The latent class model is a powerful tool for identifying latent classes within populations that share common characteristics for categorical data in social, psychological, and behavioral sciences. In this article, we propose two new algorithms to estimate a latent class model for categorical data. Our algorithms are developed by using a newly defined regularized Laplacian matrix calculated from the response matrix. We provide theoretical convergence rates of our algorithms by considering a sparsity parameter and show that our algorithms stably yield consistent latent class analysis under mild conditions. Additionally, we propose a metric to capture the strength of latent class analysis and several procedures designed based on this metric to infer how many latent classes one should use for real-world categorical data. The efficiency and accuracy of our algorithms are verified by extensive simulated experiments, and we further apply our algorithms to real-world categorical data with promising results.

This paper endeavors to augment the robustness of offline reinforcement learning (RL) in scenarios laden with heavy-tailed rewards, a prevalent circumstance in real-world applications. We propose two algorithmic frameworks, ROAM and ROOM, for robust off-policy evaluation (OPE) and offline policy optimization (OPO), respectively. Central to our frameworks is the strategic incorporation of the median-of-means method with offline RL, enabling straightforward uncertainty estimation for the value function estimator. This not only adheres to the principle of pessimism in OPO but also adeptly manages heavy-tailed rewards. Theoretical results and extensive experiments demonstrate that our two frameworks outperform existing methods on the logged dataset exhibits heavy-tailed reward distributions.

Fair Principal Component Analysis (PCA) is a problem setting where we aim to perform PCA while making the resulting representation fair in that the projected distributions, conditional on the sensitive attributes, match one another. However, existing approaches to fair PCA have two main problems: theoretically, there has been no statistical foundation of fair PCA in terms of learnability; practically, limited memory prevents us from using existing approaches, as they explicitly rely on full access to the entire data. On the theoretical side, we rigorously formulate fair PCA using a new notion called \emph{probably approximately fair and optimal} (PAFO) learnability. On the practical side, motivated by recent advances in streaming algorithms for addressing memory limitation, we propose a new setting called \emph{fair streaming PCA} along with a memory-efficient algorithm, fair noisy power method (FNPM). We then provide its {\it statistical} guarantee in terms of PAFO-learnability, which is the first of its kind in fair PCA literature. Lastly, we verify the efficacy and memory efficiency of our algorithm on real-world datasets.

Decision analysis deals with modeling and enhancing decision processes. A principal challenge in improving behavior is in obtaining a transparent description of existing behavior in the first place. In this paper, we develop an expressive, unifying perspective on inverse decision modeling: a framework for learning parameterized representations of sequential decision behavior. First, we formalize the forward problem (as a normative standard), subsuming common classes of control behavior. Second, we use this to formalize the inverse problem (as a descriptive model), generalizing existing work on imitation/reward learning -- while opening up a much broader class of research problems in behavior representation. Finally, we instantiate this approach with an example (inverse bounded rational control), illustrating how this structure enables learning (interpretable) representations of (bounded) rationality -- while naturally capturing intuitive notions of suboptimal actions, biased beliefs, and imperfect knowledge of environments.

The Wasserstein distance from optimal mass transport (OMT) is a powerful mathematical tool with numerous applications that provides a natural measure of the distance between two probability distributions. Several methods to incorporate OMT into widely used probabilistic models, such as Gaussian or Gaussian mixture, have been developed to enhance the capability of modeling complex multimodal densities of real datasets. However, very few studies have explored the OMT problems in a reproducing kernel Hilbert space (RKHS), wherein the kernel trick is utilized to avoid the need to explicitly map input data into a high-dimensional feature space. In the current study, we propose a Wasserstein-type metric to compute the distance between two Gaussian mixtures in a RKHS via the kernel trick, i.e., kernel Gaussian mixture models.

In this paper, we investigate the impact of numerical instability on the reliability of sampling, density evaluation, and evidence lower bound (ELBO) estimation in variational flows. We first empirically demonstrate that common flows can exhibit a catastrophic accumulation of error: the numerical flow map deviates significantly from the exact map -- which affects sampling -- and the numerical inverse flow map does not accurately recover the initial input -- which affects density and ELBO computations. Surprisingly though, we find that results produced by flows are often accurate enough for applications despite the presence of serious numerical instability. In this work, we treat variational flows as dynamical systems, and leverage shadowing theory to elucidate this behavior via theoretical guarantees on the error of sampling, density evaluation, and ELBO estimation. Finally, we develop and empirically test a diagnostic procedure that can be used to validate results produced by numerically unstable flows in practice.

We propose novel statistics which maximise the power of a two-sample test based on the Maximum Mean Discrepancy (MMD), by adapting over the set of kernels used in defining it. For finite sets, this reduces to combining (normalised) MMD values under each of these kernels via a weighted soft maximum. Exponential concentration bounds are proved for our proposed statistics under the null and alternative. We further show how these kernels can be chosen in a data-dependent but permutation-independent way, in a well-calibrated test, avoiding data splitting. This technique applies more broadly to general permutation-based MMD testing, and includes the use of deep kernels with features learnt using unsupervised models such as auto-encoders. We highlight the applicability of our MMD-FUSE test on both synthetic low-dimensional and real-world high-dimensional data, and compare its performance in terms of power against current state-of-the-art kernel tests.

Existing regression models tend to fall short in both accuracy and uncertainty estimation when the label distribution is imbalanced. In this paper, we propose a probabilistic deep learning model, dubbed variational imbalanced regression (VIR), which not only performs well in imbalanced regression but naturally produces reasonable uncertainty estimation as a byproduct. Different from typical variational autoencoders assuming I.I.D. representations (a data point's representation is not directly affected by other data points), our VIR borrows data with similar regression labels to compute the latent representation's variational distribution; furthermore, different from deterministic regression models producing point estimates, VIR predicts the entire normal-inverse-gamma distributions and modulates the associated conjugate distributions to impose probabilistic reweighting on the imbalanced data, thereby providing better uncertainty estimation. Experiments in several real-world datasets show that our VIR can outperform state-of-the-art imbalanced regression models in terms of both accuracy and uncertainty estimation.

We study causal representation learning, the task of inferring latent causal variables and their causal relations from high-dimensional mixtures of the variables. Prior work relies on weak supervision, in the form of counterfactual pre- and post-intervention views or temporal structure; places restrictive assumptions, such as linearity, on the mixing function or latent causal model; or requires partial knowledge of the generative process, such as the causal graph or intervention targets. We instead consider the general setting in which both the causal model and the mixing function are nonparametric. The learning signal takes the form of multiple datasets, or environments, arising from unknown interventions in the underlying causal model. Our goal is to identify both the ground truth latents and their causal graph up to a set of ambiguities which we show to be irresolvable from interventional data. We study the fundamental setting of two causal variables and prove that the observational distribution and one perfect intervention per node suffice for identifiability, subject to a genericity condition. This condition rules out spurious solutions that involve fine-tuning of the intervened and observational distributions, mirroring similar conditions for nonlinear cause-effect inference. For an arbitrary number of variables, we show that at least one pair of distinct perfect interventional domains per node guarantees identifiability. Further, we demonstrate that the strengths of causal influences among the latent variables are preserved by all equivalent solutions, rendering the inferred representation appropriate for drawing causal conclusions from new data. Our study provides the first identifiability results for the general nonparametric setting with unknown interventions, and elucidates what is possible and impossible for causal representation learning without more direct supervision.

Computing properties of molecular systems rely on estimating expectations of the (unnormalized) Boltzmann distribution. Molecular dynamics (MD) is a broadly adopted technique to approximate such quantities. However, stable simulations rely on very small integration time-steps ($10^{-15}\,\mathrm{s}$), whereas convergence of some moments, e.g. binding free energy or rates, might rely on sampling processes on time-scales as long as $10^{-1}\, \mathrm{s}$, and these simulations must be repeated for every molecular system independently. Here, we present Implict Transfer Operator (ITO) Learning, a framework to learn surrogates of the simulation process with multiple time-resolutions. We implement ITO with denoising diffusion probabilistic models with a new SE(3) equivariant architecture and show the resulting models can generate self-consistent stochastic dynamics across multiple time-scales, even when the system is only partially observed. Finally, we present a coarse-grained CG-SE3-ITO model which can quantitatively model all-atom molecular dynamics using only coarse molecular representations. As such, ITO provides an important step towards multiple time- and space-resolution acceleration of MD. Code is available at \href{https://github.com/olsson-group/ito}{https://github.com/olsson-group/ito}.