Reliable generalization in conditional latent variable models requires understanding both identifiability and extrapolation: how observed variation across attributes determines latent structure, and how that structure determines distributions at unseen attributes. However, existing identifiability and extrapolation guarantees are largely model-specific, with separate analyses in nonlinear ICA, causal representation learning, perturbation modeling, and related conditional latent variable models. We introduce concept modulation models (CMMs), an attribute-indexed class of conditional generative models with structure A→ Λ→ C→ X, where attributes select modulators, modulators induce latent concept laws, and concepts generate observed features. CMMs lift transition-based identifiability to conditional settings by showing that feature agreement on observed attributes induces a latent concept transition constrained by the CMM class. We express these constraints through attribute potentials, log-density ratios between attribute-conditioned concept laws, separating the generic lifting step from model-specific rigidity arguments. The same potentials control extrapolation: agreement at unseen attributes holds exactly when the transported attribute-potential identities extend to those attributes. This yields algebraic extrapolation criteria, identifies the common potential-based proof objects behind several existing identifiability and extrapolation results, and, when combined with the model-specific rigidity arguments in those works, recovers their stated conclusions.

Best-of-N sampling is widely used to construct pairwise preference data: N candidates are drawn from a base distribution, and the best is paired with a rejected response. Despite its widespread use, what Bradley–Terry (BT) reward learning extracts from such data, and how to choose N and the base distribution, remain unclear. We specialize a recent analysis of preference data via its induced conditional distribution to Best-of-N. For independent-reference variants, we derive closed-form reward targets as explicit functions of N and the base distribution, and show that they preserve the latent reward ranking. For the practical Best-vs-Random and Best-vs-Worst variants, chosen and rejected responses are coupled through the same candidate set, so exact BT representability generally fails; nevertheless, bounded-class minimizers approach the reference targets as N grows. Although margin and connectivity are known to govern sample efficiency in pairwise preference learning, Best-of-N couples them through N in opposing directions: larger N widens pairwise margins but reduces connectivity. This trade-off yields two design principles: use larger N when preference labels are the bottleneck, smaller N when generation is the bottleneck; and shape the base distribution to place mass between the responses whose comparison matters most at test time. Experiments on synthetic and real preference data support the predicted dependence on sample size and base-distribution shape.

Long-term ship trajectory prediction is a fundamental capability for maritime safety and autonomous navigation. While recent Transformer-based architectures have improved forecasting horizons, they predominantly rely on historical kinematic states, treating vessel motion as an isolated system. In reality, maritime navigation is profoundly modulated by extrinsic factors like weather and constrained by static vessel characteristics. Existing multimodal approaches fundamentally model the joint distribution over states and contexts, treating environmental variables as peer features rather than encoding the directional physical dependence of vessel dynamics on environmental conditions. In this work, we propose the Conditional Informer, a novel encoder-decoder architecture that formulates trajectory prediction as a conditional generation task. We employ a dedicated Conditional Attention mechanism where the vessel state explicitly queries environmental contexts through cross-attention, encoding the physical prior that weather modulates - but is not generated by - vessel dynamics. Furthermore, to address the intermittency of real-world data, we introduce a Modality Masking training strategy to prevent catastrophic degradation during sensor fallback. Extensive experiments on AIS and ERA5 data demonstrate that our approach outperforms kinematic and concatenation-based baselines by 15.4

Techniques for concept extraction, such as sparse autoencoders and transcoders, aim to extract high-level symbolic concepts from low-level nonsymbolic representations. When these extracted concepts are used for downstream tasks such as model steering and unlearning, it is essential to understand their guarantees, or lack thereof. In this work, we present a unified theoretical framework for unsupervised concept extraction, in which we frame the task of concept extraction as identifying a generative model. We present a general meta-theorem for identifiability, which reduces the problem of establishing identifiability guarantees to the problem of characterizing the intersection of two sets. As we demonstrate on a range of widely-used approaches, this meta-theorem substantially simplifies the task of proving such guarantees, thus paving the way for the development of new, principled approaches for concept extraction.

The task of compositional generation involves using a conditional generative model, trained only on a subset of the possible conditions, to produce samples from compositionally-defined target distributions such as a geometric combination of the source distributions. In this work, we argue that this task is often infeasible for vanilla conditional diffusion models: we conjecture that no inference-time technique can efficiently produce samples from the target distribution in certain well-motivated settings. This idea is supported by theory-guided generalization arguments and carefully-designed experiments on both synthetic and realistic data. In particular, while recent methods such as Feynman-Kac correction reduce inference-time approximation error, our results show that score estimation error has a more catastrophic effect on performance when the target distribution is out-of-distribution with respect to the sources, highlighting the need for a different approach to this task.

Pairwise preference learning is central to machine learning, with recent applications in aligning language models with human preferences. A typical dataset consists of triplets (x, y^+, y^-), where response y^+ is preferred over response y^- for context x. The Bradley–Terry (BT) model is the predominant approach, modeling preference probabilities as a function of latent score differences. Standard practice assumes data follows this model and learns the latent scores accordingly. However, real data may violate this assumption, and it remains unclear what BT learning recovers in such cases. Starting from triplet comparison data, we formalize the preference information it encodes through the conditional preference distribution (CPRD). We give precise conditions for when BT is appropriate for modeling the CPRD, and identify factors governing sample efficiency – namely, margin and connectivity. Together, these results offer a data-centric foundation for understanding what preference learning actually recovers.

Recent progress in flow-based generative modeling has led to models that output high-quality samples while using only a small number of function evaluations. However, at present, there is a lack of similar advances in estimating the model likelihood. In particular, most existing methods either rely on restrictive architectures that enable exact calculations, or use stochastic approximations such as Hutchinson's trace estimator that introduce substantial variance. In this work, we introduce SCAlable LikeLihood distillation of flOw maPs (SCALLOP). SCALLOP builds on the recently proposed F2D2, a likelihood flow map model that can generate samples and their densities in a small number of function evaluations. While F2D2 uses Hutchinson's estimator during training, we introduce an alternative and more scalable likelihood distillation objective that is Hutchinson-free and admits a vectorized formulation. Empirically, we demonstrate the effectiveness of SCALLOP as a Boltzmann generator in molecular science, and further validate its benefit on image datasets. SCALLOP significantly reduces both training variance and training time while consistently improving performance compared to F2D2, and is competitive with the state-of-the-art while achieving up to 10x inference speedup over the fastest baseline.

We introduce a stochastic global optimization method based on random walks on Grassmannian manifolds. To minimize a continuous objective ℓ:ℝ^d→ℝ, the method repeatedly samples random k-dimensional linear subspaces (with k≪ d), solves the resulting low-dimensional restrictions of these problems to these subspaces using an arbitrary black-box optimizer, and updates the iterate (which monotonically improves upon the previous iterate). Unlike classical optimization analyses that rely on convexity, smoothness, Lipschitz bounds, or Polyak-Lojasiewicz-type conditions, our convergence guarantees depend only on the geometric distribution of restricted minima across the k-dimensional subspaces passing through a given point in ℝ^d. We identify a gap parameter – an analogue of a spectral gap for random walks – that controls the rate at which the iterates approach the global minimum value. Finally, we argue that the same analysis yields a blind-spot robustness property: sufficiently narrow, deep dips of the loss function (small-measure regions where ℓ spikes downward) have limited influence on the algorithm's trajectory, since they are unlikely to be encountered by random subspace sampling.

Tabular data presents unique challenges for deep learning due to its heterogeneous nature, where numeric features exhibit diverse distributions, scales, and statistical properties. Although recent advances have improved how models learn from tabular data, how numeric data are transformed into model-friendly representations remains comparatively underexplored. We introduce the stretch transformation framework, which formulates numeric feature preprocessing as an optimization problem to make the target function smoother and thus more learnable. Our framework has two variants: (1) unsupervised stretch, which uniformly redistributes feature density via minimax optimization, and (2) supervised stretch, which optimizes target-aware numeric feature transformations from the perspective of target-function smoothness by minimizing the target function's Dirichlet energy in the transformed space. Our theoretical analysis further connects this framework to several popular transformations: unsupervised stretch is closely related to Piecewise Linear Encoding through a shared piecewise-linear geometry and approaches the empirical CDF transformation as the number of bins grows, while supervised stretch becomes closely related to target encoding in the fine-binning limit. Comprehensive experiments on 38 datasets from the TALENT benchmark demonstrate that supervised stretch consistently outperforms all baselines. These results show that explicitly optimizing for target function smoothness is a powerful and underexplored strategy for tabular deep learning.

Despite the empirical success of foundation models, we do not have a systematic characterization of the representations that these models learn. In this paper, we establish the contexture theory.It shows that a large class of representation learning methods can be characterized as learning from the association between the input and a context variable. Specifically, we show that many popular methods aim to approximate the top-d singular functions of the expectation operator induced by the context, in which case we say that the representation learns the contexture.We demonstrate the generality of the contexture theory by proving that representation learning within various learning paradigms -- supervised, self-supervised, and manifold learning -- can all be studied from such a perspective.We prove that representations that learn the contexture are optimal on those tasks that are compatible with the context.One important implication of our theory is that once the model is large enough to approximate the top singular functions, scaling up the model size yields diminishing returns, so further improvement requires better contexts.To this end, we study how to evaluate a context without knowing the downstream tasks. We propose a metric and show by experiments that it correlates well with the actual performance of the encoder on many real datasets.

Recent advances in representation learning reveal that widely used objectives, such as contrastive and non-contrastive, implicitly perform spectral decomposition of a contextual kernel, induced by the relationship between inputs and their contexts. Yet, these methods recover only the linear span of top eigenfunctions of the kernel, whereas exact spectral decomposition is essential for understanding feature ordering and importance. In this work, we propose a general framework to extract ordered and identifiable eigenfunctions, based on modular building blocks designed to satisfy key desiderata, including compatibility with the contextual kernel and scalability to modern settings. We then show how two main methodological paradigms, low-rank approximation and Rayleigh quotient optimization, align with this framework for eigenfunction extraction. Finally, we validate our approach on synthetic kernels and demonstrate on real-world image datasets that the recovered eigenvalues act as effective importance scores for feature selection, enabling principled efficiency-accuracy tradeoffs via adaptive-dimensional representations.

In this paper, we introduce the general framework of knowledge-enriched machine learning, for encoding and leveraging problem-specific deterministic knowledge, such as column descriptions in the tabular setting. We focus on a paradigmatic use case: supervised learning problems on tabular data. As a first step in this direction, we introduce a simple yet flexible encoding of such deterministic information in the form of concept kernels, and describe meta-algorithms which leverage this particular encoding of prior knowledge. To ground future research, we introduce KE-TALENT, a novel benchmarking suite for kernel-enriched supervised learning on tabular data, adapted from the recently-introduced TALENT benchmark to include concept kernels and other metadata for each dataset. Finally, to demonstrate the benefits of concept kernels, we provide results for several kernel-enriched versions of existing algorithms, also intended as a baseline for future research. Code is publicly available.

Unlabeled data is a key component of modern machine learning. In general, the role of unlabeled data is to impose a form of smoothness, usually from the similarity information encoded in a base kernel, such as the ϵ-neighbor kernel or the adjacency matrix of a graph. This work revisits the classical idea of spectrally transformed kernel regression (STKR), and provides a new class of general and scalable STKR estimators able to leverage unlabeled data. Intuitively, via spectral transformation, STKR exploits the data distribution for which unlabeled data can provide additional information. First, we show that STKR is a principled and general approach, by characterizing a universal type of “target smoothness”, and proving that any sufficiently smooth function can be learned by STKR. Second, we provide scalable STKR implementations for the inductive setting and a general transformation function, while prior work is mostly limited to the transductive setting. Third, we derive statistical guarantees for two scenarios: STKR with a known polynomial transformation, and STKR with kernel PCA when the transformation is unknown. Overall, we believe that this work helps deepen our understanding of how to work with unlabeled data, and its generality makes it easier to inspire new methods.

We consider the linear causal representation learning setting where we observe a linear mixing of $d$ unknown latent factors, which follow a linear structural causal model. Recent work has shown that it is possible to recover the latent factors as well as the underlying structural causal model over them, up to permutation and scaling, provided that we have at least $d$ environments, each of which corresponds to perfect interventions on a single latent node (factor). After this powerful result, a key open problem faced by the community has been to relax these conditions: allow for coarser than perfect single-node interventions, and allow for fewer than $d$ of them, since the number of latent factors $d$ could be very large. In this work, we consider precisely such a setting, where we allow a smaller than $d$ number of environments, and also allow for very coarse interventions that can very coarsely \textit{change the entire causal graph over the latent factors}. On the flip side, we relax what we wish to extract to simply the \textit{list of nodes that have shifted between one or more environments}. We provide a surprising identifiability result that it is indeed possible, under some very mild standard assumptions, to identify the set of shifted nodes. Our identifiability proof moreover is a constructive one: we explicitly provide necessary and sufficient conditions for a node to be a shifted node, and show that we can check these conditions given observed data. Our algorithm lends itself very naturally to the sample setting where instead of just interventional distributions, we are provided datasets of samples from each of these distributions. We corroborate our results on both synthetic experiments as well as an interesting psychometric dataset. The code can be found at https://github.com/TianyuCodings/iLCS.

Existing approaches to differentiable structure learning of directed acyclic graphs (DAGs) rely on strong identifiability assumptions in order to guarantee that global minimizers of the acyclicity-constrained optimization problem identifies the true DAG. Moreover, it has been observed empirically that the optimizer may exploit undesirable artifacts in the loss function. We explain and remedy these issues by studying the behavior of differentiable acyclicity-constrained programs under general likelihoods with multiple global minimizers. By carefully regularizing the likelihood, it is possible to identify the sparsest model in the Markov equivalence class, even in the absence of an identifiable parametrization. We first study the Gaussian case in detail, showing how proper regularization of the likelihood defines a score that identifies the sparsest model. Assuming faithfulness, it also recovers the Markov equivalence class. These results are then generalized to general models and likelihoods, where the same claims hold. These theoretical results are validated empirically, showing how this can be done using standard gradient-based optimizers, thus paving the way for differentiable structure learning under general models and losses.

Data augmentation is critical to the empirical success of modern self-supervised representation learning, such as contrastive learning and masked language modeling. However, a theoretical understanding of the exact role of augmentation remains limited. Recent work has built the connection between self-supervised learning and the approximation of the top eigenspace of a graph Laplacian operator, suggesting that learning a linear probe atop such representation can be connected to RKHS regression. Building on this insight, this work delves into a statistical analysis of augmentation-based pretraining. Starting from the isometry property, a geometric characterization of the target function given by the augmentation, we disentangle the effects of the model and the augmentation, and prove two generalization bounds that are free of model complexity. Our first bound works for an arbitrary encoder, where the prediction error is decomposed as the sum of an estimation error incurred by fitting a linear probe with RKHS regression, and an approximation error entailed by RKHS approximation. Our second bound specifically addresses the case where the encoder is near-optimal, that is it approximates the top-d eigenspace of the RKHS induced by the augmentation. A key ingredient in our analysis is the augmentation complexity, which we use to quantitatively compare different augmentations and analyze their impact on downstream performance.

To build intelligent machine learning systems, there are two broad approaches. One approach is to build inherently interpretable models, as endeavored by the growing field of causal representation learning. The other approach is to build highly-performant foundation models and then invest efforts into understanding how they work. In this work, we relate these two approaches and study how to learn human-interpretable concepts from data. Weaving together ideas from both fields, we formally define a notion of concepts and show that they can be provably recovered from diverse data. Experiments on synthetic data and large language models show the utility of our unified approach.

Large Language Models (LLMs) have the capacity to store and recall facts. Through experimentation with open-source models, we observe that this ability to retrieve facts can be easily manipulated by changing contexts, even without altering their factual meanings. These findings highlight that LLMs might behave like an associative memory model where certain tokens in the contexts serve as clues to retrieving facts. We mathematically explore this property by studying how transformers, the building blocks of LLMs, can complete such memory tasks. We study a simple latent concept association problem with a one-layer transformer and we show theoretically and empirically that the transformer gathers information using self-attention and uses the value matrix for associative memory.

In this paper, we demonstrate a surprising capability of large language models (LLMs): given only input feature names and a description of a prediction task, they are capable of selecting the most predictive features, with performance rivaling the standard tools of data science. Remarkably, these models exhibit this capacity across various query mechanisms. For example, we zero-shot prompt an LLM to output a numerical importance score for a feature (e.g., "blood pressure") in predicting an outcome of interest (e.g., "heart failure"), with no additional context. In particular, we find that the latest models, such as GPT-4, can consistently identify the most predictive features regardless of the query mechanism and across various prompting strategies. We illustrate these findings through extensive experiments on real-world data, where we show that LLM-based feature selection consistently achieves strong performance competitive with data-driven methods such as the LASSO, despite never having looked at the downstream training data. Our findings suggest that LLMs may be useful not only for selecting the best features for training but also for deciding which features to collect in the first place. This could potentially benefit practitioners in domains like healthcare, where collecting high-quality data comes at a high cost.