Causal discovery with latent variables is a fundamental task. Yet most existing methods rely on strong structural assumptions, such as enforcing specific indicator patterns for latents or restricting how they can interact with others. We argue that a core obstacle to a general, structural-assumption-free approach is the lack of an equivalence characterization: without knowing what can be identified, one generally cannot design methods for how to identify it. In this work, we aim to close this gap for linear non-Gaussian models. We establish the graphical criterion for when two graphs with arbitrary latent structure and cycles are distributionally equivalent, that is, they induce the same observed distribution set. Key to our approach is a new tool, edge rank constraints, which fills a missing piece in the toolbox for latent-variable causal discovery in even broader settings. We further provide a procedure to traverse the whole equivalence class and develop an algorithm to recover models from data up to such equivalence. To our knowledge, this is the first equivalence characterization with latent variables in any parametric setting without structural assumptions, and hence the first structural-assumption-free discovery method. Code and an interactive demo are available at https://equiv.cc.

In this paper, we study the angle testing problem in the context of similarity search in high-dimensional Euclidean spaces and propose two projection-based probabilistic kernel functions, one designed for angle comparison and the other for angle thresholding. Unlike existing approaches that rely on random projection vectors drawn from Gaussian distributions, our approach leverages reference angles and adopts a deterministic structure for the projection vectors. Notably, our kernel functions do not require asymptotic assumptions, such as the number of projection vectors tending to infinity, and can be theoretically and experimentally shown to outperform Gaussian-distribution-based kernel functions. We apply the proposed kernel function to Approximate Nearest Neighbor Search (ANNS) and demonstrate that our approach achieves a 2.5x--3x higher query-per-second (QPS) throughput compared to the widely-used graph-based search algorithm HNSW. Our code and data are available at https://github.com/KejingLu-810/KS.

We study several problems in differentially private domain discovery, where each user holds a subset of items from a shared but unknown domain, and the goal is to output an informative subset of items. For set union, we show that the simple baseline Weighted Gaussian Mechanism (WGM) has a near-optimal $\ell_1$ missing mass guarantee on Zipfian data as well as a distribution-free $\ell_\infty$ missing mass guarantee. We then apply the WGM as a domain-discovery precursor for existing known-domain algorithms for private top-$k$ and $k$-hitting set and obtain new utility guarantees for their unknown domain variants. Finally, experiments demonstrate that all of our WGM-based methods are competitive with or outperform existing baselines for all three problems.

Multivariate Hawkes process provides a powerful framework for modeling temporal dependencies and event-driven interactions in complex systems. While existing methods primarily focus on uncovering causal structures among observed subprocesses, real-world systems are often only partially observed, with latent subprocesses posing significant challenges. In this paper, we show that continuous-time event sequences can be represented by a discrete-time causal model as the time interval shrinks, and we leverage this insight to establish necessary and sufficient conditions for identifying latent subprocesses and the causal influences. Accordingly, we propose a two-phase iterative algorithm that alternates between inferring causal relationships among discovered subprocesses and uncovering new latent subprocesses, guided by path-based conditions that guarantee identifiability. Experiments on both synthetic and real-world datasets show that our method effectively recovers causal structures despite the presence of latent subprocesses.

Translating the internal representations and computations of models into concepts that humans can understand is a key goal of interpretability. While recent dictionary learning methods such as Sparse Autoencoders (SAEs) provide a promising route to discover human-interpretable features, they often only recover token-specific, noisy, or highly local concepts. We argue that this limitation stems from neglecting the temporal structure of language, where semantic content typically evolves smoothly over sequences. Building on this insight, we introduce Temporal Sparse Autoencoders (T-SAEs), which incorporate a novel contrastive loss encouraging consistent activations of high-level features over adjacent tokens. This simple yet powerful modification enables SAEs to disentangle semantic from syntactic features in a self-supervised manner. Across multiple datasets and models, T-SAEs recover smoother, more coherent semantic concepts without sacrificing reconstruction quality. Strikingly, they exhibit clear semantic structure despite being trained without explicit semantic signal, offering a new pathway for unsupervised interpretability in language models.

The representer theorem is a cornerstone of kernel methods, which aim to estimate latent functions in reproducing kernel Hilbert spaces (RKHSs) in a nonparametric manner. Its significance lies in converting inherently infinite-dimensional optimization problems into finite-dimensional ones over dual coefficients, thereby enabling practical and computationally tractable algorithms. In this paper, we address the problem of estimating the latent triggering kernels--functions that encode the interaction structure between events--for linear multivariate Hawkes processes based on observed event sequences within an RKHS framework. We show that, under the principle of penalized least squares minimization, a novel form of representer theorem emerges: a family of transformed kernels can be defined via a system of simultaneous integral equations, and the optimal estimator of each triggering kernel is expressed as a linear combination of these transformed kernels evaluated at the data points. Remarkably, the dual coefficients are all analytically fixed to unity, obviating the need to solve a costly optimization problem to obtain the dual coefficients. This leads to a highly efficient estimator capable of handling large-scale data more effectively than conventional nonparametric approaches. Empirical evaluations on synthetic and real-world datasets reveal that the proposed method achieves competitive predictive accuracy while substantially improving computational efficiency compared to state-of-the-art kernel method-based estimators.

We study cross-domain lossy compression, where the encoder observes a degraded source while the decoder reconstructs samples from a distinct target distribution. The problem is formulated as constrained optimal transport with two constraints on compression rate and classification loss. With shared common randomness, the one-shot setting reduces to a deterministic transport plan, and we derive closed-form distortion-rate-classification (DRC) and rate-distortion-classification (RDC) tradeoffs for Bernoulli sources under Hamming distortion. In the asymptotic regime, we establish analytic DRC/RDC expressions for Gaussian models under mean-squared error. The framework is further extended to incorporate perception divergences (Kullback-Leibler and squared Wasserstein), yielding closed-form distortion-rate-perception-classification (DRPC) functions. To validate the theory, we develop deep end-to-end compression models for super-resolution (MNIST), denoising (SVHN, CIFAR-10, ImageNet, KODAK), and inpainting (SVHN) problems, demonstrating the consistency between the theoretical results and empirical performance.