We can define the error distribution as the limiting distribution of the error between the solution $Y$ of a given stochastic differential equation (SDE) and its numerical approximation $\hat{Y}^{(m)}$, weighted by the convergence rate between the two. A goal when studying the error distribution is to provide a way of determination for error distributions for any SDE and numerical scheme that converge to the exact solution. By dividing the error into a main term and a remainder term in a particular way, the author shows that the remainder term can be negligible compared to the main term under certain suitable conditions. Under these conditions, deriving the error distribution reduces to deriving the limiting distribution of the main term. Even if the dimension is one, there are unsolved problems about the asymptotic behavior of the error when the SDE has a drift term and $0<H\leq 1/3$, but our result in the one-dimensional case can be adapted to any Hurst exponent. The main idea of the proof is to define a stochastic process $Y^{m, \rho}$ with the parameter $\rho$ interpolating between $Y$ and $\hat{Y}^{(m)}$ and to estimate the asymptotic expansion for it. Using this estimate, we determine the error distribution of the ($k$)-Milstein scheme and of the Crank-Nicholson scheme in unsolved cases.
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We formalize and interpret the geometric structure of $d$-dimensional fully connected ReLU layers in neural networks. The parameters of a ReLU layer induce a natural partition of the input domain, such that the ReLU layer can be significantly simplified in each sector of the partition. This leads to a geometric interpretation of a ReLU layer as a projection onto a polyhedral cone followed by an affine transformation, in line with the description in [doi:10.48550/arXiv.1905.08922] for convolutional networks with ReLU activations. Further, this structure facilitates simplified expressions for preimages of the intersection between partition sectors and hyperplanes, which is useful when describing decision boundaries in a classification setting. We investigate this in detail for a feed-forward network with one hidden ReLU-layer, where we provide results on the geometric complexity of the decision boundary generated by such networks, as well as proving that modulo an affine transformation, such a network can only generate $d$ different decision boundaries. Finally, the effect of adding more layers to the network is discussed.
We introduce the first cut-free nested sequent systems for first-order modal logics that admit increasing, decreasing, constant, and empty domains along with so-called general path conditions and seriality. We obtain such systems by means of two devices: 'reachability rules' and 'structural refinement'. Regarding the former device, we introduce reachability rules as special logical rules parameterized with formal grammars (viz. types of semi-Thue systems) that operate by propagating formulae and/or checking if data exists along certain paths within a nested sequent, where paths are encoded as strings generated by a parameterizing grammar. Regarding the latter device, structural refinement is a relatively new methodology used to extract nested sequent systems from labeled systems (which are ultimately obtained from a semantics) by means of eliminating structural/relational rules, introducing reachability rules, and then carrying out a notational translation. We therefore demonstrate how this method can be extended to the setting of first-order modal logics, and expose how reachability rules naturally arise from applying this method.
We consider the problem of testing whether a single coefficient is equal to zero in fixed-design linear models under a moderately high-dimensional regime, where the dimension of covariates $p$ is allowed to be in the same order of magnitude as sample size $n$. In this regime, to achieve finite-population validity, existing methods usually require strong distributional assumptions on the noise vector (such as Gaussian or rotationally invariant), which limits their applications in practice. In this paper, we propose a new method, called residual permutation test (RPT), which is constructed by projecting the regression residuals onto the space orthogonal to the union of the column spaces of the original and permuted design matrices. RPT can be proved to achieve finite-population size validity under fixed design with just exchangeable noises, whenever $p < n / 2$. Moreover, RPT is shown to be asymptotically powerful for heavy tailed noises with bounded $(1+t)$-th order moment when the true coefficient is at least of order $n^{-t/(1+t)}$ for $t \in [0,1]$. We further proved that this signal size requirement is essentially rate-optimal in the minimax sense. Numerical studies confirm that RPT performs well in a wide range of simulation settings with normal and heavy-tailed noise distributions.
The application of eigenvalue theory to dual quaternion Hermitian matrix holds significance in the realm of multi-agent formation control. In this paper, we focus on the numerical algorithm for the right eigenvalue of a dual quaternion Hermitian matrix. Rayleigh quotient iteration is proposed for computing the extreme eigenvalue with the associated eigenvector of the dual quaternion Hermitian matrix. We also derive an analysis of the convergence characteristics of the Rayleigh quotient iteration, which exhibits a local convergence rate of cubic. Numerical examples are provided to illustrate the efficiency of the proposed Rayleigh quotient iteration for the dual quaternion Hermitian eigenvalue problem.
Constructing Reed--Solomon (RS) codes that can correct insertions and deletions (insdel errors) has been considered in numerous recent works. For the special case of two-dimensional RS-codes, it is known [CST23] that an $[n,2]_q$ RS-code that can correct from $n-3$ insdel errors satisfies that $q=\Omega(n^3)$. On the other hand, there are several known constructions of $[n,2]_q$ RS-codes that can correct from $n-3$ insdel errors, where the smallest field size is $q=O(n^4)$. In this short paper, we construct $[n,2]_q$ Reed--Solomon codes that can correct $n-3$ insdel errors with $q=O(n^3)$, thereby resolving the minimum field size needed for such codes.
Data depth is an efficient tool for robustly summarizing the distribution of functional data and detecting potential magnitude and shape outliers. Commonly used functional data depth notions, such as the modified band depth and extremal depth, are estimated from pointwise depth for each observed functional observation. However, these techniques require calculating one single depth value for each functional observation, which may not be sufficient to characterize the distribution of the functional data and detect potential outliers. This paper presents an innovative approach to make the best use of pointwise depth. We propose using the pointwise depth distribution for magnitude outlier visualization and the correlation between pairwise depth for shape outlier detection. Furthermore, a bootstrap-based testing procedure has been introduced for the correlation to test whether there is any shape outlier. The proposed univariate methods are then extended to bivariate functional data. The performance of the proposed methods is examined and compared to conventional outlier detection techniques by intensive simulation studies. In addition, the developed methods are applied to simulated solar energy datasets from a photovoltaic system. Results revealed that the proposed method offers superior detection performance over conventional techniques. These findings will benefit engineers and practitioners in monitoring photovoltaic systems by detecting unnoticed anomalies and outliers.
Time series discords are a useful primitive for time series anomaly detection, and the matrix profile is capable of capturing discord effectively. There exist many research efforts to improve the scalability of discord discovery with respect to the length of time series. However, there is surprisingly little work focused on reducing the time complexity of matrix profile computation associated with dimensionality of a multidimensional time series. In this work, we propose a sketch for discord mining among multi-dimensional time series. After an initial pre-processing of the sketch as fast as reading the data, the discord mining has runtime independent of the dimensionality of the original data. On several real world examples from water treatment and transportation, the proposed algorithm improves the throughput by at least an order of magnitude (50X) and only has minimal impact on the quality of the approximated solution. Additionally, the proposed method can handle the dynamic addition or deletion of dimensions inconsequential overhead. This allows a data analyst to consider "what-if" scenarios in real time while exploring the data.
In adaptive data analysis, a mechanism gets $n$ i.i.d. samples from an unknown distribution $D$, and is required to provide accurate estimations to a sequence of adaptively chosen statistical queries with respect to $D$. Hardt and Ullman (FOCS 2014) and Steinke and Ullman (COLT 2015) showed that in general, it is computationally hard to answer more than $\Theta(n^2)$ adaptive queries, assuming the existence of one-way functions. However, these negative results strongly rely on an adversarial model that significantly advantages the adversarial analyst over the mechanism, as the analyst, who chooses the adaptive queries, also chooses the underlying distribution $D$. This imbalance raises questions with respect to the applicability of the obtained hardness results -- an analyst who has complete knowledge of the underlying distribution $D$ would have little need, if at all, to issue statistical queries to a mechanism which only holds a finite number of samples from $D$. We consider more restricted adversaries, called \emph{balanced}, where each such adversary consists of two separated algorithms: The \emph{sampler} who is the entity that chooses the distribution and provides the samples to the mechanism, and the \emph{analyst} who chooses the adaptive queries, but has no prior knowledge of the underlying distribution (and hence has no a priori advantage with respect to the mechanism). We improve the quality of previous lower bounds by revisiting them using an efficient \emph{balanced} adversary, under standard public-key cryptography assumptions. We show that these stronger hardness assumptions are unavoidable in the sense that any computationally bounded \emph{balanced} adversary that has the structure of all known attacks, implies the existence of public-key cryptography.
Standard probabilistic sparse coding assumes a Laplace prior, a linear mapping from latents to observables, and Gaussian observable distributions. We here derive a solely entropy-based learning objective for the parameters of standard sparse coding. The novel variational objective has the following features: (A) unlike MAP approximations, it uses non-trivial posterior approximations for probabilistic inference; (B) unlike for previous non-trivial approximations, the novel objective is fully analytical; and (C) the objective allows for a novel principled form of annealing. The objective is derived by first showing that the standard ELBO objective converges to a sum of entropies, which matches similar recent results for generative models with Gaussian priors. The conditions under which the ELBO becomes equal to entropies are then shown to have analytical solutions, which leads to the fully analytical objective. Numerical experiments are used to demonstrate the feasibility of learning with such entropy-based ELBOs. We investigate different posterior approximations including Gaussians with correlated latents and deep amortized approximations. Furthermore, we numerically investigate entropy-based annealing which results in improved learning. Our main contributions are theoretical, however, and they are twofold: (1) for non-trivial posterior approximations, we provide the (to the knowledge of the authors) first analytical ELBO objective for standard probabilistic sparse coding; and (2) we provide the first demonstration on how a recently shown convergence of the ELBO to entropy sums can be used for learning.
Self-driving software pipelines include components that are learned from a significant number of training examples, yet it remains challenging to evaluate the overall system's safety and generalization performance. Together with scaling up the real-world deployment of autonomous vehicles, it is of critical importance to automatically find simulation scenarios where the driving policies will fail. We propose a method that efficiently generates adversarial simulation scenarios for autonomous driving by solving an optimal control problem that aims to maximally perturb the policy from its nominal trajectory. Given an image-based driving policy, we show that we can inject new objects in a neural rendering representation of the deployment scene, and optimize their texture in order to generate adversarial sensor inputs to the policy. We demonstrate that adversarial scenarios discovered purely in the neural renderer (surrogate scene) can often be successfully transferred to the deployment scene, without further optimization. We demonstrate this transfer occurs both in simulated and real environments, provided the learned surrogate scene is sufficiently close to the deployment scene.
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