In classical canonical correlation analysis (CCA), the goal is to determine the linear transformations of two random vectors into two new random variables that are most strongly correlated. Canonical variables are pairs of these new random variables, while canonical correlations are correlations between these pairs. In this paper, we propose and study two generalizations of this classical method: (1) Instead of two random vectors we study more complex data structures that appear in important applications. In these structures, there are $L$ features, each described by $p_l$ scalars, $1 \le l \le L$. We observe $n$ such objects over $T$ time points. We derive a suitable analog of the CCA for such data. Our approach relies on embeddings into Reproducing Kernel Hilbert Spaces, and covers several related data structures as well. (2) We develop an analogous approach for multidimensional random processes. In this case, the experimental units are multivariate continuous, square-integrable functions over a given interval. These functions are modeled as elements of a Hilbert space, so in this case, we define the multiple functional canonical correlation analysis, MFCCA. We justify our approaches by their application to two data sets and suitable large sample theory. We derive consistency rates for the related transformation and correlation estimators, and show that it is possible to relax two common assumptions on the compactness of the underlying cross-covariance operators and the independence of the data.

Neural Information Processing Systems http://nips.cc/

The sparse generalized eigenvalue problem (SGEP) aims to find the leading eigen-vector with sparsity structure.

Generalized eigenvalue problems (GEPs) find applications in various fields of science and engineering. For example, principal component analysis, Fisher's discriminant analysis, and canonical correlation analysis are specific instances of GEPs and are widely used in statistical data processing. In this work, we study GEPs under generative priors, assuming that the underlying leading generalized eigenvector lies within the range of a Lipschitz continuous generative model. Under appropriate conditions, we show that any optimal solution to the corresponding optimization problems attains the optimal statistical rate. Moreover, from a computational perspective, we propose an iterative algorithm called the Projected Rayleigh Flow Method (PRFM) to approximate the optimal solution.

Autoregressive pretraining has become the de facto paradigm for learning general-purpose representations in large language models (LLMs). However, linear probe performance across downstream perception tasks shows substantial variability, suggesting that features optimized for next-token prediction do not consistently transfer well to downstream perception tasks. We demonstrate that representations learned via autoregression capture features that may lie outside the subspaces most informative for perception. To quantify the (mis)alignment between autoregressive pretraining and downstream perception, we introduce the Next Token Perception Score (NTPS)-a score derived under a linear setting that measures the overlap between autoregressive and perception feature subspaces. This metric can be easily computed in closed form from pretrained representations and labeled data, and is proven to both upper- and lower-bound the excess loss. Empirically, we show that NTPS correlates strongly with linear probe accuracy across 12 diverse NLP datasets and eight pretrained models ranging from 270M to 8B parameters, confirming its utility as a measure of alignment. Furthermore, we show that NTPS increases following low-rank adaptation (LoRA) fine-tuning, especially in large models, suggesting that LoRA aligning representations to perception tasks enhances subspace overlap and thus improves downstream performance. More importantly, we find that NTPS reliably predicts the additional accuracy gains attained by LoRA finetuning thereby providing a lightweight prescreening tool for LoRA adaptation. Our results offer both theoretical insights and practical tools for analytically assessing LLM perception skills.

Generalized eigenvalue problems (GEPs) find applications in various fields of science and engineering. For example, principal component analysis, Fisher's discriminant analysis, and canonical correlation analysis are specific instances of GEPs and are widely used in statistical data processing. In this work, we study GEPs under generative priors, assuming that the underlying leading generalized eigenvector lies within the range of a Lipschitz continuous generative model. Under appropriate conditions, we show that any optimal solution to the corresponding optimization problems attains the optimal statistical rate. Moreover, from a computational perspective, we propose an iterative algorithm called the Projected Rayleigh Flow Method (PRFM) to approximate the optimal solution. We theoretically demonstrate that under suitable assumptions, PRFM converges linearly to an estimated vector that achieves the optimal statistical rate. Numerical results are provided to demonstrate the effectiveness of the proposed method.

(This is a work in progress. Feedback is welcome) An analytical comparison is made between slow feature analysis (SFA) and the successor representation (SR). While SFA and the SR stem from distinct areas of machine learning, they share important properties, both in terms of their mathematics and the types of information they are sensitive to. This work studies their connection along these two axes. In particular, multiple variants of the SFA algorithm are explored analytically and then applied to the setting of an MDP, leading to a family of eigenvalue problems involving the SR and other related quantities. These resulting eigenvalue problems are then illustrated in the toy setting of a gridworld, where it is demonstrated that the place- and grid-like fields often associated to the SR can equally be generated using SFA.