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The precision matrix that encodes conditional linear dependency relations among a set of variables forms an important object of interest in multivariate analysis. Sparse estimation procedures for precision matrices such as the graphical lasso (Glasso) gained popularity as they facilitate interpretability, thereby separating pairs of variables that are conditionally dependent from those that are independent (given all other variables). Glasso lacks, however, robustness to outliers. To overcome this problem, one typically applies a robust plug-in procedure where the Glasso is computed from a robust covariance estimate instead of the sample covariance, thereby providing protection against outliers. In this paper, we study such estimators theoretically, by deriving and comparing their influence function, sensitivity curves and asymptotic variances.

We study the probabilistic sampling of a random variable, in which the variable is sampled only if it falls outside a given set, which is called the silence set. This helps us to understand optimal event-based sampling for the special case of IID random processes, and also to understand the design of a sub-optimal scheme for other cases. We consider the design of this probabilistic sampling for a scalar, log-concave random variable, to minimize either the mean square estimation error, or the mean absolute estimation error. We show that the optimal silence interval: (i) is essentially unique, and (ii) is the limit of an iterative procedure of centering. Further we show through numerical experiments that super-level intervals seem to be remarkably near-optimal for mean square estimation. Finally we use the Gauss inequality for scalar unimodal densities, to show that probabilistic sampling gives a mean square distortion that is less than a third of the distortion incurred by periodic sampling, if the average sampling rate is between 0.3 and 0.9 samples per tick.

We study optimal transport-based distributionally robust optimization problems where a fictitious adversary, often envisioned as nature, can choose the distribution of the uncertain problem parameters by reshaping a prescribed reference distribution at a finite transportation cost. In this framework, we show that robustification is intimately related to various forms of variation and Lipschitz regularization even if the transportation cost function fails to be (some power of) a metric. We also derive conditions for the existence and the computability of a Nash equilibrium between the decision-maker and nature, and we demonstrate numerically that nature's Nash strategy can be viewed as a distribution that is supported on remarkably deceptive adversarial samples. Finally, we identify practically relevant classes of optimal transport-based distributionally robust optimization problems that can be addressed with efficient gradient descent algorithms even if the loss function or the transportation cost function are nonconvex (but not both at the same time).

In this paper, we provide an information-theoretic perspective on Variance-Invariance-Covariance Regularization (VICReg) for self-supervised learning. To do so, we first demonstrate how information-theoretic quantities can be obtained for deterministic networks as an alternative to the commonly used unrealistic stochastic networks assumption. Next, we relate the VICReg objective to mutual information maximization and use it to highlight the underlying assumptions of the objective. Based on this relationship, we derive a generalization bound for VICReg, providing generalization guarantees for downstream supervised learning tasks and present new self-supervised learning methods, derived from a mutual information maximization objective, that outperform existing methods in terms of performance. This work provides a new information-theoretic perspective on self-supervised learning and Variance-Invariance-Covariance Regularization in particular and guides the way for improved transfer learning via information-theoretic self-supervised learning objectives.

Randomized Controlled Trials (RCTs) represent a gold standard when developing policy guidelines. However, RCTs are often narrow, and lack data on broader populations of interest. Causal effects in these populations are often estimated using observational datasets, which may suffer from unobserved confounding and selection bias. Given a set of observational estimates (e.g. from multiple studies), we propose a meta-algorithm that attempts to reject observational estimates that are biased. We do so using validation effects, causal effects that can be inferred from both RCT and observational data. After rejecting estimators that do not pass this test, we generate conservative confidence intervals on the extrapolated causal effects for subgroups not observed in the RCT. Under the assumption that at least one observational estimator is asymptotically normal and consistent for both the validation and extrapolated effects, we provide guarantees on the coverage probability of the intervals output by our algorithm. To facilitate hypothesis testing in settings where causal effect transportation across datasets is necessary, we give conditions under which a doubly-robust estimator of group average treatment effects is asymptotically normal, even when flexible machine learning methods are used for estimation of nuisance parameters. We illustrate the properties of our approach on semi-synthetic and real world datasets, and show that it compares favorably to standard meta-analysis techniques.

Sampling from Gibbs distributions $p(x) \propto \exp(-V(x)/\varepsilon)$ and computing their log-partition function are fundamental tasks in statistics, machine learning, and statistical physics. However, while efficient algorithms are known for convex potentials $V$, the situation is much more difficult in the non-convex case, where algorithms necessarily suffer from the curse of dimensionality in the worst case. For optimization, which can be seen as a low-temperature limit of sampling, it is known that smooth functions $V$ allow faster convergence rates. Specifically, for $m$-times differentiable functions in $d$ dimensions, the optimal rate for algorithms with $n$ function evaluations is known to be $O(n^{-m/d})$, where the constant can potentially depend on $m, d$ and the function to be optimized. Hence, the curse of dimensionality can be alleviated for smooth functions at least in terms of the convergence rate. Recently, it has been shown that similarly fast rates can also be achieved with polynomial runtime $O(n^{3.5})$, where the exponent $3.5$ is independent of $m$ or $d$. Hence, it is natural to ask whether similar rates for sampling and log-partition computation are possible, and whether they can be realized in polynomial time with an exponent independent of $m$ and $d$. We show that the optimal rates for sampling and log-partition computation are sometimes equal and sometimes faster than for optimization. We then analyze various polynomial-time sampling algorithms, including an extension of a recent promising optimization approach, and find that they sometimes exhibit interesting behavior but no near-optimal rates. Our results also give further insights on the relation between sampling, log-partition, and optimization problems.

Photonic integrated circuits are facilitating the development of optical neural networks, which have the potential to be both faster and more energy efficient than their electronic counterparts since optical signals are especially well-suited for implementing matrix multiplications. However, accurate programming of photonic chips for optical matrix multiplication remains a difficult challenge. Here, we describe both simple analytical models and data-driven models for offline training of optical matrix multipliers. We train and evaluate the models using experimental data obtained from a fabricated chip featuring a Mach-Zehnder interferometer mesh implementing 3-by-3 matrix multiplication. The neural network-based models outperform the simple physics-based models in terms of prediction error. Furthermore, the neural network models are also able to predict the spectral variations in the matrix weights for up to 100 frequency channels covering the C-band. The use of neural network models for programming the chip for optical matrix multiplication yields increased performance on multiple machine learning tasks.

The conditional Aalen--Johansen estimator, a general-purpose non-parametric estimator of conditional state occupation probabilities, is introduced. The estimator is applicable for any finite-state jump process and supports conditioning on external as well as internal covariate information. The conditioning feature permits for a much more detailed analysis of the distributional characteristics of the process. The estimator reduces to the conditional Kaplan--Meier estimator in the special case of a survival model and also englobes other, more recent, landmark estimators when covariates are discrete. Strong uniform consistency and asymptotic normality are established under lax moment conditions on the multivariate counting process, allowing in particular for an unbounded number of transitions.

Given univariate random variables $Y_1, \ldots, Y_n$ with the $\text{Uniform}(\theta_0 - 1, \theta_0 + 1)$ distribution, the sample midrange $\frac{Y_{(n)}+Y_{(1)}}{2}$ is the MLE for $\theta_0$ and estimates $\theta_0$ with error of order $1/n$, which is much smaller compared with the $1/\sqrt{n}$ error rate of the usual sample mean estimator. However, the sample midrange performs poorly when the data has say the Gaussian $N(\theta_0, 1)$ distribution, with an error rate of $1/\sqrt{\log n}$. In this paper, we propose an estimator of the location $\theta_0$ with a rate of convergence that can, in many settings, adapt to the underlying distribution which we assume to be symmetric around $\theta_0$ but is otherwise unknown. When the underlying distribution is compactly supported, we show that our estimator attains a rate of convergence of $n^{-\frac{1}{\alpha}}$ up to polylog factors, where the rate parameter $\alpha$ can take on any value in $(0, 2]$ and depends on the moments of the underlying distribution. Our estimator is formed by the $\ell^\gamma$-center of the data, for a $\gamma\geq2$ chosen in a data-driven way -- by minimizing a criterion motivated by the asymptotic variance. Our approach can be directly applied to the regression setting where $\theta_0$ is a function of observed features and motivates the use of $\ell^\gamma$ loss function for $\gamma > 2$ in certain settings.

When estimating an effect of an action with a randomized or observational study, that study is often not a random sample of the desired target population. Instead, estimates from that study can be transported to the target population. However, transportability methods generally rely on a positivity assumption, such that all relevant covariate patterns in the target population are also observed in the study sample. Strict eligibility criteria, particularly in the context of randomized trials, may lead to violations of this assumption. Two common approaches to address positivity violations are restricting the target population and restricting the relevant covariate set. As neither of these restrictions are ideal, we instead propose a synthesis of statistical and simulation models to address positivity violations. We propose corresponding g-computation and inverse probability weighting estimators. The restriction and synthesis approaches to addressing positivity violations are contrasted with a simulation experiment and an illustrative example in the context of sexually transmitted infection testing. In both cases, the proposed model synthesis approach accurately addressed the original research question when paired with a thoughtfully selected simulation model. Neither of the restriction approaches were able to accurately address the motivating question. As public health decisions must often be made with imperfect target population information, model synthesis is a viable approach given a combination of empirical data and external information based on the best available knowledge.