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We provide a generic algorithm for constructing formulae that distinguish behaviourally inequivalent states in systems of various transition types such as nondeterministic, probabilistic or weighted; genericity over the transition type is achieved by working with coalgebras for a set functor in the paradigm of universal coalgebra. For every behavioural equivalence class in a given system, we construct a formula which holds precisely at the states in that class. The algorithm instantiates to deterministic finite automata, transition systems, labelled Markov chains, and systems of many other types. The ambient logic is a modal logic featuring modalities that are generically extracted from the functor; these modalities can be systematically translated into custom sets of modalities in a postprocessing step. The new algorithm builds on an existing coalgebraic partition refinement algorithm. It runs in time O((m+n) log n) on systems with n states and m transitions, and the same asymptotic bound applies to the dag size of the formulae it constructs. This improves the bounds on run time and formula size compared to previous algorithms even for previously known specific instances, viz. transition systems and Markov chains; in particular, the best previous bound for transition systems was O(mn).

We study the problem of counting answers to unions of conjunctive queries (UCQs) under structural restrictions on the input query. Concretely, given a class C of UCQs, the problem #UCQ(C) provides as input a UCQ Q in C and a database D and the problem is to compute the number of answers of Q in D. Chen and Mengel [PODS'16] have shown that for any recursively enumerable class C, the problem #UCQ(C) is either fixed-parameter tractable or hard for one of the parameterised complexity classes W[1] or #W[1]. However, their tractability criterion is unwieldy in the sense that, given any concrete class C of UCQs, it is not easy to determine how hard it is to count answers to queries in C. Moreover, given a single specific UCQ Q, it is not easy to determine how hard it is to count answers to Q. In this work, we address the question of finding a natural tractability criterion: The combined conjunctive query of a UCQ $\varphi_1 \vee \dots \vee \varphi_\ell$ is the conjunctive query $\varphi_1 \wedge \dots \wedge \varphi_\ell$. We show that under natural closure properties of C, the problem #UCQ(C) is fixed-parameter tractable if and only if the combined conjunctive queries of UCQs in C, and their contracts, have bounded treewidth. A contract of a conjunctive query is an augmented structure, taking into account how the quantified variables are connected to the free variables. If all variables are free, then a conjunctive query is equal to its contract; in this special case the criterion for fixed-parameter tractability of #UCQ(C) thus simplifies to the combined queries having bounded treewidth. Finally, we give evidence that a closure property on C is necessary for obtaining a natural tractability criterion: We show that even for a single UCQ Q, the meta problem of deciding whether #UCQ({Q}) can be solved in time $O(|D|^d)$ is NP-hard for any fixed $d\geq 1$.

We consider the problem of detecting causal relationships between discrete time series, in the presence of potential confounders. A hypothesis test is introduced for identifying the temporally causal influence of $(x_n)$ on $(y_n)$, causally conditioned on a possibly confounding third time series $(z_n)$. Under natural Markovian modeling assumptions, it is shown that the null hypothesis, corresponding to the absence of temporally causal influence, is equivalent to the underlying `causal conditional directed information rate' being equal to zero. The plug-in estimator for this functional is identified with the log-likelihood ratio test statistic for the desired test. This statistic is shown to be asymptotically normal under the alternative hypothesis and asymptotically $\chi^2$ distributed under the null, facilitating the computation of $p$-values when used on empirical data. The effectiveness of the resulting hypothesis test is illustrated on simulated data, validating the underlying theory. The test is also employed in the analysis of spike train data recorded from neurons in the V4 and FEF brain regions of behaving animals during a visual attention task. There, the test results are seen to identify interesting and biologically relevant information.

Fast Hough transform is a widely used algorithm in pattern recognition. The algorithm relies on approximating lines using a specific discrete line model called dyadic lines. The worst-case deviation of a dyadic line from the ideal line it used to construct grows as $O(log(n))$, where $n$ is the linear size of the image. But few lines actually reach the worst-case bound. The present paper addresses a statistical analysis of the deviation of a dyadic line from its ideal counterpart. Specifically, our findings show that the mean deviation is zero, and the variance grows as $O(log(n))$. As $n$ increases, the distribution of these (suitably normalized) deviations converges towards a normal distribution with zero mean and a small variance. This limiting result makes an essential use of ergodic theory.

This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating appropriate training data, cost-accuracy trade-offs, and nontrivial hyperparameter tuning. The unpredictability of the accuracy of neural operators impacts their applications in downstream problems of inference, optimization, and control. A framework based on the linear variational problem that gives the correction to the prediction furnished by neural operators is considered based on earlier work in JCP 486 (2023) 112104. The operator, called Residual-based Error Corrector Operator or simply Corrector Operator, associated with the corrector problem is analyzed further. Numerical results involving a nonlinear reaction-diffusion model in two dimensions with PCANet-type neural operators show almost two orders of increase in the accuracy of approximations when neural operators are corrected using the correction scheme. Further, topology optimization involving a nonlinear reaction-diffusion model is considered to highlight the limitations of neural operators and the efficacy of the correction scheme. Optimizers with neural operator surrogates are seen to make significant errors (as high as 80 percent). However, the errors are much lower (below 7 percent) when neural operators are corrected.

Multivariate normal (MVN) probabilities arise in myriad applications, but they are analytically intractable and need to be evaluated via Monte-Carlo-based numerical integration. For the state-of-the-art minimax exponential tilting (MET) method, we show that the complexity of each of its components can be greatly reduced through an integrand parameterization that utilizes the sparse inverse Cholesky factor produced by the Vecchia approximation, whose approximation error is often negligible relative to the Monte-Carlo error. Based on this idea, we derive algorithms that can estimate MVN probabilities and sample from truncated MVN distributions in linear time (and that are easily parallelizable) at the same convergence or acceptance rate as MET, whose complexity is cubic in the dimension of the MVN probability. We showcase the advantages of our methods relative to existing approaches using several simulated examples. We also analyze a groundwater-contamination dataset with over twenty thousand censored measurements to demonstrate the scalability of our method for partially censored Gaussian-process models.

We give a damped proximal augmented Lagrangian method (DPALM) for solving problems with a weakly-convex objective and convex linear/nonlinear constraints. Instead of taking a full stepsize, DPALM adopts a damped dual stepsize to ensure the boundedness of dual iterates. We show that DPALM can produce a (near) $\vareps$-KKT point within $O(\vareps^{-2})$ outer iterations if each DPALM subproblem is solved to a proper accuracy. In addition, we establish overall iteration complexity of DPALM when the objective is either a regularized smooth function or in a regularized compositional form. For the former case, DPALM achieves the complexity of $\widetilde{\mathcal{O}}\left(\varepsilon^{-2.5} \right)$ to produce an $\varepsilon$-KKT point by applying an accelerated proximal gradient (APG) method to each DPALM subproblem. For the latter case, the complexity of DPALM is $\widetilde{\mathcal{O}}\left(\varepsilon^{-3} \right)$ to produce a near $\varepsilon$-KKT point by using an APG to solve a Moreau-envelope smoothed version of each subproblem. Our outer iteration complexity and the overall complexity either generalize existing best ones from unconstrained or linear-constrained problems to convex-constrained ones, or improve over the best-known results on solving the same-structured problems. Furthermore, numerical experiments on linearly/quadratically constrained non-convex quadratic programs and linear-constrained robust nonlinear least squares are conducted to demonstrate the empirical efficiency of the proposed DPALM over several state-of-the art methods.

Secure multi-party computation provides a wide array of protocols for mutually distrustful parties be able to securely evaluate functions of private inputs. Within recent years, many such protocols have been proposed representing a plethora of strategies to securely and efficiently handle such computation. These protocols have become increasingly efficient, but their performance still is impractical in many settings. We propose new approaches to some of these problems which are either more efficient than previous works within the same security models or offer better security guarantees with comparable efficiency. The goals of this research are to improve efficiency and security of secure multi-party protocols and explore the application of such approaches to novel threat scenarios. Some of the novel optimizations employed are dynamically switching domains of shared secrets, asymmetric computations, and advantageous functional transformations, among others. Specifically, this work presents a novel combination of Shamir and Additive secret sharing to be used in parallel which allows for the transformation of efficient protocols secure against passive adversaries to be secure against active adversaries. From this set of primitives we propose the construction of a comparison protocol which can be implemented under that approach with a complexity which is more efficient than other recent works for common domains of interest. Finally, we present a system which addresses a critical security threat for the protection and obfuscation of information which may be of high consequence.

We study inference on the long-term causal effect of a continual exposure to a novel intervention, which we term a long-term treatment, based on an experiment involving only short-term observations. Key examples include the long-term health effects of regularly-taken medicine or of environmental hazards and the long-term effects on users of changes to an online platform. This stands in contrast to short-term treatments or "shocks," whose long-term effect can reasonably be mediated by short-term observations, enabling the use of surrogate methods. Long-term treatments by definition have direct effects on long-term outcomes via continual exposure so surrogacy cannot reasonably hold. Our approach instead learns long-term temporal dynamics directly from short-term experimental data, assuming that the initial dynamics observed persist but avoiding the need for both surrogacy assumptions and auxiliary data with long-term observations. We connect the problem with offline reinforcement learning, leveraging doubly-robust estimators to estimate long-term causal effects for long-term treatments and construct confidence intervals. Finally, we demonstrate the method in simulated experiments.

The existence of representative datasets is a prerequisite of many successful artificial intelligence and machine learning models. However, the subsequent application of these models often involves scenarios that are inadequately represented in the data used for training. The reasons for this are manifold and range from time and cost constraints to ethical considerations. As a consequence, the reliable use of these models, especially in safety-critical applications, is a huge challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches, and eventually to increase the generalization capability of these models. Furthermore, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-based models with existing knowledge. The identified approaches are structured according to the categories integration, extraction and conformity. Special attention is given to applications in the field of autonomous driving.