Separation logic is used to reason locally about stateful programs. State of the art program logics for higher-order store are usually built on top of untyped operational semantics, in part because traditional denotational methods have struggled to simultaneously account for general references and parametric polymorphism. The recent discovery of simple denotational semantics for general references and polymorphism in synthetic guarded domain theory has enabled us to develop TULIP, a higher-order separation logic over the typed equational theory of higher-order store for a monadic version of System F{mu,ref}. The Tulip logic differs from operationally-based program logics in two ways: predicates range over the meanings of typed terms rather than over the raw code of untyped terms, and they are automatically invariant under the equational congruence of higher-order store, which applies even underneath a binder. As a result, "pure" proof steps that conventionally require focusing the Hoare triple on an operational redex are replaced by a simple equational rewrite in Tulip. We have evaluated Tulip against standard examples involving linked lists in the heap, comparing our abstract equational reasoning with more familiar operational-style reasoning. Our main result is the soundness of Tulip, which we establish by constructing a BI-hyperdoctrine over the denotational semantics of F{mu,ref} in an impredicative version of synthetic guarded domain theory.
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The Linguistic Matrix Theory programme introduced by Kartsaklis, Ramgoolam and Sadrzadeh is an approach to the statistics of matrices that are generated in type-driven distributional semantics, based on permutation invariant polynomial functions which are regarded as the key observables encoding the significant statistics. In this paper we generalize the previous results on the approximate Gaussianity of matrix distributions arising from compositional distributional semantics. We also introduce a geometry of observable vectors for words, defined by exploiting the graph-theoretic basis for the permutation invariants and the statistical characteristics of the ensemble of matrices associated with the words. We describe successful applications of this unified framework to a number of tasks in computational linguistics, associated with the distinctions between synonyms, antonyms, hypernyms and hyponyms.
A non-intrusive proper generalized decomposition (PGD) strategy, coupled with an overlapping domain decomposition (DD) method, is proposed to efficiently construct surrogate models of parametric linear elliptic problems. A parametric multi-domain formulation is presented, with local subproblems featuring arbitrary Dirichlet interface conditions represented through the traces of the finite element functions used for spatial discretization at the subdomain level, with no need for additional auxiliary basis functions. The linearity of the operator is exploited to devise low-dimensional problems with only few active boundary parameters. An overlapping Schwarz method is used to glue the local surrogate models, solving a linear system for the nodal values of the parametric solution at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the overlapping region. The proposed DD-PGD methodology relies on a fully algebraic formulation allowing for real-time computation based on the efficient interpolation of the local surrogate models in the parametric space, with no additional problems to be solved during the execution of the Schwarz algorithm. Numerical results for parametric diffusion and convection-diffusion problems are presented to showcase the accuracy of the DD-PGD approach, its robustness in different regimes and its superior performance with respect to standard high-fidelity DD methods.
This paper introduces novel weighted conformal p-values and methods for model-free selective inference. The problem is as follows: given test units with covariates $X$ and missing responses $Y$, how do we select units for which the responses $Y$ are larger than user-specified values while controlling the proportion of false positives? Can we achieve this without any modeling assumptions on the data and without any restriction on the model for predicting the responses? Last, methods should be applicable when there is a covariate shift between training and test data, which commonly occurs in practice. We answer these questions by first leveraging any prediction model to produce a class of well-calibrated weighted conformal p-values, which control the type-I error in detecting a large response. These p-values cannot be passed on to classical multiple testing procedures since they may not obey a well-known positive dependence property. Hence, we introduce weighted conformalized selection (WCS), a new procedure which controls false discovery rate (FDR) in finite samples. Besides prediction-assisted candidate selection, WCS (1) allows to infer multiple individual treatment effects, and (2) extends to outlier detection with inlier distributions shifts. We demonstrate performance via simulations and applications to causal inference, drug discovery, and outlier detection datasets.
The problem of sequential anomaly detection and identification is considered in the presence of a sampling constraint. Specifically, multiple data streams are generated by distinct sources and the goal is to quickly identify those that exhibit ``anomalous'' behavior, when it is not possible to sample every source at each time instant. Thus, in addition to a stopping rule, which determines when to stop sampling, and a decision rule, which indicates which sources to identify as anomalous upon stopping, one needs to specify a sampling rule that determines which sources to sample at each time instant. The focus of this work is on ordering sampling rules, which sample the data sources, among those currently estimated as anomalous (resp. non-anomalous), for which the corresponding local test statistics have the smallest (resp. largest) values. It is shown that with an appropriate design, which is specified explicitly, an ordering sampling rule leads to the optimal expected time for stopping, among all policies that satisfy the same sampling and error constraints, to a first-order asymptotic approximation as the false positive and false negative error rates under control both go to zero. This is the first asymptotic optimality result for ordering sampling rules when multiple sources can be sampled per time instant. Moreover, this is established under a general setup where the number of anomalies is not required to be a priori known. A novel proof technique is introduced, which unifies different versions of the problem regarding the homogeneity of the sources and prior information on the number of anomalies.
The main topic of this paper are algorithms for computing Nash equilibria. We cast our particular methods as instances of a general algorithmic abstraction, namely, a method we call {\em algorithmic boosting}, which is also relevant to other fixed-point computation problems. Algorithmic boosting is the principle of computing fixed points by taking (long-run) averages of iterated maps and it is a generalization of exponentiation. We first define our method in the setting of nonlinear maps. Secondly, we restrict attention to convergent linear maps (for computing dominant eigenvectors, for example, in the PageRank algorithm) and show that our algorithmic boosting method can set in motion {\em exponential speedups in the convergence rate}. Thirdly, we show that algorithmic boosting can convert a (weak) non-convergent iterator to a (strong) convergent one. We also consider a {\em variational approach} to algorithmic boosting providing tools to convert a non-convergent continuous flow to a convergent one. Then, by embedding the construction of averages in the design of the iterated map, we constructively prove the existence of Nash equilibria (and, therefore, Brouwer fixed points). We then discuss implementations of averaging and exponentiation, an important matter even for the scalar case. We finally discuss a relationship between dominant (PageRank) eigenvectors and Nash equilibria.
We propose a new class of models for variable clustering called Asymptotic Independent block (AI-block) models, which defines population-level clusters based on the independence of the maxima of a multivariate stationary mixing random process among clusters. This class of models is identifiable, meaning that there exists a maximal element with a partial order between partitions, allowing for statistical inference. We also present an algorithm for recovering the clusters of variables without specifying the number of clusters \emph{a priori}. Our work provides some theoretical insights into the consistency of our algorithm, demonstrating that under certain conditions it can effectively identify clusters in the data with a computational complexity that is polynomial in the dimension. This implies that groups can be learned nonparametrically in which block maxima of a dependent process are only sub-asymptotic. To further illustrate the significance of our work, we applied our method to neuroscience and environmental real-datasets. These applications highlight the potential and versatility of the proposed approach.
Discovering causal relationships from observational data is a fundamental yet challenging task. In some applications, it may suffice to learn the causal features of a given response variable, instead of learning the entire underlying causal structure. Invariant causal prediction (ICP, Peters et al., 2016) is a method for causal feature selection which requires data from heterogeneous settings. ICP assumes that the mechanism for generating the response from its direct causes is the same in all settings and exploits this invariance to output a subset of the causal features. The framework of ICP has been extended to general additive noise models and to nonparametric settings using conditional independence testing. However, nonparametric conditional independence testing often suffers from low power (or poor type I error control) and the aforementioned parametric models are not suitable for applications in which the response is not measured on a continuous scale, but rather reflects categories or counts. To bridge this gap, we develop ICP in the context of transformation models (TRAMs), allowing for continuous, categorical, count-type, and uninformatively censored responses (we show that, in general, these model classes do not allow for identifiability when there is no exogenous heterogeneity). We propose TRAM-GCM, a test for invariance of a subset of covariates, based on the expected conditional covariance between environments and score residuals which satisfies uniform asymptotic level guarantees. For the special case of linear shift TRAMs, we propose an additional invariance test, TRAM-Wald, based on the Wald statistic. We implement both proposed methods in the open-source R package "tramicp" and show in simulations that under the correct model specification, our approach empirically yields higher power than nonparametric ICP based on conditional independence testing.
Surface defect inspection is of great importance for industrial manufacture and production. Though defect inspection methods based on deep learning have made significant progress, there are still some challenges for these methods, such as indistinguishable weak defects and defect-like interference in the background. To address these issues, we propose a transformer network with multi-stage CNN (Convolutional Neural Network) feature injection for surface defect segmentation, which is a UNet-like structure named CINFormer. CINFormer presents a simple yet effective feature integration mechanism that injects the multi-level CNN features of the input image into different stages of the transformer network in the encoder. This can maintain the merit of CNN capturing detailed features and that of transformer depressing noises in the background, which facilitates accurate defect detection. In addition, CINFormer presents a Top-K self-attention module to focus on tokens with more important information about the defects, so as to further reduce the impact of the redundant background. Extensive experiments conducted on the surface defect datasets DAGM 2007, Magnetic tile, and NEU show that the proposed CINFormer achieves state-of-the-art performance in defect detection.
We propose an approach to compute inner and outer-approximations of the sets of values satisfying constraints expressed as arbitrarily quantified formulas. Such formulas arise for instance when specifying important problems in control such as robustness, motion planning or controllers comparison. We propose an interval-based method which allows for tractable but tight approximations. We demonstrate its applicability through a series of examples and benchmarks using a prototype implementation.
In large-scale systems there are fundamental challenges when centralised techniques are used for task allocation. The number of interactions is limited by resource constraints such as on computation, storage, and network communication. We can increase scalability by implementing the system as a distributed task-allocation system, sharing tasks across many agents. However, this also increases the resource cost of communications and synchronisation, and is difficult to scale. In this paper we present four algorithms to solve these problems. The combination of these algorithms enable each agent to improve their task allocation strategy through reinforcement learning, while changing how much they explore the system in response to how optimal they believe their current strategy is, given their past experience. We focus on distributed agent systems where the agents' behaviours are constrained by resource usage limits, limiting agents to local rather than system-wide knowledge. We evaluate these algorithms in a simulated environment where agents are given a task composed of multiple subtasks that must be allocated to other agents with differing capabilities, to then carry out those tasks. We also simulate real-life system effects such as networking instability. Our solution is shown to solve the task allocation problem to 6.7% of the theoretical optimal within the system configurations considered. It provides 5x better performance recovery over no-knowledge retention approaches when system connectivity is impacted, and is tested against systems up to 100 agents with less than a 9% impact on the algorithms' performance.
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