Optimal approximation of piecewise smooth functions using deep ReLU neural networks

We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in $L^2$. As a model class, we consider the set $\mathcal{E}^\beta (\mathbb R^d)$ of possibly discontinuous piecewise $C^\beta$ functions $f : [-1/2, 1/2]^d \to \mathbb R$, where the different smooth regions of $f$ are separated by $C^\beta$ hypersurfaces. For dimension $d \geq 2$, regularity $\beta > 0$, and accuracy $\varepsilon > 0$, we construct artificial neural networks with ReLU activation function that approximate functions from $\mathcal{E}^\beta(\mathbb R^d)$ up to $L^2$ error of $\varepsilon$. The constructed networks have a fixed number of layers, depending only on $d$ and $\beta$, and they have $O(\varepsilon^{-2(d-1)/\beta})$ many nonzero weights, which we prove to be optimal. In addition to the optimality in terms of the number of weights, we show that in order to achieve the optimal approximation rate, one needs ReLU networks of a certain depth. Precisely, for piecewise $C^\beta(\mathbb R^d)$ functions, this minimal depth is given---up to a multiplicative constant---by $\beta/d$. Up to a log factor, our constructed networks match this bound. This partly explains the benefits of depth for ReLU networks by showing that deep networks are necessary to achieve efficient approximation of (piecewise) smooth functions. Finally, we analyze approximation in high-dimensional spaces where the function $f$ to be approximated can be factorized into a smooth dimension reducing feature map $\tau$ and classifier function $g$---defined on a low-dimensional feature space---as $f = g \circ \tau$. We show that in this case the approximation rate depends only on the dimension of the feature space and not the input dimension.Comment: Generalized some estimates to $L^p$ norms for $0<p<\infty

Approximation in Lp(μ)L^p(\mu) with deep ReLU neural networks

We discuss the expressive power of neural networks which use the non-smooth ReLU activation function $\varrho(x) = \max\{0,x\}$ by analyzing the approximation theoretic properties of such networks. The existing results mainly fall into two categories: approximation using ReLU networks with a fixed depth, or using ReLU networks whose depth increases with the approximation accuracy. After reviewing these findings, we show that the results concerning networks with fixed depth--- which up to now only consider approximation in $L^p(\lambda)$ for the Lebesgue measure $\lambda$--- can be generalized to approximation in $L^p(\mu)$, for any finite Borel measure $\mu$. In particular, the generalized results apply in the usual setting of statistical learning theory, where one is interested in approximation in $L^2(\mathbb{P})$, with the probability measure $\mathbb{P}$ describing the distribution of the data.Comment: Accepted for presentation at SampTA 201

Der Wille zur Demokratie und die Gewalt der Macht: Ein erster Versuch, die türkischen Protestereignisse zu fassen

Seit Ende Mai protestieren in verschiedenen türkischen Städten immer wieder Bürger gegen die bevormundende Politik der konservativen AKP-Regierung und den unverhältnismäßigen Umgang der Sicherheitskräfte mit Demonstranten. Die Forderungen nach stärkerer partizipativer Politik, bürgerlicher Mitsprache bei öffentlichen Entscheidungen und dem Schutz der Bürgerrechte sind Ausdruck eines Willens zur Demokratie. Aber was ist eigentlich demokratisch an diesen Protesten

Assessment of regional analgesia in clinical practice and research

Assessment of pain and sensory function during regional analgesia contributes to a better understanding of the mechanisms underlying the action of drugs and techniques, and provides information on the effectiveness of regional analgesia in daily practice. Sensory tests only partially mimic clinical pain, mainly because they are artificial and reproduce only a part of the complex experience of pain. Therefore information gained by sensory tests should not be uncritically generalized to clinical pain conditions. Studies using experimental pain models are not in competition with studies performed under clinical conditions, but complement them. In order to mirror clinical pain, experimental models ideally stimulate muscles and viscera, induce peripheral and central sensitization, and evoke temporal and spatial summation. These methods are available, but are underused. Test modalities used in clinical practice have limited validity. In recent years almost no research has been performed to develop better test modalities that are suitable for daily practic

Security for the Industrial IoT: The Case for Information-Centric Networking

Industrial production plants traditionally include sensors for monitoring or documenting processes, and actuators for enabling corrective actions in cases of misconfigurations, failures, or dangerous events. With the advent of the IoT, embedded controllers link these `things' to local networks that often are of low power wireless kind, and are interconnected via gateways to some cloud from the global Internet. Inter-networked sensors and actuators in the industrial IoT form a critical subsystem while frequently operating under harsh conditions. It is currently under debate how to approach inter-networking of critical industrial components in a safe and secure manner. In this paper, we analyze the potentials of ICN for providing a secure and robust networking solution for constrained controllers in industrial safety systems. We showcase hazardous gas sensing in widespread industrial environments, such as refineries, and compare with IP-based approaches such as CoAP and MQTT. Our findings indicate that the content-centric security model, as well as enhanced DoS resistance are important arguments for deploying Information Centric Networking in a safety-critical industrial IoT. Evaluation of the crypto efforts on the RIOT operating system for content security reveal its feasibility for common deployment scenarios.Comment: To be published at IEEE WF-IoT 201

Xenon does not reduce opioid requirement for orthopedic surgery

Purpose: Is to test the hypothesis that 70% xenon has a relevant opioid sparing effect compared to a minimum alveolar concentration (MAC)-equivalent combination of N2O and desflurane. Methods: In this randomized, controlled study of 30 patients undergoing major orthopedic surgery we determined the plasma alfentanil concentration required to suppress response to skin incision in 50% of patients (Cp50) anesthetized with xenon (70%) or a combination of N2O (70%) and desflurane (2%). A response was defined as movement, pressor response > 15 mmHg, heart rate > 90 beats · min−1, autonomic reactions or a combination of these. At skin incision, alfentanil was administered at a randomly selected target plasma concentration thereafter the concentration was increased or decreased according to the patient's response. After skin incision, desflurane was adjusted to maintain the bispectral index below 60 and prevent responsiveness in both groups. Results: The Cp50 (± standard error) of alfentanil was 83 ± 48 ng · mL−1 with xenon and 49 ± 26 ng · mL−1 with N2O/desflurane (P = 0.451). During surgery five xenon and 15 N2O/desflurane patients were given desflurane at 1.0 ± 0.5 volume % and 2.5 ± 0.7 volume %. The total age adjusted MAC was 0.97 ± 0.07 and 0.94 ± 0.07 respectively (P = 0.217). The intraoperative plasma alfentanil concentrations were 95 ± 80 and 93 ± 60 ng · mL−1 respectively (mean ± SD;P = 0.451). Patients given xenon were slightly more bradycardic, whereas blood pressure was similar. Conclusion: Xenon compared to a MAC-equivalent combination of N2O and desflurane does not substantially reduce opioid requirement for orthopedic surgery. A small but clinically irrelevant difference cannot be excluded, howeve

Mean curvature flow in asymptotically flat product spacetimes

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold $M\times\mathbb{R}$, where $M$ is asymptotically flat. If the initial hypersurface $F_0\subset M\times\mathbb{R}$ is uniformly spacelike and asymptotic to $M\times\left\{s\right\}$ for some $s\in\mathbb{R}$ at infinity, we show that a mean curvature flow starting at $F_0$ exists for all times and converges uniformly to $M\times\left\{s\right\}$ as $t\to \infty$.Comment: 23 pages, final versio