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The concept of signatures and expected signatures is vital in data science, especially for sequential data analysis. The signature transform, a Cartan type development, translates paths into high-dimensional feature vectors, capturing their intrinsic characteristics. Under natural conditions, the expectation of the signature determines the law of the signature, providing a statistical summary of the data distribution. This property facilitates robust modeling and inference in machine learning and stochastic processes. Building on previous work by the present authors [Unified signature cumulants and generalized Magnus expansions, FoM Sigma '22] we here revisit the actual computation of expected signatures, in a general semimartingale setting. Several new formulae are given. A log-transform of (expected) signatures leads to log-signatures (signature cumulants), offering a significant reduction in complexity., Comment: arXiv admin note: text overlap with arXiv:2102.03345

Clinicians spend a significant amount of time reviewing medical images and transcribing their findings regarding patient diagnosis, referral and treatment in text form. Vision-language models (VLMs), which automatically interpret images and summarize their findings as text, have enormous potential to alleviate clinical workloads and increase patient access to high-quality medical care. While foundational models have stirred considerable interest in the medical community, it is unclear whether their general capabilities translate to real-world clinical utility. In this work, we show that foundation VLMs markedly underperform compared to practicing ophthalmologists on specialist tasks crucial to the care of patients with age-related macular degeneration (AMD). To address this, we initially identified the essential capabilities required for image-based clinical decision-making, and then developed a curriculum to selectively train VLMs in these skills. The resulting model, RetinaVLM, can be instructed to write reports that significantly outperform those written by leading foundation medical VLMs in disease staging (F1 score of 0.63 vs. 0.11) and patient referral (0.67 vs. 0.39), and approaches the diagnostic performance of junior ophthalmologists (who achieve 0.77 and 0.78 on the respective tasks). Furthermore, in a reader study involving two senior ophthalmologists with up to 32 years of experience, RetinaVLM's reports were found to be similarly correct (78.6% vs. 82.1%) and complete (both 78.6%) as reports written by junior ophthalmologists with up to 10 years of experience. These results demonstrate that our curriculum-based approach provides a blueprint for specializing generalist foundation medical VLMs to handle real-world clinical tasks., Comment: Submitted to Nature Medicine

We consider both $N$-player and mean-field games of optimal portfolio liquidation in which the players are not allowed to change the direction of trading. Players with an initially short position of stocks are only allowed to buy while players with an initially long position are only allowed to sell the stock. Under suitable conditions on the model parameters we show that the games are equivalent to games of timing where the players need to determine the optimal times of market entry and exit. We identify the equilibrium entry and exit times and prove that equilibrium mean-trading rates can be characterized in terms of the solutions to a highly non-linear higher-order integral equation with endogenous terminal condition. We prove the existence of a unique solution to the integral equation from which we obtain the existence of a unique equilibrium both in the mean-field and the $N$-player game.

Analyzing temporal developments is crucial for the accurate prognosis of many medical conditions. Temporal changes that occur over short time scales are key to assessing the health of physiological functions, such as the cardiac cycle. Moreover, tracking longer term developments that occur over months or years in evolving processes, such as age-related macular degeneration (AMD), is essential for accurate prognosis. Despite the importance of both short and long term analysis to clinical decision making, they remain understudied in medical deep learning. State of the art methods for spatiotemporal representation learning, developed for short natural videos, prioritize the detection of temporal constants rather than temporal developments. Moreover, they do not account for varying time intervals between acquisitions, which are essential for contextualizing observed changes. To address these issues, we propose two approaches. First, we combine clip-level contrastive learning with a novel temporal embedding to adapt to irregular time series. Second, we propose masking and predicting latent frame representations of the temporal sequence. Our two approaches outperform all prior methods on temporally-dependent tasks including cardiac output estimation and three prognostic AMD tasks. Overall, this enables the automated analysis of temporal patterns which are typically overlooked in applications of deep learning to medicine.

Uncertainty estimation, which provides a means of building explainable neural networks for medical imaging applications, have mostly been studied for single deep learning models that focus on a specific task. In this paper, we propose a method to propagate uncertainty through cascades of deep learning models in medical imaging pipelines. This allows us to aggregate the uncertainty in later stages of the pipeline and to obtain a joint uncertainty measure for the predictions of later models. Additionally, we can separately report contributions of the aleatoric, data-based, uncertainty of every component in the pipeline. We demonstrate the utility of our method on a realistic imaging pipeline that reconstructs undersampled brain and knee magnetic resonance (MR) images and subsequently predicts quantitative information from the images, such as the brain volume, or knee side or patient's sex. We quantitatively show that the propagated uncertainty is correlated with input uncertainty and compare the proportions of contributions of pipeline stages to the joint uncertainty measure.

The electrocardiogram (ECG) is a widely available diagnostic tool that allows for a cost-effective and fast assessment of the cardiovascular health. However, more detailed examination with expensive cardiac magnetic resonance (CMR) imaging is often preferred for the diagnosis of cardiovascular diseases. While providing detailed visualization of the cardiac anatomy, CMR imaging is not widely available due to long scan times and high costs. To address this issue, we propose the first self-supervised contrastive approach that transfers domain-specific information from CMR images to ECG embeddings. Our approach combines multimodal contrastive learning with masked data modeling to enable holistic cardiac screening solely from ECG data. In extensive experiments using data from 40,044 UK Biobank subjects, we demonstrate the utility and generalizability of our method. We predict the subject-specific risk of various cardiovascular diseases and determine distinct cardiac phenotypes solely from ECG data. In a qualitative analysis, we demonstrate that our learned ECG embeddings incorporate information from CMR image regions of interest. We make our entire pipeline publicly available, including the source code and pre-trained model weights.

Medical datasets and especially biobanks, often contain extensive tabular data with rich clinical information in addition to images. In practice, clinicians typically have less data, both in terms of diversity and scale, but still wish to deploy deep learning solutions. Combined with increasing medical dataset sizes and expensive annotation costs, the necessity for unsupervised methods that can pretrain multimodally and predict unimodally has risen. To address these needs, we propose the first self-supervised contrastive learning framework that takes advantage of images and tabular data to train unimodal encoders. Our solution combines SimCLR and SCARF, two leading contrastive learning strategies, and is simple and effective. In our experiments, we demonstrate the strength of our framework by predicting risks of myocardial infarction and coronary artery disease (CAD) using cardiac MR images and 120 clinical features from 40,000 UK Biobank subjects. Furthermore, we show the generalizability of our approach to natural images using the DVM car advertisement dataset. We take advantage of the high interpretability of tabular data and through attribution and ablation experiments find that morphometric tabular features, describing size and shape, have outsized importance during the contrastive learning process and improve the quality of the learned embeddings. Finally, we introduce a novel form of supervised contrastive learning, label as a feature (LaaF), by appending the ground truth label as a tabular feature during multimodal pretraining, outperforming all supervised contrastive baselines., Comment: Accepted in CVPR 2023

We consider a novel class of portfolio liquidation games with market drop-out ("absorption"). More precisely, we consider mean-field and finite player liquidation games where a player drops out of the market when her position hits zero. In particular round-trips are not admissible. This can be viewed as a no statistical arbitrage condition. In a model with only sellers we prove that the absorption condition is equivalent to a short selling constraint. We prove that equilibria (both in the mean-field and the finite player game) are given as solutions to a non-linear higher-order integral equation with endogenous terminal condition. We prove the existence of a unique solution to the integral equation from which we obtain the existence of a unique equilibrium in the MFG and the existence of a unique equilibrium in the $N$-player game. We establish the convergence of the equilibria in the finite player games to the obtained mean-field equilibrium and illustrate the impact of the drop-out constraint on equilibrium trading rates.

We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process $X$. We consider classic and randomized stopping times represented by linear and non-linear functionals of the rough path signature $\mathbb{X}^{<\infty}$ associated to $X$, and prove that maximizing over these classes of signature stopping times, in fact, solves the original optimal 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 $\mathbb{E}\left[ \mathbb{X}^{\le N}_{0,T} \right]$. By applying a deep neural network approach to approximate the non-linear signature functionals, we can efficiently solve the optimal stopping problem numerically. The only assumption on the process $X$ is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion, which fail to be either semi-martingales or Markov processes, and can be used, in particular, for American-type option pricing in fractional models, e.g. on financial or electricity markets., Comment: 39 pages, 1 figure

The signature of a path can be described as its full non-commutative exponential. Following T. Lyons we regard its expectation, the expected signature, as path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant. We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants, and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions; with motivation ranging from financial mathematics to statistical physics. From an affine process perspective, the functional relation may be interpreted as infinite-dimensional, non-commutative ("Hausdorff") variation of Riccati's equation. Many examples are given., Comment: 42 pages, 2 figures

Least squares Monte Carlo methods are a popular numerical approximation method for solving stochastic control problems. Based on dynamic programming, their key feature is the approximation of the conditional expectation of future rewards by linear least squares regression. Hence, the choice of basis functions is crucial for the accuracy of the method. Earlier work by some of us [Belomestny, Schoenmakers, Spokoiny, Zharkynbay. Commun.~Math.~Sci., 18(1):109-121, 2020](arXiv:1808.02341) proposes to reinforce the basis functions in the case of optimal stopping problems by already computed value functions for later times, thereby considerably improving the accuracy with limited additional computational cost. We extend the reinforced regression method to a general class of stochastic control problems, while considerably improving the method's efficiency, as demonstrated by substantial numerical examples as well as theoretical analysis.

We consider a family of fractional Brownian fields $\{B^{H}\}_{H\in (0,1)}$ on $\mathbb{R}^{d}$, where $H$ denotes their Hurst parameter. We first define a rich class of normalizing kernels $\psi$ such that the covariance of $$ X^{H}(x) = \Gamma(H)^{\frac{1}{2}} \left( B^{H}(x) - \int_{\mathbb{R}^{d}} B^{H}(u) \psi(u, x)du\right), $$ converges to the covariance of a log-correlated Gaussian field when $H \downarrow 0$. We then use Berestycki's ``good points'' approach in order to derive the limiting measure of the so-called multiplicative chaos of the fractional Brownian field $$ M^{H}_\gamma(dx) = e^{\gamma X^{H}(x) - \frac{\gamma^{2}}{2} E[X^{H}(x)^{2}] }dx, $$ as $H\downarrow 0$ for all $\gamma \in (0,\gamma^{*}(d)]$, where $\gamma^{*}(d)>\sqrt{\frac{7}{4}d}$. As a corollary we establish the $L^{2}$ convergence of $M^{H}_\gamma$ over the sets of ``good points'', where the field $X^H$ has a typical behaviour. As a by-product of the convergence result, we prove that for log-normal rough volatility models with small Hurst parameter, the volatility process is supported on the sets of ``good points'' with probability close to $1$. Moreover, on these sets the volatility converges in $L^2$ to the volatility of multifractal random walks., Comment: 48 pages, 1 figure

In this paper we study randomized optimal stopping problems and consider corresponding forward and backward Monte Carlo based optimisation algorithms. In particular we prove the convergence of the proposed algorithms and derive the corresponding convergence rates.

Increasing the mobility of service providers, including professionals, via mutual recognition (of regulatory systems) agreements (MRAs) has become a significant issue worldwide. Despite increasing interest in MRAs, it may be argued that MRAs are but one of a larger range of major developments that have fueled current interest in occupational competence descriptors and professional practice. Other important developments are as follows: rapid and accelerating change; public dissatisfaction with the professions; the limited understanding of the nature of professional practice; the emergence of a knowledge society; globalization leading to a corporate model of service provision; and moves to alternative and flexible modes of service delivery. The claim that mutual recognition is but one issue among these major developments may be illustrated by the examples of the following accreditation schemes: (1) the Law Society of New South Wales Specialist Accreditation Scheme; (2) the regulatory model recently developed in the United States by the National Council of State Boards of Nursing; (3) the competence-based assessment strategy developed by the Dietitians Association of Australia; and (4) various practices being used to assess competence in 17 Australian professions. These examples demonstrate that the role of professional competence standards in MRAs and in professionals' mobility is far from clear. (Contains 25 references.) (MN)

The hypothesis that making judgments is the basis for workplace learning was examined by studying work in a range of occupations with a view to identifying and analyzing the main judgments involved in individuals' work performance. To test the hypothesis, six practicing professionals (three men and three women) were interviewed. Two of interviewees were public servants with division-level responsibilities. The occupations of the other four were as follows: psychiatrist; private school principal; nurse turned ambulance officer; and corporate consultant. During the interviews, the professionals were asked to reflect on the judgments they make in critical work situations. The interviews were recorded, transcribed, and analyzed in the context of theories regarding workplace learning and logic of action. It was concluded that growth in capacity to make judgments is evident in the following abilities and actions: (1) an ability to separate the stage of presentation of the initial need to make a judgment from the stage of the contiguous actualization of that judgment; (2) an ability to "read" the conative, emotive, and ethical considerations in the light of that separability; and (3) the decentering of the practitioner's sense of identity at one of these stages, but not in both of them. (Contains 34 references.) (MN)

A project investigated the integration of key competencies into workplace training in Australia. Part 1 of the project researched the work of personnel engaged in on-the-job training (OJT) curriculum development in five industries to find out how key competencies were being incorporated into OJT curricula and to suggest models to guide this process. Part 2 used action research at 22 sites to test ways to incorporate key competencies into workplace training and develop models for incorporating them into OJT delivery, assessment, and reporting. Findings indicated a typical work activity is composed of both industry-specific work skills and several generic skills. In a holistic approach, the development of key competencies would be a process of the trainee acquiring more integrated combinations of competencies while moving from school into the workplace. Key competencies were not all equally applicable or relevant to all jobs. They needed to be made explicit in training activities. The variables in a training culture and context had a significant effect on the nature of teaching key competencies. Recommended approaches to training in key competencies stressed the importance of integrated, context-specific training for developing skills and expertise and used key competencies to link on- and off-the-job training. The project identified a need for research to further understanding of key competencies in workplace training. (Appendixes include 60 references, project information papers, exemplars from curriculum case studies, and 22 case studies.) (YLB)

The state of research and development (R&D) in technical and further education (TAFE) in Australia was reviewed. Special attention was paid to the following: current R&D expenditure and output; shortcomings in Australia's current research; organization and funding of research in TAFE; and dissemination of R&D. It was concluded that a stronger research base would benefit TAFE by providing a better information base, critical analysis and accountability, improved cost-effectiveness, varied perspectives, better understanding of education and training processes, and a higher profile for vocational education and training (VET). Research on VET was receiving only about half the funding received for research in other categories of education, and the existing research in VET was fragmented, not focused on general issues, and not being fully used. Seventeen principles related to priorities and funding, R&D organization, and R&D utilization were proposed as underlying principles of a new national research strategy designed to increase the likelihood that research in VET will be focused and easily implemented. It was recommended that the highest priority be given to research projects examining the economic benefits of VET, the assumptions underlying competency-based training, the nature of workplace learning, management in TAFE systems, community perceptions of VET, and approaches to the dissemination and use of research in VET. (Sixteen tables/figures are included. The bibliography contains 152 references.) (MN)