Dualities in Convex Algebraic Geometry

Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article compares three notions of duality that are relevant in these contexts: duality of convex bodies, duality of projective varieties, and the Karush-Kuhn-Tucker conditions derived from Lagrange duality. We show that the optimal value of a polynomial program is an algebraic function whose minimal polynomial is expressed by the hypersurface projectively dual to the constraint set. We give an exposition of recent results on the boundary structure of the convex hull of a compact variety, we contrast this to Lasserre's representation as a spectrahedral shadow, and we explore the geometric underpinnings of semidefinite programming duality.Comment: 48 pages, 11 figure

On Approximate Nonlinear Gaussian Message Passing On Factor Graphs

Factor graphs have recently gained increasing attention as a unified framework for representing and constructing algorithms for signal processing, estimation, and control. One capability that does not seem to be well explored within the factor graph tool kit is the ability to handle deterministic nonlinear transformations, such as those occurring in nonlinear filtering and smoothing problems, using tabulated message passing rules. In this contribution, we provide general forward (filtering) and backward (smoothing) approximate Gaussian message passing rules for deterministic nonlinear transformation nodes in arbitrary factor graphs fulfilling a Markov property, based on numerical quadrature procedures for the forward pass and a Rauch-Tung-Striebel-type approximation of the backward pass. These message passing rules can be employed for deriving many algorithms for solving nonlinear problems using factor graphs, as is illustrated by the proposition of a nonlinear modified Bryson-Frazier (MBF) smoother based on the presented message passing rules

A Comprehensive Mathematical Model of Motor Unit Pool Organization, Surface Electromyography, and Force Generation

Neuromuscular physiology is a vibrant research field that has recently seen exciting advances. Previous publications have focused on thorough analyses of particular aspects of neuromuscular physiology, yet an integration of the various novel findings into a single, comprehensive model is missing. In this article, we provide a unified description of a comprehensive mathematical model of surface electromyographic (EMG) measurements and the corresponding force signal in skeletal muscles, both consolidating and extending the results of previous studies regarding various components of the neuromuscular system. The model comprises motor unit (MU) pool organization, recruitment and rate coding, intracellular action potential generation and the resulting EMG measurements, as well as the generated muscular force during voluntary isometric contractions. Mathematically, it consists of a large number of linear PDEs, ODEs, and various stochastic nonlinear relationships, some of which are solved analytically, others numerically. A parameterization of the electrical and mechanical components of the model is proposed that ensures a physiologically meaningful EMG-force relation in the simulated signals, in particular taking the continuous, size-dependent distribution of MU parameters into account. Moreover, a novel nonlinear transformation of the common drive model input is proposed, which ensures that the model force output equals the desired target force. On a physiological level, this corresponds to adjusting the rate coding model to the force generating capabilities of the simulated muscle, while from a control theoretic point of view, this step is equivalent to an exact linearizing transformation of the controlled neuromuscular system. Finally, an alternative analytical formulation of the EMG model is proposed, which renders the physiological meaning of the model more clear and facilitates a mathematical proof that muscle fibers in this model at no point in time represent a net current source or sink. A consistent description of a complete physiological model as presented here, including thorough justification of model component choices, will facilitate the use of these advanced models in future research. Results of a numerical simulation highlight the model's capability to reproduce many physiological effects observed in experimental measurements, and to produce realistic synthetic data that are useful for the validation of signal processing algorithms

Anatomy-guided domain adaptation for 3D in-bed human pose estimation

3D human pose estimation is a key component of clinical monitoring systems. The clinical applicability of deep pose estimation models, however, is limited by their poor generalization under domain shifts along with their need for sufficient labeled training data. As a remedy, we present a novel domain adaptation method, adapting a model from a labeled source to a shifted unlabeled target domain. Our method comprises two complementary adaptation strategies based on prior knowledge about human anatomy. First, we guide the learning process in the target domain by constraining predictions to the space of anatomically plausible poses. To this end, we embed the prior knowledge into an anatomical loss function that penalizes asymmetric limb lengths, implausible bone lengths, and implausible joint angles. Second, we propose to filter pseudo labels for self-training according to their anatomical plausibility and incorporate the concept into the Mean Teacher paradigm. We unify both strategies in a point cloud-based framework applicable to unsupervised and source-free domain adaptation. Evaluation is performed for in-bed pose estimation under two adaptation scenarios, using the public SLP dataset and a newly created dataset. Our method consistently outperforms various state-of-the-art domain adaptation methods, surpasses the baseline model by 31%/66%, and reduces the domain gap by 65%/82%. Source code is available at https://github.com/multimodallearning/da-3dhpe-anatomy.Comment: submitted to Medical Image Analysi

Robust predictive control for respiratory CO2 gas removal in closed-loop mechanical ventilation: an in-silico study

In this study a physiological closed-loop system for arterial CO2 partial pressure control was designed and comprehensively tested using a set of models of the respiratory CO2 gas exchange. The underlying preclinical data were collected from 12 pigs in presence of severe changes in hemodynamic and pulmonary condition. A minimally complex nonlinear state space model of CO2 gas exchange was identified post hoc in different lung conditions. The control variable was measured noninvasively using the endtidal CO2 partial pressure. For the simulation study the output signal of the controller was defined as the alveolar minute volume set value of an underlying adaptive lung protective ventilation mode. A linearisation of the two-compartment CO2 gas exchange model was used for the design of a model predictive controller (MPC). It was augmented by a tube based controller suppressing prediction errors due to model uncertainties. The controller was subject to comparative testing in interaction with each of the CO2 gas exchange models previously identified on the preclinical study data. The performance was evaluated for the system response towards the following five tests in comparison to a PID controller: recruitment maneuver, PEEP titration maneuver, stepwise change in the CO2 production, breath-hold maneuver and a step in the reference signal. A root mean square error of 2.69 mmHg between arterial CO2 partial pressure and the reference signal was achieved throughout the trial. The reference-variable response of the model predictive controller was superior regarding overshoot and settling time

Responsible and Regulatory Conform Machine Learning for Medicine: A Survey of Challenges and Solutions

Machine learning is expected to fuel significant improvements in medical care. To ensure that fundamental principles such as beneficence, respect for human autonomy, prevention of harm, justice, privacy, and transparency are respected, medical machine learning systems must be developed responsibly. Many high-level declarations of ethical principles have been put forth for this purpose, but there is a severe lack of technical guidelines explicating the practical consequences for medical machine learning. Similarly, there is currently considerable uncertainty regarding the exact regulatory requirements placed upon medical machine learning systems. This survey provides an overview of the technical and procedural challenges involved in creating medical machine learning systems responsibly and in conformity with existing regulations, as well as possible solutions to address these challenges. First, a brief review of existing regulations affecting medical machine learning is provided, showing that properties such as safety, robustness, reliability, privacy, security, transparency, explainability, and nondiscrimination are all demanded already by existing law and regulations - albeit, in many cases, to an uncertain degree. Next, the key technical obstacles to achieving these desirable properties are discussed, as well as important techniques to overcome these obstacles in the medical context. We notice that distribution shift, spurious correlations, model underspecification, uncertainty quantification, and data scarcity represent severe challenges in the medical context. Promising solution approaches include the use of large and representative datasets and federated learning as a means to that end, the careful exploitation of domain knowledge, the use of inherently transparent models, comprehensive out-of-distribution model testing and verification, as well as algorithmic impact assessments

Improving Systems Dynamics by Means of Advanced Signal Processing -Mathematical, Laboratorial and Clinical Evaluation of Propofol Monitoring in Breathing Gas

Abstract: Electrochemical sensors are used in various gas measurement applications and are available for different gases. Depending on the application, the sensor might need to be installed far away from the actual measurement site, requiring the use of long sampling lines. Examples are portable gas measurement devices in which remote locations like tanks and chemical reactors need to be monitored. But also medical applications, where the sensors cannot be positioned in close vicinity to the patient, are common like, e.g., the side-stream measurement of breathing gas. Due to the characteristics of electrochemical sensors and to the adsorption and desorption behavior of sampling lines for different gases, the electrical sensor signal may indicate long response times. In this paper, we propose an on-line signal processing algorithm which is capable to significantly improve the performance. After characterizing the dynamic behavior of the sensor system, a properly designed deconvolution filter is used to reduce response time and signal noise. Within this article, we also provide an http://www.sensorsportal.com/HTML/DIGEST/P_2747.htm Sensors & Transducers, Vol. 193, Issue 10, October 2015, pp. 145-153 146 example of this algorithm for a novel electrochemical sensor for the measurement of the anesthetic agent propofol in exhaled air. For this application, the acceleration is prerequisite for the measurement chain to be of practical use in a clinical setting. Our goals, to establish measurement dynamics to record the physiologic parameter and to reduce non-physiological disturbances, were achieved with additional reserves. This article is based o