Minimal Assumptions Research Articles

Deep learning for inverse problems with unknown operator

We consider ill-posed inverse problems where the forward operator T is unknown, and instead we have access to training data consisting of functions fi and their noisy images Tfi. This is a practically relevant and challenging problem which current methods are able to solve only under strong assumptions on the training set. Here we propose a new method that requires minimal assumptions on the data, and prove reconstruction rates that depend on the number of training points and the noise level. We show that, in the regime of “many” training data, the method is minimax optimal. The proposed method employs a type of convolutional neural networks (U-nets) and empirical risk minimization in order to “fit” the unknown operator. In a nutshell, our approach is based on two ideas: the first is to relate U-nets to multiscale decompositions such as wavelets, thereby linking them to the existing theory, and the second is to use the hierarchical structure of U-nets and the low number of parameters of convolutional neural nets to prove entropy bounds that are practically useful. A significant difference with the existing works on neural networks in nonparametric statistics is that we use them to approximate operators and not functions, which we argue is mathematically more natural and technically more convenient.

A Global Model Approach to Robust Few-Shot SAR Automatic Target Recognition

In real-world scenarios, it may not always be possible to collect hundreds of labeled samples per class for training deep learning-based SAR Automatic Target Recognition (ATR) models. This work specifically tackles the few-shot SAR ATR problem, where only a handful of labeled samples may be available to support the task of interest. Our approach is composed of two stages. In the first, a global representation model is trained via self-supervised learning on a large pool of diverse and unlabeled SAR data. In the second stage, the global model is used as a fixed feature extractor and a classifier is trained to partition the feature space given the few-shot support samples, while simultaneously being calibrated to detect anomalous inputs. Unlike competing approaches which require a pristine labeled dataset for pretraining via meta-learning, our approach learns highly transferable features from unlabeled data that have little-to-no relation to the downstream task. We evaluate our method in standard and extended MSTAR operating conditions and find it to achieve high accuracy and robust out-of-distribution detection in many different few-shot settings. Our results are particularly significant because they show the merit of a global model approach to SAR ATR, which makes minimal assumptions, and provides many axes for extendability.

On the optimal configuration of a square array group testing algorithm

Up to date, only lower and upper bounds for the optimal configuration of a Square Array (A2) Group Testing (GT) algorithm are known. We establish exact analytical formulae and provide a couple of applications of our result. First, we compare the A2 GT scheme to several other classical GT schemes in terms of the gain per specimen attained at optimal configuration. Second, operating under objective Bayesian framework with the loss designed to attain minimum at optimal GT configuration, we suggest the preferred choice of the group size under natural minimal assumptions: the prior information regarding the prevalence suggests that grouping and application of A2 is better than individual testing. The same suggestion is provided for the Minimax strategy.

Non-occurrence of gap for one-dimensional non-autonomous functionals

Let \(F(y){{:}{=}}\int \nolimits _t^TL(s, y(s), y'(s))\,ds\) be a positive functional defined on the space \(W^{1,p}([t,T]; {{\mathbb {R}}}^n)\) (\(p\ge 1\)) of Sobolev functions with, possibly, one or both end point conditions. It is important, especially for the applications, to be able to approximate the infimum of F with the values of F along a sequence of Lipschitz functions satisfying the same boundary condition(s). Sometimes this is not possible, i.e., the so called Lavrentiev phenomenon occurs. This is the case of the seemingly innocent Manià’s Lagrangian \(L(s,y,y')=(y^3-s)^2(y')^6\) and boundary data \(y(0)=0, y(1)=1\); nevertheless in this situation the phenomenon does not occur with just the end point condition \(y(1)=1\). The paper focuses about the different set of conditions that are needed to avoid the Lavrentiev phenomenon for problems depending on the number of end point conditions that are considered. Under minimal assumptions on the (possibly) extended value, Lagrangian, we ensure the non-occurrence of the Lavrentiev phenomenon with just one end point condition, thus extending a milestone result by Alberti and Serra Cassano to non-autonomous case. We then introduce an additional hypothesis, satisfied when the Lagrangian is bounded on bounded sets, in order to ensure the non-occurrence of the phenomenon when dealing with both end point conditions; the result gives some new light even in the autonomous case.

Genetic and environmental influence on resting state networks in young male and female adults: a cartographer mapping study.

We propose a unique, minimal assumption, approach based on variance analyses (compared with standard approaches) to investigate genetic influence on individual differences on the functional connectivity of the brain using 65 monozygotic and 65 dizygotic healthy young adult twin pairs' low-frequency oscillation resting state functional Magnetic Resonance Imaging (fMRI) data from the Human Connectome Project. Overall, we found high number of genetically-influenced functional (GIF) connections involving posterior to posterior brain regions (occipital/temporal/parietal) implicated in low-level processes such as vision, perception, motion, categorization, dorsal/ventral stream visuospatial, and long-term memory processes, as well as high number across midline brain regions (cingulate) implicated in attentional processes, and emotional responses to pain. We found low number of GIF connections involving anterior to anterior/posterior brain regions (frontofrontal > frontoparietal, frontotemporal, frontooccipital) implicated in high-level processes such as working memory, reasoning, emotional judgment, language, and action planning. We found very low number of GIF connections involving subcortical/noncortical networks such as basal ganglia, thalamus, brainstem, and cerebellum. In terms of sex-specific individual differences, individual differences in males were more genetically influenced while individual differences in females were more environmentally influenced in terms of the interplay of interactions of Task positive networks (brain regions involved in various task-oriented processes and attending to and interacting with environment), extended Default Mode Network (a central brain hub for various processes such as internal monitoring, rumination, and evaluation of self and others), primary sensorimotor systems (vision, audition, somatosensory, and motor systems), and subcortical/noncortical networks. There were >8.5-19.1 times more GIF connections in males than females. These preliminary (young adult cohort-specific) findings suggest that individual differences in the resting state brain may be more genetically influenced in males and more environmentally influenced in females; furthermore, standard approaches may suggest that it is more substantially nonadditive genetics, rather than additive genetics, which contribute to the differences in sex-specific individual differences based on this young adult (male and female) specific cohort. Finally, considering the preliminary cohort-specific results, based on standard approaches, environmental influences on individual differences may be substantially greater than that of genetics, for either sex, frontally and brain-wide. [Correction added on 10 May 2023, after first online publication: added: functional Magnetic Resonance Imaging. Added: individual differences in, twice. Added statement between furthermore … based on standard approaches.].

Evolution of self-organised division of labour driven by stigmergy in leaf-cutter ants

Social insects owe their widespread success to their ability to efficiently coordinate behaviour to carry out complex tasks. Several leaf-cutter ant species employ an advanced type of division of labour known as task partitioning, where the task of retrieving leaves is distributed between workers that cut and drop and those that collect the fallen leaves. It is not entirely clear how such highly coordinated behaviour can evolve, as it would seem to require the simultaneous mutations of multiple traits during the same generation. Here, we use an agent-based simulation model to show how task partitioning in leaf-cutter ants can gradually evolve by exploiting stigmergy (indirect coordination through the environment) through gravity (leaves falling from the treetop on the ground forming a cache). Our simple model allows independent variation in two core behavioural dimensions: the tendency to drop leaves and the tendency to pick up dropped leaves. Task partitioning readily evolves even under these minimal assumptions through adaptation to an arboreal environment where traveling up and down the tree is costly. Additionally, we analyse ant movement dynamics to demonstrate how the ants achieve efficient task allocation through task switching and negative feedback control.

Comparing T cell receptor repertoires using optimal transport.

The complexity of entire T cell receptor (TCR) repertoires makes their comparison a difficult but important task. Current methods of TCR repertoire comparison can incur a high loss of distributional information by considering overly simplistic sequence- or repertoire-level characteristics. Optimal transport methods form a suitable approach for such comparison given some distance or metric between values in the sample space, with appealing theoretical and computational properties. In this paper we introduce a nonparametric approach to comparing empirical TCR repertoires that applies the Sinkhorn distance, a fast, contemporary optimal transport method, and a recently-created distance between TCRs called TCRdist. We show that our methods identify meaningful differences between samples from distinct TCR distributions for several case studies, and compete with more complicated methods despite minimal modeling assumptions and a simpler pipeline.

Sup-norm adaptive drift estimation for multivariate nonreversible diffusions

We consider the question of estimating the drift for a large class of ergodic multivariate and possibly nonreversible diffusion processes, based on continuous observations, in sup-norm loss. Nonparametric classes of smooth functions of unknown order are considered, and we suggest an adaptive approach which allows to construct drift estimators attaining optimal sup-norm rates of convergence. Reversibility structures and related functional inequalities are known to be key tools for these estimation problems. We can discard such restrictions by making use of mixing properties which are satisfied for the very general class of processes under consideration. Analysing diffusions, the scalar case is very distinct from the general multivariate setting. Therefore, we treat scalar and multivariate processes separately which leads to in several aspects improved univariate results. While we consider drift estimation on bounded domains for exponentially β-mixing multivariate processes, for scalar diffusion processes we work under minimal assumptions that allow estimation of unbounded drift terms over the entire real line, and we provide classical minimax results (including lower bounds) which cannot be obtained under state-of-the-art conditions in the multivariate case. In addition, we prove a Donsker theorem for the classical kernel estimator of the invariant density in the scalar setting and establish its semiparametric efficiency.

Testing for the rank of a covariance operator

How can we discern whether the covariance operator of a stochastic process is of reduced rank, and if so, what its precise rank is? And how can we do so at a given level of confidence? This question is central to a great deal of methods for functional data, which require low-dimensional representations whether by functional PCA or other methods. The difficulty is that the determination is to be made on the basis of i.i.d. replications of the process observed discretely and with measurement error contamination. This adds a ridge to the empirical covariance, obfuscating the underlying dimension. We build a matrix-completion inspired test statistic that circumvents this issue by measuring the best possible least square fit of the empirical covariance’s off-diagonal elements, optimised over covariances of given finite rank. For a fixed grid of sufficiently large size, we determine the statistic’s asymptotic null distribution as the number of replications grows. We then use it to construct a bootstrap implementation of a stepwise testing procedure controlling the familywise error rate corresponding to the collection of hypotheses formalising the question at hand. Under minimal regularity assumptions, we prove that the procedure is consistent and that its bootstrap implementation is valid. The procedure circumvents smoothing and associated smoothing parameters, is indifferent to measurement error heteroskedasticity, and does not assume a low-noise regime. An extensive simulation study reveals an excellent practical performance, stably across a wide range of settings and the procedure is further illustrated by means of two data analyses.

Universal Learning Waveform Selection Strategies for Adaptive Target Tracking

Online selection of optimal waveforms for target tracking with active sensors has long been a problem of interest. Many conventional solutions utilize an estimation-theoretic interpretation, in which a waveform-specific Cramér–Rao lower bound on measurement error is used to select the optimal waveform for each tracking step. However, this approach is only valid in the high SNR regime, and requires a rather restrictive set of assumptions regarding the target motion and measurement models. Furthermore, due to computational concerns, many traditional approaches are limited to near-term, or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">myopic</i> , optimization, even though radar scenes exhibit strong temporal correlation. More recently, reinforcement learning has been proposed for waveform selection, in which the problem is framed as a Markov decision process, allowing for long-term planning. However, a major limitation of reinforcement learning is that the memory length of the underlying Markov process is often unknown for realistic target and channel dynamics, and a more general framework is desirable. This work develops a universal sequential waveform selection scheme which asymptotically achieves Bellman optimality in any radar scene, which can be modeled as a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$U{\text{th}}$</tex-math></inline-formula> -order Markov process for a finite, but unknown, integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$U$</tex-math></inline-formula> . Our approach is based on well-established tools from the field of universal source coding, where a stationary source is parsed into variable length phrases in order to build a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">context-tree</i> , which is used as a probabilistic model for the scene’s behavior. We show that an algorithm based on a multialphabet version of the context-tree weighting method can be used to optimally solve a broad class of waveform-agile tracking problems while making minimal assumptions about the environment’s behavior.

Using mechanistic model-based inference to understand and project epidemic dynamics with time-varying contact and vaccination rates

Epidemiological models range in complexity from relatively simple statistical models that make minimal assumptions about the variables driving epidemic dynamics to more mechanistic models that include effects such as vaccine-derived and infection-derived immunity, population structure and heterogeneity. The former are often fitted to data in real-time and used for short-term forecasting, while the latter are more suitable for comparing longer-term scenarios under differing assumptions about control measures or other factors. Here, we present a mechanistic model of intermediate complexity that can be fitted to data in real-time but is also suitable for investigating longer-term dynamics. Our approach provides a bridge between primarily empirical approaches to forecasting and assumption-driven scenario models. The model was developed as a policy advice tool for New Zealand’s 2021 outbreak of the Delta variant of SARS-CoV-2 and includes the effects of age structure, non-pharmaceutical interventions, and the ongoing vaccine rollout occurring during the time period studied. We use an approximate Bayesian computation approach to infer the time-varying transmission coefficient from real-time data on reported cases. We then compare projections of the model with future, out-of-sample data. We find that this approach produces a good fit with in-sample data and reasonable forward projections given the inherent limitations of predicting epidemic dynamics during periods of rapidly changing policy and behaviour. Results from the model helped inform the New Zealand Government’s policy response throughout the outbreak.

Markov risk mappings and risk-sensitive optimal prediction

We formulate a probabilistic Markov property in discrete time under a dynamic risk framework with minimal assumptions. This is useful for recursive solutions to risk-sensitive versions of dynamic optimisation problems such as optimal prediction, where at each stage the recursion depends on the whole future. The property holds for standard measures of risk used in practice, and is formulated in several equivalent versions including a representation via acceptance sets, a strong version, and a dual representation.

A Multifidelity Monte Carlo Method for Realistic Computational Budgets

A method for the multifidelity Monte Carlo (MFMC) estimation of statistical quantities is proposed which is applicable to computational budgets of any size. Based on a sequence of optimization problems each with a globally minimizing closed-form solution, this method extends the usability of a well known MFMC algorithm, recovering it when the computational budget is large enough. Theoretical results verify that the proposed approach is at least as optimal as its namesake and retains the benefits of multifidelity estimation with minimal assumptions on the budget or amount of available data, providing a notable reduction in variance over simple Monte Carlo estimation.

Uniform in time solutions for a chemotaxis with potential consumption model

In this work we investigate the following chemo-attraction with consumption model in bounded domains of RN (N=1,2,3): ∂tu−Δu=−∇⋅(u∇v),∂tv−Δv=−usv where s≥1, endowed with isolated boundary conditions and initial conditions for (u,v). The main novelty in the model is the nonlinear potential consumption term usv. Through the convergence of solutions of an adequate truncated model, two main results are established; existence of uniform in time weak solutions in 3D domains, and uniqueness and regularity in 2D (or 1D) domains. Both results are proved imposing minimal regularity assumptions on the boundary of the domain.

Implicit Elastoplastic Finite Element Analysis of a Wheel Bearing Shaft Clinching Process Using the Multi-Body Function

An implicit, elastoplastic, finite element method (FEM) with multi-body treatment function was applied to accurately analyze the real-world shaft clinching of a duplex-pair tapered roller (DPTR) wheel-bearing unit (WBU) under minimal assumptions during modeling. The inner races were viewed as elastoplastically deformable and were fitted to the hub shaft before clinching by imposing a thermal load reflecting the mechanical load of press-fitting. The forming roller (i.e., the power source) was considered to be force-prescribed, similar to the approach on real shop floors. The predictions focused on the homogenizing stage, during which the two inner races bear the preload. At this time, local plastic deformation occurred at the end of the hub shaft and in the armpit area and the cavity was either maintained or enlarged. The predicted cavity size in case of force-prescribed forming roller increased, compared with the velocity-prescribed forming roller. The residual stress became axisymmetric and was divided into two parts by the cavity. These findings allow engineers to control the pre-stresses imparted to the inner races of tapered roller bearing assemblies.

Addressing positivity violations in causal effect estimation using Gaussian process priors.

In observational studies, causal inference relies on several key identifying assumptions. One identifiability condition is the positivity assumption, which requires the probability of treatment be bounded away from 0 and 1. That is, for every covariate combination, it should be possible to observe both treated and control subjects the covariate distributions should overlap between treatment arms. If the positivity assumption is violated, population-level causal inference necessarily involves some extrapolation. Ideally, a greater amount of uncertainty about the causal effect estimate should be reflected in such situations. With that goal in mind, we construct a Gaussian process model for estimating treatment effects in the presence of practical violations of positivity. Advantages of our method include minimal distributional assumptions, a cohesive model for estimating treatment effects, and more uncertainty associated with areas in the covariate space where there is less overlap. We assess the performance of our approach with respect to bias and efficiency using simulation studies. The method is then applied to a study of critically ill female patients to examine the effect of undergoing right heart catheterization.

Robots, Trade, and Luddism: A Sufficient Statistic Approach to Optimal Technology Regulation

Abstract Technological change, from the advent of robots to expanded trade opportunities, creates winners and losers. How should government policy respond? We provide a general theory of optimal technology regulation in a second-best world, with rich heterogeneity across households, linear taxes on the subset of firms affected by technological change, and a non-linear tax on labour income. Our first set of results consists of optimal tax formulas, with minimal structural assumptions, involving sufficient statistics that can be implemented using evidence on the distributional impact of new technologies, such as robots and trade. Our final results are comparative static exercises illustrating, among other things, that while distributional concerns create a rationale for non-zero taxes on robots and trade, the magnitude of these taxes may decrease as the process of automation and globalization deepens and inequality increases.

Constrained Deep Reinforcement Based Functional Split Optimization in Virtualized RANs

In virtualized radio access network (vRAN), the base station (BS) functions are decomposed into virtualized components that can be hosted at the centralized unit or distributed units through functional splits. Such flexibility has many benefits; however, it also requires solving the problem of finding the optimal splits of functions of the BSs in such a way that minimizes the total network cost. The underlying vRAN system is complex and precise modelling of it is not trivial. Formulating the functional split problem to minimize the cost results in a combinatorial problem that is provably NP-hard, and solving it is computationally expensive. In this paper, a constrained deep reinforcement learning (RL) approach is proposed to solve the problem with minimal assumptions about the underlying system. Since in deep RL, the action selection is the outcome of inference of a neural network, it can be done in real-time while training to update the neural networks can be done in the background. However, since the problem is combinatorial, even for a small number of functions, the action space of the RL problem becomes large. Therefore, to deal with such a large action space, a chain rule-based stochastic policy is exploited in which a long short-term memory (LSTM) network-based sequence-to-sequence model is applied to estimate the policy that is selecting the functional split actions. However, the utilized policy is still limited to an unconstrained problem, and each split decision is bounded by vRAN’s constraint requirements. Hence, a constrained policy gradient method is leveraged to train and guide the policy toward constraint satisfaction. Further, a search strategy by greedy decoding or temperature sampling is utilized to improve the optimality performance at the test time. Simulations are performed to evaluate the performance of the proposed solution using synthetic and real network datasets. Our numerical results show that the proposed RL solution architecture successfully learns to make optimal functional split decisions with the accuracy of the solution is up to 0.05% of the optimality gap. Moreover, our solution can achieve considerable cost savings compared to C-RAN or D-RAN systems and a faster computational time than the optimal baseline.

ZAP:Z-Value Adaptive Procedures for False Discovery Rate Control with Side Information

AbstractAdaptive multiple testing with covariates is an important research direction that has gained major attention in recent years. It has been widely recognised that leveraging side information provided by auxiliary covariates can improve the power of false discovery rate (FDR) procedures. Currently, most such procedures are devised with p-values as their main statistics. However, for two-sided hypotheses, the usual data processing step that transforms the primary statistics, known as p-values, into p-values not only leads to a loss of information carried by the main statistics, but can also undermine the ability of the covariates to assist with the FDR inference. We develop a p-value based covariate-adaptive (ZAP) methodology that operates on the intact structural information encoded jointly by the p-values and covariates. It seeks to emulate the oracle p-value procedure via a working model, and its rejection regions significantly depart from those of the p-value adaptive testing approaches. The key strength of ZAP is that the FDR control is guaranteed with minimal assumptions, even when the working model is misspecified. We demonstrate the state-of-the-art performance of ZAP using both simulated and real data, which shows that the efficiency gain can be substantial in comparison with p-value-based methods. Our methodology is implemented in the R package zap.

Sparse Monte Carlo Method for Nonlocal Diffusion Problems

A class of evolution equations with nonlocal diffusion is considered in this work. These are integro-differential equations arising as models of propagation phenomena in continuum media with nonlocal interactions including neural tissue, porous media, peridynamics, and models with fractional diffusion, as well as continuum limits of interacting dynamical systems. The principal challenge of numerical integration of nonlocal systems stems from the lack of spatial regularity of the data and solutions intrinsic to nonlocal models. To overcome this problem we propose a semidiscrete numerical scheme based on the combination of sparse Monte Carlo and discontinuous Galerkin methods. Our method requires minimal assumptions on the regularity of the data. In particular, the kernel of the nonlocal diffusivity is assumed to be a square integrable function and may be singular or discontinuous. An important feature of our method is the use of sparsity. Sparse sampling of points in the Monte Carlo approximation of the nonlocal term allows us to use fewer discretization points without compromising the accuracy. For kernels with singularities, more points are selected automatically in the regions near the singularities. We prove convergence of the numerical method and estimate the rate of convergence. There are two principal ingredients in the error of the numerical method related to the use of Monte Calro and Galerkin approximations, respectively. We analyze both errors. Two representative examples of discontinuous kernels are presented. The first example features a kernel with a singularity, while the kernel in the second example experiences jump discontinuity. We show how the information about the singularity in the former case and the geometry of the discontinuity set in the latter translate into the rate of convergence of the numerical procedure. In addition, we illustrate the rate of convergence estimate with a numerical example of an initial value problem, for which an explicit analytic solution is available. Numerical results are consistent with analytical estimates.