We study a nonlinear factor model in which observed responses depend on low-rank latent factors through an unknown monotone link function. This setting is challenging and largely underexplored due to severe nonconvexity and identifiability issues. The link function is assumed to lie in a reproducing kernel Hilbert space (RKHS), enabling flexible nonparametric modeling while preserving identifiability. We formulate the problem as the joint recovery of the low-rank factors, loadings, and the nonlinear link function from possibly incomplete and noisy observations and propose a projected block coordinate descent (BCD) algorithm with explicit regularization to address scale and rotational ambiguities. Under mild incoherence of factors and standard sampling conditions, we establish convergence guarantees in both noiseless and noisy regimes, along with sublinear regret bounds for the link-function updates. Our results extend classical linear factor models to a broad nonlinear regime and provide a principled framework for learning nonlinear latent structures. We evaluate the proposed approach using controlled synthetic experiments, indicating promising performance.

Systems neuroscience aims to understand how networks of neurons distributed throughout the brain mediate computational tasks. One popular approach to identify those networks is to first calculate measures of neural activity (e.g.

Systems neuroscience aims to understand how networks of neurons distributed throughout the brain mediate computational tasks. One popular approach to identify those networks is to first calculate measures of neural activity (e.g. Critically, despite the established role of directed communication between brain regions in neural computation, measures of directed communication have been rarely utilized in network estimation because they are incompatible with the implicit assumptions of the linear factor model approach. Here, we develop a novel spectral measure of directed communication called the Directed Spectrum (DS). We prove that it is compatible with the implicit assumptions of linear factor models, and we provide a method to estimate the DS.

This paper studies identification of average treatment effects in a panel data setting. It introduces a novel nonparametric factor model and proves identification of average treatment effects. The identification proof is based on the introduction of a consistent estimator. Underlying the proof is a result that there is a consistent estimator for the expected outcome in the absence of the treatment for each unit and time period; this result can be applied more broadly, for example in problems of decompositions of group-level differences in outcomes, such as the much-studied gender wage gap.

Systems neuroscience aims to understand how networks of neurons distributed throughout the brain mediate computational tasks. One popular approach to identify those networks is to first calculate measures of neural activity (e.g. Critically, despite the established role of directed communication between brain regions in neural computation, measures of directed communication have been rarely utilized in network estimation because they are incompatible with the implicit assumptions of the linear factor model approach. Here, we develop a novel spectral measure of directed communication called the Directed Spectrum (DS). We prove that it is compatible with the implicit assumptions of linear factor models, and we provide a method to estimate the DS.

Synthetic control methods (SCMs) have become a crucial tool for causal inference in comparative case studies. The fundamental idea of SCMs is to estimate counterfactual outcomes for a treated unit by using a weighted sum of observed outcomes from untreated units. The accuracy of the synthetic control (SC) is critical for estimating the causal effect, and hence, the estimation of SC weights has been the focus of much research. In this paper, we first point out that existing SCMs suffer from an implicit endogeneity problem, which is the correlation between the outcomes of untreated units and the error term in the model of a counterfactual outcome. We show that this problem yields a bias in the causal effect estimator. We then propose a novel SCM based on density matching, assuming that the density of outcomes of the treated unit can be approximated by a weighted average of the densities of untreated units (i.e., a mixture model). Based on this assumption, we estimate SC weights by matching moments of treated outcomes and the weighted sum of moments of untreated outcomes. Our proposed method has three advantages over existing methods. First, our estimator is asymptotically unbiased under the assumption of the mixture model. Second, due to the asymptotic unbiasedness, we can reduce the mean squared error for counterfactual prediction. Third, our method generates full densities of the treatment effect, not only expected values, which broadens the applicability of SCMs. We provide experimental results to demonstrate the effectiveness of our proposed method.

We propose kernel PCA as a method for analyzing the dependence structure of multivariate extremes and demonstrate that it can be a powerful tool for clustering and dimension reduction. Our work provides some theoretical insight into the preimages obtained by kernel PCA, demonstrating that under certain conditions they can effectively identify clusters in the data. We build on these new insights to characterize rigorously the performance of kernel PCA based on an extremal sample, i.e., the angular part of random vectors for which the radius exceeds a large threshold. More specifically, we focus on the asymptotic dependence of multivariate extremes characterized by the angular or spectral measure in extreme value theory and provide a careful analysis in the case where the extremes are generated from a linear factor model. We give theoretical guarantees on the performance of kernel PCA preimages of such extremes by leveraging their asymptotic distribution together with Davis-Kahan perturbation bounds. Our theoretical findings are complemented with numerical experiments illustrating the finite sample performance of our methods.

We propose a spectral clustering algorithm for analyzing the dependence structure of multivariate extremes. More specifically, we focus on the asymptotic dependence of multivariate extremes characterized by the angular or spectral measure in extreme value theory. Our work studies the theoretical performance of spectral clustering based on a random $k$-nearest neighbor graph constructed from an extremal sample, i.e., the angular part of random vectors for which the radius exceeds a large threshold. In particular, we derive the asymptotic distribution of extremes arising from a linear factor model and prove that, under certain conditions, spectral clustering can consistently identify the clusters of extremes arising in this model. Leveraging this result we propose a simple consistent estimation strategy for learning the angular measure. Our theoretical findings are complemented with numerical experiments illustrating the finite sample performance of our methods.