Worst-case Error Research Articles

Coresets for the Average Case Error for Finite Query Sets.

Coreset is usually a small weighted subset of an input set of items, that provably approximates their loss function for a given set of queries (models, classifiers, hypothesis). That is, the maximum (worst-case) error over all queries is bounded. To obtain smaller coresets, we suggest a natural relaxation: coresets whose average error over the given set of queries is bounded. We provide both deterministic and randomized (generic) algorithms for computing such a coreset for any finite set of queries. Unlike most corresponding coresets for the worst-case error, the size of the coreset in this work is independent of both the input size and its Vapnik–Chervonenkis (VC) dimension. The main technique is to reduce the average-case coreset into the vector summarization problem, where the goal is to compute a weighted subset of the n input vectors which approximates their sum. We then suggest the first algorithm for computing this weighted subset in time that is linear in the input size, for , where is the approximation error, improving, e.g., both [ICML’17] and applications for principal component analysis (PCA) [NIPS’16]. Experimental results show significant and consistent improvement also in practice. Open source code is provided.

Random sections of ellipsoids and the power of random information

We study the circumradius of the intersection of an m m -dimensional ellipsoid E \mathcal E with semi-axes σ 1 ≥ ⋯ ≥ σ m \sigma _1\geq \dots \geq \sigma _m with random subspaces of codimension n n , where n n can be much smaller than m m . We find that, under certain assumptions on σ \sigma , this random radius R n = R n ( σ ) \mathcal {R}_n=\mathcal {R}_n(\sigma ) is of the same order as the minimal such radius σ n + 1 \sigma _{n+1} with high probability. In other situations R n \mathcal {R}_n is close to the maximum σ 1 \sigma _1 . The random variable R n \mathcal {R}_n naturally corresponds to the worst-case error of the best algorithm based on random information for L 2 L_2 -approximation of functions from a compactly embedded Hilbert space H H with unit ball E \mathcal E . In particular, σ k \sigma _k is the k k th largest singular value of the embedding H ↪ L 2 H\hookrightarrow L_2 . In this formulation, one can also consider the case m = ∞ m=\infty and we prove that random information behaves very differently depending on whether σ ∈ ℓ 2 \sigma \in \ell _2 or not. For σ ∉ ℓ 2 \sigma \notin \ell _2 we get E [ R n ] = σ 1 \mathbb {E}[\mathcal {R}_n] = \sigma _1 and random information is completely useless. For σ ∈ ℓ 2 \sigma \in \ell _2 the expected radius tends to zero at least at rate o ( 1 / n ) o(1/\sqrt {n}) as n → ∞ n\to \infty . In the important case \[ σ k ≍ k − α ln − β ⁡ ( k + 1 ) , \sigma _k \asymp k^{-\alpha } \ln ^{-\beta }(k+1), \] where α > 0 \alpha > 0 and β ∈ R \beta \in \mathbb {R} (which corresponds to various Sobolev embeddings), we prove E [ R n ( σ ) ] ≍ { σ 1 a m p ; if α > 1 / 2 \, or \, β ≤ α = 1 / 2 , 2 m m σ n + 1 ln ⁡ ( n + 1 ) a m p ; if β > α = 1 / 2 , 2 m m σ n + 1 a m p ; if α > 1 / 2. \begin{equation*} \mathbb E [\mathcal {R}_n(\sigma )] \asymp \left \{\begin {array}{cl} \sigma _1 & \text {if} \quad \alpha >1/2 \text {\, or \,} \beta \leq \alpha =1/2, {2mm} \\ \sigma _{n+1} \, \sqrt {\ln (n+1)} \quad & \text {if} \quad \beta >\alpha =1/2, {2mm} \\ \sigma _{n+1} & \text {if} \quad \alpha >1/2. \end{array}\right . \end{equation*} In the proofs we use a comparison result for Gaussian processes à la Gordon, exponential estimates for sums of chi-squared random variables, and estimates for the extreme singular values of (structured) Gaussian random matrices. The upper bound is constructive. It is proven for the worst case error of a least squares estimator.

Detection of Juvenile Myoclonic Epilepsy Based on Deep Transferred CNN and Structural Connectivity Maps

Acoustic emission offers various data concerning the fracture behavior of different materials. In this paper AE study was conducted on 14 numbers of tensile specimens made of Al6061 reinforced with silicon carbide particles in which tensile loading is done up to failure with a 100 kN universal testing machine. AE parameters are filtered from the specimens and solely 60 % of actual failure loading parameters are considered for further analysis. Parameters like count, energy, duration, rise time and amplitude are used to characterize the fractures occurred on metal matrix composites due to the low matrix cavitations, particle cracking, interfacial debonding and the transition of mode from tensile to shear. The three individual artificial neural networks generated with the AE parameters like hits, felicity ratio and rise angle were trained properly towards the ultimate strength as the anticipated output. The network was able to predict the worst case error of – 3.51 %,-4.73 %, and -2.73 %. The failure prediction accuracy by using rise angle as input found to be better, however the three trained artificial neural networks have also proven its significance towards the prediction exercise.

Closed‐form analysis of undetected range errors due to ionospheric impacts for GBAS category I operations

Ionospheric anomalies may cause large differential range errors in Ground-Based Augmentation System (GBAS) users. To mitigate those integrity threats, worst-case ionosphere-induced position errors for potentially usable satellite geometries must be bounded by the GBAS ground facility. This mitigation method requires us to compute the worst-case range error for each satellite affected by a hypothetical ionospheric front. This paper presents a simulation-based method for deriving a closed-form expression of undetected ionosphere-induced range errors. Two types of ionospheric impact scenarios are defined in terms of the motion of an ionospheric front. Explicit expressions for outputs of the code-carrier smoothing filter and the code-carrier divergence monitor are derived to reduce the computational load of ionospheric impact simulations. An exhaustive search algorithm is applied to generate the worst undetected range error among all possible ionospheric impact conditions. Finally, a closed-form expression that bounds the maximum ionospheric range errors is determined as a linear function of the magnitude of gradient and the relative speed of the ionospheric front.

Worst-case Recovery Guarantees for Least Squares Approximation Using Random Samples

We construct a least squares approximation method for the recovery of complex-valued functions from a reproducing kernel Hilbert space on D subset mathbb {R}^d. The nodes are drawn at random for the whole class of functions, and the error is measured in L_2(D,varrho _{D}). We prove worst-case recovery guarantees by explicitly controlling all the involved constants. This leads to new preasymptotic recovery bounds with high probability for the error of hyperbolic Fourier regression on multivariate data. In addition, we further investigate its counterpart hyperbolic wavelet regression also based on least squares to recover non-periodic functions from random samples. Finally, we reconsider the analysis of a cubature method based on plain random points with optimal weights and reveal near-optimal worst-case error bounds with high probability. It turns out that this simple method can compete with the quasi-Monte Carlo methods in the literature which are based on lattices and digital nets.

Optimal Linear Encryption Against Stealthy Attacks on Remote State Estimation

Defending against malicious attacks has become increasingly important in various cyber-physical systems. This article presents an encryption-based countermeasure against stealthy attacks on remote state estimation. Smart sensors transmit data to a remote estimator through a wireless communication network, in which data packets can be intercepted and compromised by attackers. The remote end is equipped with a false data detector that monitors the system. To avoid being detected, the attack should follow the stealthiness constraint. A linear encryption scheme is proposed to reduce the influence of potential stealthy attacks. For arbitrary linear encryption, the worst-case linear attack that yields the largest estimation error is derived. Accordingly, the optimal linear encryption, which minimizes the worst-case estimation error, is designed based on the Stackelberg game analysis. The above optimal strategies are considered in both the complete and partial measurement information scenarios for the attacker. Moreover, the generalization to nonlinear encryption strategies is also discussed. Comparisons of attack and encryption strategies through numerical examples are provided to illustrate the theoretical results.

L_2-norm sampling discretization and recovery of functions from RKHS with finite trace

In this paper we study L_2-norm sampling discretization and sampling recovery of complex-valued functions in RKHS on D subset mathbb {R}^d based on random function samples. We only assume the finite trace of the kernel (Hilbert–Schmidt embedding into L_2) and provide several concrete estimates with precise constants for the corresponding worst-case errors. In general, our analysis does not need any additional assumptions and also includes the case of non-Mercer kernels and also non-separable RKHS. The fail probability is controlled and decays polynomially in n, the number of samples. Under the mild additional assumption of separability we observe improved rates of convergence related to the decay of the singular values. Our main tool is a spectral norm concentration inequality for infinite complex random matrices with independent rows complementing earlier results by Rudelson, Mendelson, Pajor, Oliveira and Rauhut.

An accurate digital converter for lossy capacitive sensors

This paper presents an incremental delta-sigma based capacitance-to-digital converter for lossy capacitive sensors. The proposed converter consists of a capacitance-to-voltage converter followed by a single-bit incremental delta-sigma converter. Unlike conventional incremental-delta sigma converter where a fixed reference voltage signal is utilized, the proposed converter utilizes a reference signal which depends on the amplitude and frequency of the input excitation signal. The generated reference signal for the incremental-delta sigma converter reduces the error in the capacitance measurement due to variation in the excitation signal frequency and amplitude. In addition, the proposed converter utilizes a frequency-independent quadrature phase shifter circuit for the generation of reference π/2 phase-shifted signal for the phase-sensitive detection operation. Moreover, a modified clock counting approach is utilized to measure the sensor capacitance accurately. A prototype of the proposed circuit is fabricated using commercial off-the-shelf components and tested. The experimental results demonstrate that the circuit is able to measure the sensor capacitor in the range from 20 pF to 163 pF. The worst-case error in sensor capacitance measurement was less than 0.2%. Furthermore, the proposed converter experimentally provides a signal-to-noise ratio of 56.66 dB.

A balanced sensor scheduling for multitarget localization in a distributed multiple-input multiple-output radar network

In this article, we consider the problem of optimally selecting a subset of transmitters from a transmitter set available to a multiple-input and multiple-output radar network. The aim is to minimize the location estimation error of underlying targets under a power constraint. We formulate it as a minimum-variance estimation problem and show that the underlying variance reduction function is submodular. From the properties of submodularity, we present a balanced selection policy which minimizes the worst-case error value using a minimax strategy. A greedy algorithm with guaranteed performance with respect to optimal solutions is given to efficiently implement the scheduling policy. The effectiveness and the efficiency of the proposed algorithm are demonstrated in simulated examples.

Integration in reproducing kernel Hilbert spaces of Gaussian kernels

The Gaussian kernel plays a central role in machine learning, uncertainty quantification and scattered data approximation, but has received relatively little attention from a numerical analysis standpoint. The basic problem of finding an algorithm for efficient numerical integration of functions reproduced by Gaussian kernels has not been fully solved. In this article we construct two classes of algorithms that use N N evaluations to integrate d d -variate functions reproduced by Gaussian kernels and prove the exponential or super-algebraic decay of their worst-case errors. In contrast to earlier work, no constraints are placed on the length-scale parameter of the Gaussian kernel. The first class of algorithms is obtained via an appropriate scaling of the classical Gauss–Hermite rules. For these algorithms we derive lower and upper bounds on the worst-case error of the forms exp ⁡ ( − c 1 N 1 / d ) N 1 / ( 4 d ) \exp (-c_1 N^{1/d}) N^{1/(4d)} and exp ⁡ ( − c 2 N 1 / d ) N − 1 / ( 4 d ) \exp (-c_2 N^{1/d}) N^{-1/(4d)} , respectively, for positive constants c 1 > c 2 c_1 > c_2 . The second class of algorithms we construct is more flexible and uses worst-case optimal weights for points that may be taken as a nested sequence. For these algorithms we derive upper bounds of the form exp ⁡ ( − c 3 N 1 / ( 2 d ) ) \exp (-c_3 N^{1/(2d)}) for a positive constant c 3 c_3 .

Class-wise Calibration: A Case Study on COVID-19 Hate Speech

Proper calibration of deep-learning models is critical for many high-stakes problems. In this paper, we show that existing calibration metrics fail to pay attention to miscalibration on individual classes, hence overlooking minority classes and causing significant issues on imbalanced classification problems. Using a COVID-19 hate-speech dataset, we first discover that in imbalanced datasets, miscalibration error on an individual class varies greatly, and error on minority classes can be magnitude times worse than what is suggested by the overall calibration performance. To address this issue, we propose a new metric based on expected miscalibration error, named as Contraharmonic Expected Calibration Error (CECE), which punishes severe miscalibration on individual classes. We further devise a novel variant of temperature scaling for imbalanced data to improve class-wise miscalibration, which re-weights the loss function by the inverse class count to tune the scaling parameter to reduce worst-case minority calibration error. Our case study on a benchmarking COVID-19 hate speech task shows the effectiveness of our calibration metric and our temperature scaling strategy.

Time-dependent unbounded Hamiltonian simulation with vector norm scaling

The accuracy of quantum dynamics simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the Hamiltonian. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. However, the operator norm measures the worst-case error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. In this sense, our result outperforms all previous error bounds in the quantum simulation literature. Our result extends that of [Jahnke, Lubich, BIT Numer. Math. 2000] to the time-dependent setting. We also clarify the existence and the importance of commutator scalings of Trotter and generalized Trotter methods for time-dependent Hamiltonian simulations.

Error propagation dynamics of velocimetry-based pressure field calculations (2): on the error profile

A recent study investigated the propagation of error in a velocimetry-based pressure (V-pressure) field reconstruction problem by directly analyzing the properties of the pressure Poisson equation (Pan et al 2016 Meas. Sci. Technol. 27 084012). In the present work, we extend these results by quantifying the effect of the error profile in the data field (shape/structure of the error in space) on the resultant error in the reconstructed pressure field. We first calculate the mode of the error in the data that maximizes error in the pressure field, which is the most dangerous error (called the worst error in the present work). This calculation of the worst error is equivalent to finding the principle mode of, for example, an Euler–Bernoulli beam problem in one-dimension and the Kirchhoff–Love plate in two-dimensions, thus connecting the V-pressure problem from experimental fluid mechanics to buckling elastic bodies from elastic mechanics. Taking advantage of this analogy, we then analyze how the error profile (e.g. spatial frequency of the error and the location of the most concentrated error) in the data field coupled with fundamental features of the flow domain (i.e. size, shape, and dimension of the domain, and the configuration of boundary conditions) significantly affects the error propagation from data to the reconstructed pressure. Our analytical results lend to practical applications in two ways. First, minimization of error propagation can be achieved by avoiding low frequency error profiles in data similar to the worst case scenarios and error concentrated at sensitive locations. Second, small amounts of the error in the data, if the error profile is similar to the worst error case, can cause significant error in the reconstructed pressure field; such a synthetic error can be used to benchmark V-pressure algorithms.

Weighted integration over a hyperrectangle based on digital nets and sequences

Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to approximate integrals over the unit cube with respect to the uniform measure. In this paper we discuss QMC integration with respect to general product measures defined on an arbitrary hyperrectangle. We only require that the cumulative distribution function is invertible. We develop a worst-case error bound and study the dependence of the error on the number of points and the dimension for digital nets and sequences as well as polynomial lattice point sets, which are mapped to the domain using the inverse cumulative distribution function. We do not require any smoothness properties of the probability density function and the worst-case error does not depend on the particular choice of density function and its smoothness. The component-by-component construction of polynomial lattice rules is based on a criterion which depends only on the size of the hyperrectangle but is otherwise independent of the product measure.

One-bit tensor completion via transformed tensor singular value decomposition

This paper considers the problem of low-tubal-rank tensor completion from incomplete one-bit observations. Our work is inspired by the recently proposed invertible linear transforms based tensor-tensor product and transformed tensor singular value decomposition (t-SVD). Under this framework, a tensor nuclear norm constrained maximum log-likelihood estimation model is proposed, which is convex and efficiently solved. The feasibility of the model is proved with an upper bound of the estimation error obtained. We also show a lower bound of the worst-case estimation error, which combing with the obtained upper bound demonstrates that the estimation error is nearly order-optimal. Furthermore, an algorithm based on the alternating direction multipliers method (ADMM) and non-monotone spectral projected-gradient (SPG) method is designed to solve the estimation model. Simulations are performed to show the effectiveness of the proposed method, and the applications to real-world data demonstrate the promising performance of the proposed method.

Sensitivity of a low-frequency polarimetric radio interferometer

Context.The sensitivity of a radio interferometer is a key figure of merit (FoM) for a radio telescope. The sensitivity of a single polarized interferometer is typically given as an antenna effective area over a system temperature,Ae/Tsys, assuming an unpolarized source. For a dual-polarized polarimetric interferometer intended to observe sources of unknown polarization, the state of polarization must not be assumed a priori. Furthermore, in contrast to the narrow field of view (FoV) of dish-based interferometers, the sensitivity of a polarimetric low-frequency radio interferometer warrants a careful review because of the very wide FoV of the dual-polarized antennas. A revision of this key FoM is particularly needed in the context of the Low-Frequency Square Kilometre Array (SKA-Low) where the sensitivity requirements are currently stated usingAe/Tsysfor a single-polarized antenna system, which produces ambiguity for off-zenith angles.Aims.This paper aims to derive an expression for the sensitivity of a polarimetric radio interferometer that is valid for all-sky observations of arbitrarily polarized sources, with neither a restriction on FoV nor with any a priori assumption regarding the polarization state of the source. We verify the resulting formula with an all-sky observation using the Murchison Widefield Array telescope.Methods.The sensitivity expression was developed from first principles by applying the concept of system equivalent flux density (SEFD) to a polarimetric radio interferometer (not by computingAe/Tsys). The SEFD was calculated from the standard deviation of the noisy flux density estimate for a target source due to system noise.Results.The SEFD for a polarimetric radio interferometer is generally not 1/√2 of a single-polarized interferometer as is often assumed for narrow FoV. This assumption can lead to significant errors for a dual-polarized dipole based system, which is common in low-frequency radio astronomy: up to ∼15% for a zenith angle with a coverage of 45° and up to ∼45% for 60° coverage. The worst case errors occur in the diagonal planes of the dipole for very wide FoV. This is demonstrated through theory, simulation, and observations. Furthermore, using the resulting formulation, the calculation of the off-zenith sensitivity is straightforward and unambiguous.Conclusions.For wide FoV observations pertinent to a low-frequency radio interferometer such as SKA-Low, the narrow FoV and the single-polarized sensitivity expressions are not correct and should be replaced by the formula derived in this paper.

Hmcode-2020: improved modelling of non-linear cosmological power spectra with baryonic feedback

ABSTRACT We present an updated version of the hmcode augmented halo model that can be used to make accurate predictions of the non-linear matter power spectrum over a wide range of cosmologies. Major improvements include modelling of baryon-acoustic oscillation (BAO) damping in the power spectrum and an updated treatment of massive neutrinos. We fit our model to simulated power spectra and show that we can match the results with an root mean square (RMS) error of 2.5 per cent across a range of cosmologies, scales $k \lt 10\, h\, \mathrm{Mpc}^{-1}$, and redshifts z < 2. The error rarely exceeds 5 per cent and never exceeds 16 per cent. The worst-case errors occur at z ≃ 2, or for cosmologies with unusual dark energy equations of state. This represents a significant improvement over previous versions of hmcode, and over other popular fitting functions, particularly for massive-neutrino cosmologies with high neutrino mass. We also present a simple halo model that can be used to model the impact of baryonic feedback on the power spectrum. This six-parameter physical model includes gas expulsion by active galactic nuclei (AGN) feedback and encapsulates star formation. By comparing this model to data from hydrodynamical simulations, we demonstrate that the power spectrum response to feedback is matched at the <1 per cent level for z < 1 and $k\lt 20\, h\, \mathrm{Mpc}^{-1}$. We also present a single-parameter variant of this model, parametrized in terms of feedback strength, which is only slightly less accurate. We make code available for our non-linear and baryon models at https://github.com/alexander-mead/HMcode and it is also available within camb and soon within class.

On quasi-Monte Carlo methods in weighted ANOVA spaces

In the present paper we study quasi-Monte Carlo rules for approximating integrals over the $d$-dimensional unit cube for functions from weighted Sobolev spaces of regularity one. While the properties of these rules are well understood for anchored Sobolev spaces, this is not the case for the ANOVA spaces, which are another very important type of reference spaces for quasi-Monte Carlo rules. Using a direct approach we provide a formula for the worst case error of quasi-Monte Carlo rules for functions from weighted ANOVA spaces. As a consequence we bound the worst case error from above in terms of weighted discrepancy of the employed integration nodes. On the other hand we also obtain a general lower bound in terms of the number $n$ of used integration nodes. For the one-dimensional case our results lead to the optimal integration rule and also in the two-dimensional case we provide rules yielding optimal convergence rates.

The Minimax Estimator of the Average Treatment Effect, among Linear Combinations of Conditional Average Treatment Effects Estimators

I consider the estimation of the average treatment effect (ATE), in a population that can be divided into $G$ groups, and such that one has unbiased and uncorrelated estimators of the conditional average treatment effect (CATE) in each group. These conditions are for instance met in stratified randomized experiments. I assume that the outcome is homoscedastic, and that each CATE is bounded in absolute value by $B$ standard deviations of the outcome, for some known constant $B$. I derive, across all linear combinations of the CATEs' estimators, the estimator of the ATE with the lowest worst-case mean-squared error. This optimal estimator assigns a weight equal to group $g$'s share in the population to the most precisely estimated CATEs, and a weight proportional to one over the CATE's variance to the least precisely estimated CATEs. This optimal estimator is feasible: the weights only depend on known quantities. I then allow for positive covariances known up to the outcome's variance between the estimators. This condition is met in differences-in-differences designs, if errors are homoscedastic and uncorrelated. Under those assumptions, I show that the minimax estimator is still feasible and can easily be computed.

Improving Analytical Delay Modelingfor CMOS Inverters

Analytical methods for gate delay estimation are very useful to speedup timing analysis of digital integrated circuits. This work presents a novel approach to analytically estimate the CMOS inverter delay. The proposed method considers the influence of input slope, output load and I/O coupling capacitance, as well as relevant effects such as channel length modulation and drain induced barrier lowering. Experimental results are on good agreement with HSPICE simulations, showing significant accuracy improvement compared to published related work. The delay model error has an average value of 3%, and the worst case error is smaller than 10%.