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Bundle adjustment (BA) is the standard way to optimise camera poses and to produce sparse representations of a scene. However, as the number of camera poses and features grows, refinement through bundle adjustment becomes inefficient. Inspired by global motion averaging methods, we propose a new bundle adjustment objective which does not rely on image features' reprojection errors yet maintains precision on par with classical BA. Our method averages over relative motions while implicitly incorporating the contribution of the structure in the adjustment. To that end, we weight the objective function by local hessian matrices - a by-product of local bundle adjustments performed on relative motions (e.g., pairs or triplets) during the pose initialisation step. Such hessians are extremely rich as they encapsulate both the features' random errors and the geometric configuration between the cameras. These pieces of information propagated to the global frame help to guide the final optimisation in a more rigorous way. We argue that this approach is an upgraded version of the motion averaging approach and demonstrate its effectiveness on both photogrammetric datasets and computer vision benchmarks.

In practice, the use of rounding is ubiquitous. Although researchers have looked at the implications of rounding continuous random variables, rounding may be applied to functions of discrete random variables as well. For example, to infer on suicide excess deaths after a national emergency, authorities may provide a rounded average of deaths before and after the emergency started. Suicide rates tend to be relatively low around the world and such rounding may seriously affect inference on the change of suicide rate. In this paper, we study the scenario when a rounded to nearest integer average is used as a proxy for a non-negative discrete random variable. Specifically, our interest is in drawing inference on a parameter from the pmf of Y , when we get U = n[Y /n] as a proxy for Y . The probability generating function of U , E(U ), and Var(U ) capture the effect of the coarsening of the support of Y . Also, moments and estimators of distribution parameters are explored for some special cases. We show that under certain conditions, there is little impact from rounding. However, we also find scenarios where rounding can significantly affect statistical inference as demonstrated in three examples. The simple methods we propose are able to partially counter rounding error effects. While for some probability distributions it may be difficult to derive maximum likelihood estimators as a function of U , we provide a framework to obtain an estimator numerically.

Sensitivity analysis for the unconfoundedness assumption is a crucial component of observational studies. The marginal sensitivity model has become increasingly popular for this purpose due to its interpretability and mathematical properties. As the basis of $L^\infty$-sensitivity analysis, it assumes the logit difference between the observed and full data propensity scores is uniformly bounded. In this article, we introduce a new $L^2$-sensitivity analysis framework which is flexible, sharp and efficient. We allow the strength of unmeasured confounding to vary across units and only require it to be bounded marginally for partial identification. We derive analytical solutions to the optimization problems under our $L^2$-models, which can be used to obtain sharp bounds for the average treatment effect (ATE). We derive efficient influence functions and use them to develop efficient one-step estimators in both analyses. We show that multiplier bootstrap can be applied to construct simultaneous confidence bands for our ATE bounds. In a real-data study, we demonstrate that $L^2$-analysis relaxes the interpretation of $L^\infty$-analysis and provides a much more reliable calibration process using observed covariates. Finally, we provide an extension of our theoretical results to the conditional average treatment effect (CATE).

In this paper we propose an estimator of spot covariance matrix which ensure symmetric positive semi-definite estimations. The proposed estimator relies on a suitable modification of the Fourier covariance estimator in Malliavin and Mancino (2009) and it is consistent for suitable choices of the weighting kernel. The accuracy and the ability of the estimator to produce positive semi-definite covariance matrices is evaluated with an extensive numerical study, in comparison with the competitors present in the literature. The results of the simulation study are confirmed under many scenarios, that consider the dimensionality of the problem, the asynchronicity of data and the presence of several specification of market microstructure noise.

To estimate causal effects, analysts performing observational studies in health settings utilize several strategies to mitigate bias due to confounding by indication. There are two broad classes of approaches for these purposes: use of confounders and instrumental variables (IVs). Because such approaches are largely characterized by untestable assumptions, analysts must operate under an indefinite paradigm that these methods will work imperfectly. In this tutorial, we formalize a set of general principles and heuristics for estimating causal effects in the two approaches when the assumptions are potentially violated. This crucially requires reframing the process of observational studies as hypothesizing potential scenarios where the estimates from one approach are less inconsistent than the other. While most of our discussion of methodology centers around the linear setting, we touch upon complexities in non-linear settings and flexible procedures such as target minimum loss-based estimation (TMLE) and double machine learning (DML). To demonstrate the application of our principles, we investigate the use of donepezil off-label for mild cognitive impairment (MCI). We compare and contrast results from confounder and IV methods, traditional and flexible, within our analysis and to a similar observational study and clinical trial.

We investigate the high-dimensional linear regression problem in situations where there is noise correlated with Gaussian covariates. In regression models, the phenomenon of the correlated noise is called endogeneity, which is due to unobserved variables and others, and has been a major problem setting in causal inference and econometrics. When the covariates are high-dimensional, it has been common to assume sparsity on the true parameters and estimate them using regularization, even with the endogeneity. However, when sparsity does not hold, it has not been well understood to control the endogeneity and high dimensionality simultaneously. In this paper, we demonstrate that an estimator without regularization can achieve consistency, i.e., benign overfitting, under certain assumptions on the covariance matrix. Specifically, we show that the error of this estimator converges to zero when covariance matrices of the correlated noise and instrumental variables satisfy a condition on their eigenvalues. We consider several extensions to relax these conditions and conduct experiments to support our theoretical findings. As a technical contribution, we utilize the convex Gaussian minimax theorem (CGMT) in our dual problem and extend the CGMT itself.

This paper reconsiders end-to-end learning approaches to the Optimal Power Flow (OPF). Existing methods, which learn the input/output mapping of the OPF, suffer from scalability issues due to the high dimensionality of the output space. This paper first shows that the space of optimal solutions can be significantly compressed using principal component analysis (PCA). It then proposes Compact Learning, a new method that learns in a subspace of the principal components before translating the vectors into the original output space. This compression reduces the number of trainable parameters substantially, improving scalability and effectiveness. Compact Learning is evaluated on a variety of test cases from the PGLib with up to 30,000 buses. The paper also shows that the output of Compact Learning can be used to warm-start an exact AC solver to restore feasibility, while bringing significant speed-ups.

Making causal inferences from observational studies can be challenging when confounders are missing not at random. In such cases, identifying causal effects is often not guaranteed. Motivated by a real example, we consider a treatment-independent missingness assumption under which we establish the identification of causal effects when confounders are missing not at random. We propose a weighted estimating equation (WEE) approach for estimating model parameters and introduce three estimators for the average causal effect, based on regression, propensity score weighting, and doubly robust estimation. We evaluate the performance of these estimators through simulations, and provide a real data analysis to illustrate our proposed method.

This PhD thesis contains several contributions to the field of statistical causal modeling. Statistical causal models are statistical models embedded with causal assumptions that allow for the inference and reasoning about the behavior of stochastic systems affected by external manipulation (interventions). This thesis contributes to the research areas concerning the estimation of causal effects, causal structure learning, and distributionally robust (out-of-distribution generalizing) prediction methods. We present novel and consistent linear and non-linear causal effects estimators in instrumental variable settings that employ data-dependent mean squared prediction error regularization. Our proposed estimators show, in certain settings, mean squared error improvements compared to both canonical and state-of-the-art estimators. We show that recent research on distributionally robust prediction methods has connections to well-studied estimators from econometrics. This connection leads us to prove that general K-class estimators possess distributional robustness properties. We, furthermore, propose a general framework for distributional robustness with respect to intervention-induced distributions. In this framework, we derive sufficient conditions for the identifiability of distributionally robust prediction methods and present impossibility results that show the necessity of several of these conditions. We present a new structure learning method applicable in additive noise models with directed trees as causal graphs. We prove consistency in a vanishing identifiability setup and provide a method for testing substructure hypotheses with asymptotic family-wise error control that remains valid post-selection. Finally, we present heuristic ideas for learning summary graphs of nonlinear time-series models.

We introduce an effective model to overcome the problem of mode collapse when training Generative Adversarial Networks (GAN). Firstly, we propose a new generator objective that finds it better to tackle mode collapse. And, we apply an independent Autoencoders (AE) to constrain the generator and consider its reconstructed samples as "real" samples to slow down the convergence of discriminator that enables to reduce the gradient vanishing problem and stabilize the model. Secondly, from mappings between latent and data spaces provided by AE, we further regularize AE by the relative distance between the latent and data samples to explicitly prevent the generator falling into mode collapse setting. This idea comes when we find a new way to visualize the mode collapse on MNIST dataset. To the best of our knowledge, our method is the first to propose and apply successfully the relative distance of latent and data samples for stabilizing GAN. Thirdly, our proposed model, namely Generative Adversarial Autoencoder Networks (GAAN), is stable and has suffered from neither gradient vanishing nor mode collapse issues, as empirically demonstrated on synthetic, MNIST, MNIST-1K, CelebA and CIFAR-10 datasets. Experimental results show that our method can approximate well multi-modal distribution and achieve better results than state-of-the-art methods on these benchmark datasets. Our model implementation is published here: //github.com/tntrung/gaan