We consider the following decision problems: given a finite, rational Markov chain, source and target states, and a rational threshold, does there exist an n such that the probability of reaching the target from the source in n steps is equal to the threshold (resp. crosses the threshold)? These problems are known to be equivalent to the Skolem (resp. Positivity) problems for Linear Recurrence Sequences (LRS). These are number-theoretic problems whose decidability has been open for decades. We present a short, self-contained, and elementary reduction from LRS to Markov Chains that improves the state of the art as follows: (a) We reduce to ergodic Markov Chains, a class that is widely used in Model Checking. (b) We reduce LRS to Markov Chains of significantly lower order than before. We thus get sharper hardness results for a more ubiquitous class of Markov Chains. Immediate applications include problems in modeling biological systems, and regular automata-based counting problems.
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The approximate stabilizer rank of a quantum state is the minimum number of terms in any approximate decomposition of that state into stabilizer states. Bravyi and Gosset showed that the approximate stabilizer rank of a so-called "magic" state like $|T\rangle^{\otimes n}$, up to polynomial factors, is an upper bound on the number of classical operations required to simulate an arbitrary quantum circuit with Clifford gates and $n$ number of $T$ gates. As a result, an exponential lower bound on this quantity seems inevitable. Despite this intuition, several attempts using various techniques could not lead to a better than a linear lower bound on the "exact" rank of ${|T\rangle}^{\otimes n}$, meaning the minimal size of a decomposition that exactly produces the state. For the "approximate" rank, which is more realistically related to the cost of simulating quantum circuits, no lower bound better than $\tilde \Omega(\sqrt n)$ has been known. In this paper, we improve the lower bound on the approximate rank to $\tilde \Omega (n^2)$ for a wide range of the approximation parameters. An immediate corollary of our result is the existence of polynomial time computable functions which require a super-linear number of terms in any decomposition into exponentials of quadratic forms over $\mathbb{F}_2$, resolving a question in [Wil18]. Our approach is based on a strong lower bound on the approximate rank of a quantum state sampled from the Haar measure, a step-by-step analysis of the approximate rank of a magic-state teleportation protocol to sample from the Haar measure, and a result about trading Clifford operations with $T$ gates by [LKS18].
Value at Risk (VaR) and Conditional Value at Risk (CVaR) have become the most popular measures of market risk in Financial and Insurance fields. However, the estimation of both risk measures is challenging, because it requires the knowledge of the tail of the distribution. Therefore, tools from Extreme Value Theory are usually employed, considering that the tail data follow a Generalized Pareto distribution (GPD). Using the existing relations from the parameters of the baseline distribution and the limit GPD's parameters, we define highly informative priors that incorporate all the information available for the whole set of observations. We show how to perform Metropolis-Hastings (MH) algorithm to estimate VaR and CVaR employing the highly informative priors, in the case of exponential, stable and Gamma distributions. Afterwards, we perform a thorough simulation study to compare the accuracy and precision provided by three different methods. Finally, data from a real example is analyzed to show the practical application of the methods.
We propose a test of fairness in score-based ranking systems called matched pair calibration. Our approach constructs a set of matched item pairs with minimal confounding differences between subgroups before computing an appropriate measure of ranking error over the set. The matching step ensures that we compare subgroup outcomes between identically scored items so that measured performance differences directly imply unfairness in subgroup-level exposures. We show how our approach generalizes the fairness intuitions of calibration from a binary classification setting to ranking and connect our approach to other proposals for ranking fairness measures. Moreover, our strategy shows how the logic of marginal outcome tests extends to cases where the analyst has access to model scores. Lastly, we provide an example of applying matched pair calibration to a real-word ranking data set to demonstrate its efficacy in detecting ranking bias.
Learning the graphical structure of Bayesian networks is key to describing data-generating mechanisms in many complex applications but poses considerable computational challenges. Observational data can only identify the equivalence class of the directed acyclic graph underlying a Bayesian network model, and a variety of methods exist to tackle the problem. Under certain assumptions, the popular PC algorithm can consistently recover the correct equivalence class by reverse-engineering the conditional independence (CI) relationships holding in the variable distribution. The dual PC algorithm is a novel scheme to carry out the CI tests within the PC algorithm by leveraging the inverse relationship between covariance and precision matrices. By exploiting block matrix inversions we can also perform tests on partial correlations of complementary (or dual) conditioning sets. The multiple CI tests of the dual PC algorithm proceed by first considering marginal and full-order CI relationships and progressively moving to central-order ones. Simulation studies show that the dual PC algorithm outperforms the classic PC algorithm both in terms of run time and in recovering the underlying network structure, even in the presence of deviations from Gaussianity. Additionally, we show that the dual PC algorithm applies for Gaussian copula models, and demonstrate its performance in that setting.
Evaluation of motion comfort using advanced active human body models and efficient simplified models
Active muscles are crucial for maintaining postural stability when seated in a moving vehicle. Advanced active 3D non-linear full body models have been developed for impact and comfort simulation, including large numbers of individual muscle elements, and detailed non-linear models of the joint structures. While such models have an apparent potential to provide insight into postural stabilization, they are computationally demanding, making them less practical in particular for driving comfort where long time periods are to be studied. In vibration comfort and in general biomechanical research, linearized models are effectively used. This paper evaluates the effectiveness of simplified 3D full body human models to capture vibration comfort. An efficient seated human body model is developed and validated using experimental data. We evaluate the required complexity in terms of joints and degrees of freedom for the spine, and explore how well linear spring-damper models can approximate reflexive postural stabilization. Results indicate that linear stiffness and damping models can capture well the human response. However, the results are improved by adding proportional integral derivative (PID) controllers to maintain the defined initial body posture. The integrator is shown to be essential to prevent drift from the defined posture. The joint angular relative displacement is used as the input reference to each PID controller. With this model, a faster than real time solution is obtained when used with a simple seat model. The paper also discusses advantages and disadvantages of various models and provides insight into which models are more appropriate for motion comfort analysis. The results of this paper can provide valuable insights for designers, and researchers in the automotive and seating industries to improve the comfort and safety of seats and vehicles occupants.
Hilbert and Ackermann asked for a method to consistently extend incomplete theories to complete theories. G\"odel essentially proved that any theory capable of encoding its own statements and their proofs contains statements that are true but not provable. Hilbert did not accept that G\"odel's construction answered his question, and in his late writings and lectures, G\"odel agreed that it did not, since theories can be completed incrementally, by adding axioms to prove ever more true statements, as science normally does, with completeness as the vanishing point. This pragmatic view of validity is familiar not only to scientists who conjecture test hypotheses but also to real estate agents and other dealers, who conjure claims, albeit invalid, as necessary to close a deal, confident that they will be able to conjure other claims, albeit invalid, sufficient to make the first claims valid. We study the underlying logical process and describe the trajectories leading to testable but unfalsifiable theories to which bots and other automated learners are likely to converge.
Learning latent causal models from data has many important applications such as robustness, model extrapolation, and counterfactuals. Most prior theoretic work has focused on full causal discovery (i.e., recovering the true latent variables) but requires strong assumptions such as linearity or fails to have any analysis of the equivalence class of solutions (e.g., IRM). Instead of full causal discovery, we focus on a specific type of causal query called the domain counterfactual, which hypothesizes what a sample would have looked like if it had been generated in a different domain (or environment). Concretely, we assume domain-specific invertible latent structural causal models and a shared invertible observation function, both of which are less restrictive assumptions than prior theoretic works. Under these assumptions, we define domain counterfactually equivalent models and prove that any model can be transformed into an equivalent model via two invertible functions. This constructive property provides a tight characterization of the domain counterfactual equivalence classes. Building upon this result, we prove that every equivalence class contains a model where all intervened variables are at the end when topologically sorted by the causal DAG, i.e., all non-intervened variables have non-intervened ancestors. This surprising result suggests that an algorithm that only allows intervention in the last $k$ latent variables may improve model estimation for counterfactuals. In experiments, we enforce the sparse intervention hypothesis via this theoretic result by constraining that the latent SCMs can only differ in the last few causal mechanisms and demonstrate the feasibility of this algorithm in simulated and image-based experiments.
The separate tasks of denoising, conditional expectation and manifold learning can often be posed in a common setting of finding the conditional expectations arising from a product of two random variables. This paper focuses on this more general problem and describes an operator theoretic approach to estimating the conditional expectation. Kernel integral operators are used as a compactification tool, to set up the estimation problem as a linear inverse problem in a reproducing kernel Hilbert space. This equation is shown to have solutions that are stable to numerical approximation, thus guaranteeing the convergence of data-driven implementations. The overall technique is easy to implement, and their successful application to some real-world problems are also shown.
Automatic structures are structures whose universe and relations can be represented as regular languages. It follows from the standard closure properties of regular languages that the first-order theory of an automatic structure is decidable. While existential quantifiers can be eliminated in linear time by application of a homomorphism, universal quantifiers are commonly eliminated via the identity $\forall\,x\,.\,\Phi \equiv \neg (\exists\,x\,.\,\neg \Phi)$. If $\Phi$ is represented in the standard way as an NFA, a priori this approach results in a doubly exponential blow-up. However, the recent literature has shown that there are classes of automatic structures for which universal quantifiers can be eliminated by different means without this blow-up by treating them as first-class citizens and not resorting to double complementation. While existing lower bounds for some classes of automatic structures show that a singly exponential blow-up is unavoidable when eliminating a universal quantifier, it is not known whether there may be better approaches that avoid the na\"ive doubly exponential blow-up, perhaps at least in restricted settings. In this paper, we answer this question negatively and show that there is a family of NFA representing automatic relations for which the minimal NFA recognising the language after eliminating a single universal quantifier is doubly exponential, and deciding whether this language is empty is ExpSpace-complete.
Although Transformer-based methods have significantly improved state-of-the-art results for long-term series forecasting, they are not only computationally expensive but more importantly, are unable to capture the global view of time series (e.g. overall trend). To address these problems, we propose to combine Transformer with the seasonal-trend decomposition method, in which the decomposition method captures the global profile of time series while Transformers capture more detailed structures. To further enhance the performance of Transformer for long-term prediction, we exploit the fact that most time series tend to have a sparse representation in well-known basis such as Fourier transform, and develop a frequency enhanced Transformer. Besides being more effective, the proposed method, termed as Frequency Enhanced Decomposed Transformer ({\bf FEDformer}), is more efficient than standard Transformer with a linear complexity to the sequence length. Our empirical studies with six benchmark datasets show that compared with state-of-the-art methods, FEDformer can reduce prediction error by $14.8\%$ and $22.6\%$ for multivariate and univariate time series, respectively. the code will be released soon.
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