Showing papers on "Upper and lower bounds published in 2019"

An Algorithm for the Traveling Salesman Problem

Abstract: A “branch and bound” algorithm is presented for solving the traveling salesman problem. The set of all tours feasible solutions is broken up into increasingly small subsets by a procedure called branching. For each subset a lower bound on the length of the tours therein is calculated. Eventually, a subset is found that contains a single tour whose length is less than or equal to some lower bound for every tour. The motivation of the branching and the calculation of the lower bounds are based on ideas frequently used in solving assignment problems. Computationally, the algorithm extends the size of problem that can reasonably be solved without using methods special to the particular problem.

Large Intelligent Surface-Assisted Wireless Communication Exploiting Statistical CSI

Abstract: Large intelligent surface (LIS)-assisted wireless communications have drawn attention worldwide. With the use of low-cost LIS on building walls, signals can be reflected by the LIS and sent out along desired directions by controlling its phases, thereby providing supplementary links for wireless communication systems. In this paper, we evaluate the performance of an LIS-assisted large-scale antenna system by formulating a tight upper bound of the ergodic spectral efficiency and investigate the effect of the phase shifts on the ergodic spectral efficiency in different propagation scenarios. In particular, we propose an optimal phase shift design based on the upper bound of the ergodic spectral efficiency and statistical channel state information. Furthermore, we derive the requirement on the quantization bits of the LIS to promise an acceptable spectral efficiency degradation. Numerical results show that using the proposed phase shift design can achieve the maximum ergodic spectral efficiency, and a 2-bit quantizer is sufficient to ensure spectral efficiency degradation of no more than 1 bit/s/Hz.

Theoretically Principled Trade-off between Robustness and Accuracy

Abstract: We identify a trade-off between robustness and accuracy that serves as a guiding principle in the design of defenses against adversarial examples. Although this problem has been widely studied empirically, much remains unknown concerning the theory underlying this trade-off. In this work, we decompose the prediction error for adversarial examples (robust error) as the sum of the natural (classification) error and boundary error, and provide a differentiable upper bound using the theory of classification-calibrated loss, which is shown to be the tightest possible upper bound uniform over all probability distributions and measurable predictors. Inspired by our theoretical analysis, we also design a new defense method, TRADES, to trade adversarial robustness off against accuracy. Our proposed algorithm performs well experimentally in real-world datasets. The methodology is the foundation of our entry to the NeurIPS 2018 Adversarial Vision Challenge in which we won the 1st place out of ~2,000 submissions, surpassing the runner-up approach by $11.41\%$ in terms of mean $\ell_2$ perturbation distance.

Theoretically Principled Trade-off between Robustness and Accuracy

Abstract: We identify a trade-off between robustness and accuracy that serves as a guiding principle in the design of defenses against adversarial examples. Although this problem has been widely studied empirically, much remains unknown concerning the theory underlying this trade-off. In this work, we decompose the prediction error for adversarial examples (robust error) as the sum of the natural (classification) error and boundary error, and provide a differentiable upper bound using the theory of classification-calibrated loss, which is shown to be the tightest possible upper bound uniform over all probability distributions and measurable predictors. Inspired by our theoretical analysis, we also design a new defense method, TRADES, to trade adversarial robustness off against accuracy. Our proposed algorithm performs well experimentally in real-world datasets. The methodology is the foundation of our entry to the NeurIPS 2018 Adversarial Vision Challenge in which we won the 1st place out of ~2,000 submissions, surpassing the runner-up approach by $11.41\%$ in terms of mean $\ell_2$ perturbation distance.

Nearly-tight VC-dimension and Pseudodimension Bounds for Piecewise Linear Neural Networks

Abstract: We prove new upper and lower bounds on the VC-dimension of deep neural networks with the ReLU activation function. These bounds are tight for almost the entire range of parameters. Letting $W$ be the number of weights and $L$ be the number of layers, we prove that the VC-dimension is $O(W L \log(W))$, and provide examples with VC-dimension $\Omega( W L \log(W/L) )$. This improves both the previously known upper bounds and lower bounds. In terms of the number $U$ of non-linear units, we prove a tight bound $\Theta(W U)$ on the VC-dimension. All of these bounds generalize to arbitrary piecewise linear activation functions, and also hold for the pseudodimensions of these function classes. Combined with previous results, this gives an intriguing range of dependencies of the VC-dimension on depth for networks with different non-linearities: there is no dependence for piecewise-constant, linear dependence for piecewise-linear, and no more than quadratic dependence for general piecewise-polynomial.

A Threshold-Parameter-Dependent Approach to Designing Distributed Event-Triggered $H_{\infty}$ Consensus Filters Over Sensor Networks

Abstract: This paper is concerned with distributed event-triggered $\boldsymbol {H_{\infty }}$ consensus filtering for a discrete-time linear system over a sensor network. Different from some existing event-triggered communication schemes (ETCSs), a new distributed ETCS is first developed to reduce the communication frequency of neighboring sensors, where the threshold parameter in an event triggering condition is time-varying with attainable upper and lower bounds. Then a threshold-parameter-dependent approach is proposed to derive criteria for designing the desired $\boldsymbol {H_{\infty }}$ consensus filters and the ETCS such that the resultant filtering error system is asymptotically stable with the prescribed $\boldsymbol {H_{\infty }}$ performance while maintaining satisfactory resource efficiency. Furthermore, a polytope-like transformation with regard to time-varying threshold parameters is performed and a recursive algorithm is presented to determine the threshold-parameter-dependent filter matrix sequences and event triggering weighting matrix sequence. Two illustrative examples are employed to show the effectiveness of the developed approach.

The string swampland constraints require multi-field inflation

Abstract: An important unsolved problem that affects practically all attempts to connect string theory to cosmology and phenomenology is how to distinguish effective field theories belonging to the string landscape from those that are not consistent with a quantum theory of gravity at high energies (the "string swampland"). It was recently proposed that potentials of the string landscape must satisfy at least two conditions, the "swampland criteria", that severely restrict the types of cosmological dynamics they can sustain. The first criterion states that the (multi-field) effective field theory description is only valid over a field displacement Δ ≤ Δ ~ (1) (in units where the Planck mass is 1), measured as a distance in the target space geometry. A second, more recent, criterion asserts that, whenever the potential V is positive, its slope must be bounded from below, and suggests |∇ V| / V ≥ c ~ (1). A recent analysis concluded that these two conditions taken together practically rule out slow-roll models of inflation. In this note we show that the two conditions rule out inflationary backgrounds that follow geodesic trajectories in field space, but not those following curved, non-geodesic, trajectories (which are parametrized by a non-vanishing bending rate Ω of the multi-field trajectory). We derive a universal lower bound on Ω (relative to the Hubble parameter H) as a function of Δ, c and the number of efolds Ne, assumed to be at least of order 60. If later studies confirm c and Δ to be strictly (1), the bound implies strong turns with Ω / H ≥ 3 Ne ~ 180. Slow-roll inflation in the landscape is not ruled out, but it is strongly multi-field.

Adaptive Neural State-Feedback Tracking Control of Stochastic Nonlinear Switched Systems: An Average Dwell-Time Method

Abstract: In this paper, the problem of adaptive neural state-feedback tracking control is considered for a class of stochastic nonstrict-feedback nonlinear switched systems with completely unknown nonlinearities. In the design procedure, the universal approximation capability of radial basis function neural networks is used for identifying the unknown compounded nonlinear functions, and a variable separation technique is employed to overcome the design difficulty caused by the nonstrict-feedback structure. The most outstanding novelty of this paper is that individual Lyapunov function of each subsystem is constructed by flexibly adopting the upper and lower bounds of the control gain functions of each subsystem. Furthermore, by combining the average dwell-time scheme and the adaptive backstepping design, a valid adaptive neural state-feedback controller design algorithm is presented such that all the signals of the switched closed-loop system are in probability semiglobally uniformly ultimately bounded, and the tracking error eventually converges to a small neighborhood of the origin in probability. Finally, the availability of the developed control scheme is verified by two simulation examples.

Lower Bounds for Non-Convex Stochastic Optimization.

Abstract: We lower bound the complexity of finding $\epsilon$-stationary points (with gradient norm at most $\epsilon$) using stochastic first-order methods. In a well-studied model where algorithms access smooth, potentially non-convex functions through queries to an unbiased stochastic gradient oracle with bounded variance, we prove that (in the worst case) any algorithm requires at least $\epsilon^{-4}$ queries to find an $\epsilon$ stationary point. The lower bound is tight, and establishes that stochastic gradient descent is minimax optimal in this model. In a more restrictive model where the noisy gradient estimates satisfy a mean-squared smoothness property, we prove a lower bound of $\epsilon^{-3}$ queries, establishing the optimality of recently proposed variance reduction techniques.

Quantum Noise Theory of Exceptional Point Amplifying Sensors.

Abstract: Open quantum systems can have exceptional points (EPs), degeneracies where both eigenvalues and eigenvectors coalesce. Recently, it has been proposed and demonstrated that EPs can enhance the performance of sensors in terms of amplification of a detected signal. However, typically amplification of signals also increases the system noise, and it has not yet been shown that an EP sensor can have improved signal-to-noise performance. We develop a quantum noise theory to calculate the signal-to-noise performance of an EP sensor. We use the quantum Fisher information to extract a lower bound for the signal-to-noise ratio (SNR) and show that parametrically improved SNR is possible. Finally, we construct a specific experimental protocol for sensing using an EP amplifier near its lasing threshold and heterodyne signal detection that achieves the optimal scaling predicted by the Fisher bound. Our results can be generalized to higher order EPs for any bosonic non-Hermitian system with linear interactions.

Network-Based Quantized Control for Fuzzy Singularly Perturbed Semi-Markov Jump Systems and its Application

Abstract: This paper deals with the quantized control problem for nonlinear semi-Markov jump systems subject to singular perturbation under a network-based framework. The nonlinearity of the system is well solved by applying Takagi–Sugeno (T-S) fuzzy theory. The semi-Markov jump process with the memory matrix of transition probability is introduced, for which the obtained results are more reasonable and less limiting. In addition, the packet dropouts governed by a Bernoulli variable and the signal quantization associated with a logarithmic quantizer are deeply studied. The major goal is to devise a fuzzy controller, which not only assures the mean-square $\bar { \sigma }$ -error stability of the corresponding system but also allows a higher upper bound of the singularly perturbed parameter. Sufficient conditions are developed to make sure that the applicable controller could be found. The further examination to demonstrate the feasibility of the presented method is given by designing a controller of a series DC motor model.

Sample-Optimal Parametric Q-Learning Using Linearly Additive Features.

Abstract: Consider a Markov decision process (MDP) that admits a set of state-action features, which can linearly express the process's probabilistic transition model. We propose a parametric Q-learning algorithm that finds an approximate-optimal policy using a sample size proportional to the feature dimension $K$ and invariant with respect to the size of the state space. To further improve its sample efficiency, we exploit the monotonicity property and intrinsic noise structure of the Bellman operator, provided the existence of anchor state-actions that imply implicit non-negativity in the feature space. We augment the algorithm using techniques of variance reduction, monotonicity preservation, and confidence bounds. It is proved to find a policy which is $\epsilon$-optimal from any initial state with high probability using $\widetilde{O}(K/\epsilon^2(1-\gamma)^3)$ sample transitions for arbitrarily large-scale MDP with a discount factor $\gamma\in(0,1)$. A matching information-theoretical lower bound is proved, confirming the sample optimality of the proposed method with respect to all parameters (up to polylog factors).

Multimessenger parameter estimation of GW170817

Abstract: We combine gravitational wave (GW) and electromagnetic (EM) data to perform a Bayesian parameter estimation of the binary neutron star (NS) merger GW170817. The EM likelihood is constructed from a fit to a large number of numerical relativity simulations which we combine with a lower bound on the mass of the remnant’s accretion disk inferred from the modeling of the EM light curve. In comparison with previous works, our analysis yields a more precise determination of the tidal deformability of the binary, for which the EM data provide a lower bound, and of the mass ratio of the binary, with the EM data favoring a smaller mass asymmetry. The 90% credible interval for the areal radius of a $ 1.4 M_\odot$ NS is found to be $ 12.2^{+1.0}_{-0.8} \pm 0.2$ km (statistical and systematic uncertainties).

Mapping Quantum Circuits to IBM QX Architectures Using the Minimal Number of SWAP and H Operations

Abstract: The recent progress in the physical realization of quantum computers (the first publicly available ones---IBM's QX architectures---have been launched in 2017) has motivated research on automatic methods that aid users in running quantum circuits on them. Here, certain physical constraints given by the architectures which restrict the allowed interactions of the involved qubits have to be satisfied. Thus far, this has been addressed by inserting SWAP and H operations. However, it remains unknown whether existing methods add a minimum number of SWAP and H operations or, if not, how far they are away from that minimum---an NP-complete problem. In this work, weaddress this by formulating the mapping task as a symbolic optimization problem that is solved using reasoning engines like Boolean satisfiability solvers. By this, we do not only provide a method that maps quantum circuits to IBM's QX architectures with a minimal number of SWAP and H operations, but also show by experimental evaluation that the number of operations added by IBM's heuristic solution exceeds the lower bound by more than 100% on average. An implementation of the proposed methodology is publicly available at http://iic.jku.at/eda/research/ibm_qx_mapping.

Couplings and quantitative contraction rates for Langevin dynamics

Abstract: We introduce a new probabilistic approach to quantify convergence to equilibrium for (kinetic) Langevin processes. In contrast to previous analytic approaches that focus on the associated kinetic Fokker-Planck equation, our approach is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. It yields contractions in a particular Wasserstein distance, and it provides rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. In particular, we are able to recover kinetic behavior in terms of explicit lower bounds for the contraction rate. For example, for a rescaled double-well potential with local minima at distance a, we obtain a lower bound for the contraction rate of order Ω(a−1) provided the friction coefficient is of order Θ(a−1)

Sharp phase transition for the random-cluster and Potts models via decision trees

Abstract: We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that 1. For the Potts model on transitive graphs, correlations decay exponentially fast for $\beta<\beta_c$. 2. For the random-cluster model with cluster weight $q\geq1$ on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the supercritical regime. 3. For the random-cluster models with cluster weight $q\geq1$ on planar quasi-transitive graphs $\mathbb{G}$, $$\frac{p_c(\mathbb{G})p_c(\mathbb{G}^*)}{(1-p_c(\mathbb{G}))(1-p_c(\mathbb{G}^*))}~=~q.$$ As a special case, we obtain the value of the critical point for the square, triangular and hexagonal lattices (this provides a short proof of the result of Beffara and Duminil-Copin [Probability Theory and Related Fields, 153(3-4):511--542, 2012]). These results have many applications for the understanding of the subcritical (respectively disordered) phase of all these models. The techniques developed in this paper have potential to be extended to a wide class of models including the Ashkin-Teller model, continuum percolation models such as Voronoi percolation and Boolean percolation, super-level sets of massive Gaussian Free Field, and random-cluster and Potts model with infinite range interactions.

$H_{\infty}$ State Estimation for Discrete-Time Nonlinear Singularly Perturbed Complex Networks Under the Round-Robin Protocol

Abstract: This paper investigates the $H_{\infty }$ state estimation problem for a class of discrete-time nonlinear singularly perturbed complex networks (SPCNs) under the Round-Robin (RR) protocol. A discrete-time nonlinear SPCN model is first devised on two time scales with their discrepancies reflected by a singular perturbation parameter (SPP). The network measurement outputs are transmitted via a communication network where the data transmissions are scheduled by the RR protocol with hope to avoid the undesired data collision. The error dynamics of the state estimation is governed by a switched system with a periodic switching parameter. A novel Lyapunov function is constructed that is dependent on both the transmission order and the SPP. By establishing a key lemma specifically tackling the SPP, sufficient conditions are obtained such that, for any SPP less than or equal to a predefined upper bound, the error dynamics of the state estimation is asymptotically stable and satisfies a prescribed $H_{\infty }$ performance requirement. Furthermore, the explicit parameterization of the desired state estimator is given by means of the solution to a set of matrix inequalities, and the upper bound of the SPP is then evaluated in the feasibility of these matrix inequalities. Moreover, the corresponding results for linear discrete-time SPCNs are derived as corollaries. A numerical example is given to illustrate the effectiveness of the proposed state estimator design scheme.

Complexity of Linear Regions in Deep Networks

Abstract: It is well-known that the expressivity of a neural network depends on its architecture, with deeper networks expressing more complex functions. In the case of networks that compute piecewise linear functions, such as those with ReLU activation, the number of distinct linear regions is a natural measure of expressivity. It is possible to construct networks with merely a single region, or for which the number of linear regions grows exponentially with depth; it is not clear where within this range most networks fall in practice, either before or after training. In this paper, we provide a mathematical framework to count the number of linear regions of a piecewise linear network and measure the volume of the boundaries between these regions. In particular, we prove that for networks at initialization, the average number of regions along any one-dimensional subspace grows linearly in the total number of neurons, far below the exponential upper bound. We also find that the average distance to the nearest region boundary at initialization scales like the inverse of the number of neurons. Our theory suggests that, even after training, the number of linear regions is far below exponential, an intuition that matches our empirical observations. We conclude that the practical expressivity of neural networks is likely far below that of the theoretical maximum, and that this gap can be quantified.

Benefits of V2V Communication for Autonomous and Connected Vehicles

Abstract: In this paper, we investigate the benefits of vehicle-to-vehicle (V2V) communication for autonomous vehicles and provide results on how V2V information helps reduce employable time headway in the presence of parasitic lags. For a string of vehicles adopting a constant time headway policy and availing the on-board information of predecessor’s vehicle position and velocity, the minimum employable time headway ( $h_{\min }$ ) must be lower bounded by $2\tau _{0}$ for string stability, where $\tau _{0}$ is the maximum parasitic actuation lag. In this paper, we quantify the benefits of using V2V communication in terms of a reduction in the employable time headway: 1) If the position and velocity information of $r$ immediately preceding vehicles is used, then $h_{\min }$ can be reduced to ${4\tau _{0}}/{(1+r)}$ ; 2) furthermore, if the acceleration of ‘ $r$ ’ immediately preceding vehicles is used, then $h_{\min }$ can be reduced to ${2\tau _{0}}/{(1+r)}$ ; and 3) if the position, velocity, and acceleration of the immediate and the $r$ -th predecessors are used, then $h_{\min } \ge {2\tau _{0}}/{(1+r)}$ . Note that cases (2) and (3) provide the same lower bound on the minimum employable time headway; however, case (3) requires much less communicated information.

Robust Federated Learning in a Heterogeneous Environment.

Abstract: We study a recently proposed large-scale distributed learning paradigm, namely Federated Learning, where the worker machines are end users' own devices. Statistical and computational challenges arise in Federated Learning particularly in the presence of heterogeneous data distribution (i.e., data points on different devices belong to different distributions signifying different clusters) and Byzantine machines (i.e., machines that may behave abnormally, or even exhibit arbitrary and potentially adversarial behavior). To address the aforementioned challenges, first we propose a general statistical model for this problem which takes both the cluster structure of the users and the Byzantine machines into account. Then, leveraging the statistical model, we solve the robust heterogeneous Federated Learning problem \emph{optimally}; in particular our algorithm matches the lower bound on the estimation error in dimension and the number of data points. Furthermore, as a by-product, we prove statistical guarantees for an outlier-robust clustering algorithm, which can be considered as the Lloyd algorithm with robust estimation. Finally, we show via synthetic as well as real data experiments that the estimation error obtained by our proposed algorithm is significantly better than the non-Byzantine-robust algorithms; in particular, we gain at least by 53\% and 33\% for synthetic and real data experiments, respectively, in typical settings.

Scalable Verified Training for Provably Robust Image Classification

Abstract: Recent work has shown that it is possible to train deep neural networks that are provably robust to norm-bounded adversarial perturbations. Most of these methods are based on minimizing an upper bound on the worst-case loss over all possible adversarial perturbations. While these techniques show promise, they often result in difficult optimization procedures that remain hard to scale to larger networks. Through a comprehensive analysis, we show how a simple bounding technique, interval bound propagation (IBP), can be exploited to train large provably robust neural networks that beat the state-of-the-art in verified accuracy. While the upper bound computed by IBP can be quite weak for general networks, we demonstrate that an appropriate loss and clever hyper-parameter schedule allow the network to adapt such that the IBP bound is tight. This results in a fast and stable learning algorithm that outperforms more sophisticated methods and achieves state-of-the-art results on MNIST, CIFAR-10 and SVHN. It also allows us to train the largest model to be verified beyond vacuous bounds on a downscaled version of IMAGENET.

Functional Variational Bayesian Neural Networks

Abstract: Variational Bayesian neural networks (BNNs) perform variational inference over weights, but it is difficult to specify meaningful priors and approximate posteriors in a high-dimensional weight space. We introduce functional variational Bayesian neural networks (fBNNs), which maximize an Evidence Lower BOund (ELBO) defined directly on stochastic processes, i.e. distributions over functions. We prove that the KL divergence between stochastic processes equals the supremum of marginal KL divergences over all finite sets of inputs. Based on this, we introduce a practical training objective which approximates the functional ELBO using finite measurement sets and the spectral Stein gradient estimator. With fBNNs, we can specify priors entailing rich structures, including Gaussian processes and implicit stochastic processes. Empirically, we find fBNNs extrapolate well using various structured priors, provide reliable uncertainty estimates, and scale to large datasets.

Exponential Stabilization of Takagi–Sugeno Fuzzy Systems With Aperiodic Sampling: An Aperiodic Adaptive Event-Triggered Method

Abstract: In this paper, we study the exponential stabilization problem for continuous-time Takagi–Sugeno fuzzy systems subject to aperiodic sampling. By aiming to transmission reduction, an appropriate aperiodic event-triggered communication scheme with adaptive mechanism is put forward, which covers the existing periodic mechanisms as special cases. For the sake of reduction in design conservativeness, both the available information of sampling behavior and threshold error are fully acquired by constructing a novel time-dependent Lyapunov functional. Then, a new exponential stability criterion is presented to establish the quantitative relationship among the adaptive adjusted event threshold, the decay rate, the upper bound, and the lower bound of variable sampling period, simultaneously. By resorting to a matrix transformation, the corresponding stabilization criterion is further derived by which the sampled-data controller can be obtained. Finally, two illustrative examples are provided to demonstrate the virtue and applicability of proposed design method.

On Exclusion of Positive Cosmological Constant

Abstract: Some time ago we have suggested that positive vacuum energy exhibits a finite quantum break time, which can be a signal that a positive cosmological constant is inconsistent. From the requirement that the Universe never undergoes through quantum breaking, we have derived an absolute lower bound on the speed of variation of positive vacuum energy. The same suggestion about exclusion of positive cosmological constant was made recently. We show that the new bound represents a particular string theoretic version of the old bound, which is more general. In this light, we show that the existing window still provides a large room for the inflationary and dark energy model building. In particular, the inflationary models with gravitational strength interactions, are protected against fast quantum breaking.

The reachability problem for Petri nets is not elementary

Abstract: Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modelling and analysis of hardware, software and database systems, as well as chemical, biological and business processes. The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of verification. Decidability was proved by Mayr in his seminal STOC 1981 work, and the currently best published upper bound is non-primitive recursive Ackermannian of Leroux and Schmitz from LICS 2019. We establish a non-elementary lower bound, i.e. that the reachability problem needs a tower of exponentials of time and space. Until this work, the best lower bound has been exponential space, due to Lipton in 1976. The new lower bound is a major breakthrough for several reasons. Firstly, it shows that the reachability problem is much harder than the coverability (i.e., state reachability) problem, which is also ubiquitous but has been known to be complete for exponential space since the late 1970s. Secondly, it implies that a plethora of problems from formal languages, logic, concurrent systems, process calculi and other areas, that are known to admit reductions from the Petri nets reachability problem, are also not elementary. Thirdly, it makes obsolete the currently best lower bounds for the reachability problems for two key extensions of Petri nets: with branching and with a pushdown stack. At the heart of our proof is a novel gadget so called the factorial amplifier that, assuming availability of counters that are zero testable and bounded by k, guarantees to produce arbitrarily large pairs of values whose ratio is exactly the factorial of k. We also develop a novel construction that uses arbitrarily large pairs of values with ratio R to provide zero testable counters that are bounded by R. Repeatedly composing the factorial amplifier with itself by means of the construction then enables us to compute in linear time Petri nets that simulate Minsky machines whose counters are bounded by a tower of exponentials, which yields the non-elementary lower bound. By refining this scheme further, we in fact establish hardness for h-exponential space already for Petri nets with h + 13 counters.