present in their inbox or sent folder. A modern email client can proactively retrieve relevant attachable items from the user's past emails based on the context of the current conversation, and recommend them for inclusion, to reduce the time and effort involved in composing the response. In this paper, we propose a weakly supervised learning framework for recommending attachable items to the user. As email search systems are commonly available, we constrain the recommendation task to formulating effective search queries from the context of the conversations. The query is submitted to an existing IR system to retrieve relevant items for attachment. We also present a novel strategy for generating labels from an email corpus---without the need for manual annotations---that can be used to train and evaluate the query formulation model. In addition, we describe a deep convolutional neural network that demonstrates satisfactory performance on this query formulation task when evaluated on the publicly available Avocado dataset and a proprietary dataset of internal emails obtained through an employee participation program., Comment: CIKM2017. Proceedings of the 26th ACM International Conference on Information and Knowledge Management. 2017

It is well understood that dynamic instability is among the primary drivers of forecast uncertainty in chaotic, physical systems. Data assimilation techniques have been designed to exploit this phenomena, reducing the effective dimension of the data assimilation problem to the directions of rapidly growing errors. Recent mathematical work has, moreover, provided formal proofs of the central hypothesis of the Assimilation in the Unstable Subspace methodology of Anna Trevisan and her collaborators: for filters and smoothers in perfect, linear, Gaussian models, the distribution of forecast errors asymptotically conforms to the unstable-neutral subspace. Specifically, the column span of the forecast and posterior error covariances asymptotically align with the span of backward Lyapunov vectors with non-negative exponents. Earlier mathematical studies have focused on perfect models, and this current work now explores the relationship between dynamical instability, the precision of observations and the evolution of forecast error in linear models with additive model error. We prove bounds for the asymptotic uncertainty, explicitly relating the rate of dynamical expansion, model precision and observational accuracy. Formalizing this relationship, we provide a novel, necessary criterion for the boundedness of forecast errors. Furthermore, we numerically explore the relationship between observational design, dynamical instability and filter boundedness. Additionally, we include a detailed introduction to the Multiplicative Ergodic Theorem and to the theory and construction of Lyapunov vectors.

The large size of multiscale, distribution and transmission, power grids hinder fast system-wide estimation and real-time control and optimization of operations. This paper studies graph reduction methods of power grids that are favorable for fast simulations and follow-up applications. While the classical Kron reduction has been successful in reduced order modeling of power grids with traditional, hierarchical design, the selection of reference nodes for the reduced model in a multiscale, distribution and transmission, network becomes ambiguous. In this work we extend the use of the iterative Kron reduction by utilizing the electric grid's graph topology for the selection of reference nodes, consistent with the design features of multiscale networks. Additionally, we propose further reductions by aggregation of coherent subnetworks of triangular meshes, based on the graph topology and network characteristics, in order to preserve currents and build another power-flow equivalent network. Our reductions are achieved through the use of iterative aggregation of sub-graphs that include general tree structures, lines, and triangles. Important features of our reduction algorithms include that: (i) the reductions are, either, equivalent to the Kron reduction, or otherwise produce a power-flow equivalent network; (ii) due to the former mentioned power-flow equivalence, the reduced network can model the dynamic of the swing equations for a lossless, inductive, steady state network; (iii) the algorithms efficiently utilize hash-tables to store the sequential reduction steps.

The characteristics of the model dynamics are critical in the performance of (ensemble) Kalman filters. In particular, as emphasized in the seminal work of Anna Trevisan and co-authors, the error covariance matrix is asymptotically supported by the unstable-neutral subspace only, i.e., it is spanned by the backward Lyapunov vectors with non-negative exponents. This behavior is at the core of algorithms known as Assimilation in the Unstable Subspace, although a formal proof was still missing. This paper provides the analytical proof of the convergence of the Kalman filter covariance matrix onto the unstable-neutral subspace when the dynamics and the observation operator are linear and when the dynamical model is error-free, for any, possibly rank-deficient, initial error covariance matrix. The rate of convergence is provided as well. The derivation is based on an expression that explicitly relates the error covariances at an arbitrary time to the initial ones. It is also shown that if the unstable and neutral directions of the model are sufficiently observed and if the column space of the initial covariance matrix has a non-zero projection onto all of the forward Lyapunov vectors associated with the unstable and neutral directions of the dynamics, the covariance matrix of the Kalman filter collapses onto an asymptotic sequence which is independent of the initial covariances. Numerical results are also shown to illustrate and support the theoretical findings.

This work demonstrates an innovative numerical method for counting and locating eigenvalues with the Evans function. Utilizing the geometric phase in the Hopf bundle, the technique calculates the winding of the Evans function about a contour in the spectral plane, describing the eigenvalues enclosed by the contour for the Hocking-Stewartson pulse of the complex Ginzburg-Landau equation. Locating eigenvalues with the geometric phase in the Hopf bundle was proposed by Way, and proven by Grudzien, Bridges & Jones. Way demonstrated his proposed method for the Hocking-Stewartson pulse, and this manuscript redevelops this example as in the proof of the method, modifying his numerical shooting argument, and introduces new numerical results concerning the phase transition.

We prove that for linear, discrete, time-varying, deterministic system (perfect model) with noisy outputs, the Riccati transformation in the Kalman filter asymptotically bounds the rank of the forecast and the analysis error covariance matrices to be less than or equal to the number of non-negative Lyapunov exponents of the system. Further, the support of these error covariance matrices is shown to be confined to the space spanned by the unstable-neutral backward Lyapunov vectors, providing the theoretical justification for the methodology of the algorithms that perform assimilation only in the unstable-neutral subspace. The equivalent property of the autonomous system is investigated as a special case.

We develop a stability index for the travelling waves of non-linear reaction diffusion equations using the geometric phase induced on the Hopf bundle $S^{2n-1} \subset \mathbb{C}^n$. This can be viewed as an alternative formulation of the winding number calculation of the Evans function, whose zeroes correspond to the eigenvalues of the linearization of reaction diffusion operators about the wave. The stability of a travelling wave can be determined by the existence of eigenvalues of positive real part for the linear operator. Our method of geometric phase for locating and counting eigenvalues is inspired by the numerical results in Way's Dynamics in the Hopf bundle, the geometric phase and implications for dynamical systems. We provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on $\mathbb{C}^2$ and sketch the proof of the method of geometric phase for $\mathbb{C}^n$ and its generalization to boundary-value problems. Implementing the numerical method, modified from Way's work, we conclude with open questions inspired from the results.