Recursive architectures such as Tiny Recursive Models (TRMs) perform implicit reasoning through iterative latent computation, yet the geometric structure of these reasoning trajectories remains poorly understood. We investigate the activation manifold of TRMs during recursive unrolling and find that activations occupy an effectively linear, low-dimensional subspace whose principal directions can be tracked dynamically with cheap power iterations. This suggests that weight-sharing concentrates iterative computation along a small number of dominant eigendirections, and we find that this concentration varies sharply across computational sites. We exploit this structure through LASER (Low-Rank Activation SVD for Efficient Recursion), a dynamic compression framework that maintains an evolving low-rank basis via matrix-free subspace tracking with a fidelity-triggered reset mechanism, achieving ${\sim}60\%$ activation memory savings with no statistically significant accuracy degradation. Our analysis raises questions about how recursive architectures allocate representational capacity during implicit reasoning, and whether this concentration can be exploited to improve the efficiency and stability of latent computation.

Data-driven approximations of the infinite-dimensional Koopman operator rely on finite-dimensional projections, where the predictive accuracy of the resulting models hinges heavily on the invariance of the chosen subspace. Subspace pruning systematically discards geometrically misaligned directions to enhance this invariance proximity, which formally corresponds to the largest principal angle between the subspace and its image under the operator. Yet, existing techniques are largely restricted to Euclidean settings. To bridge this gap, this paper presents an approach for computing principal angles and vectors to enable Koopman subspace pruning within a Reproducing Kernel Hilbert Space (RKHS) geometry. We first outline an exact computational routine, which is subsequently scaled for large datasets using randomized Nystrom approximations. Based on these foundations, we introduce the Kernel-SPV and Approximate Kernel-SPV algorithms for targeted subspace refinement via principal vectors. Simulation results validate our approach.

Neural Collapse is a phenomenon that helps identify sparse and low rank structures in deep classifiers. Recent work has extended the definition of neural collapse to regression problems, albeit only measuring the phenomenon at the last layer. In this paper, we establish that Neural Regression Collapse (NRC) also occurs below the last layer across different types of models. We show that in the collapsed layers of neural regression models, features lie in a subspace that corresponds to the target dimension, the feature covariance aligns with the target covariance, the input subspace of the layer weights aligns with the feature subspace, and the linear prediction error of the features is close to the overall prediction error of the model. In addition to establishing Deep NRC, we also show that models that exhibit Deep NRC learn the intrinsic dimension of low rank targets and explore the necessity of weight decay in inducing Deep NRC. This paper provides a more complete picture of the simple structure learned by deep networks in the context of regression.

Quantum principal component analysis (qPCA) is commonly formulated as the extraction of eigenvalues and eigenvectors of a covariance-encoded density operator. Yet in many qPCA settings, the practical objective is simpler: projecting data onto the dominant spectral subspace. In this work, we introduce a projection-first framework, the Filtered Spectral Projection Algorithm (FSPA), which bypasses explicit eigenvalue estimation while preserving the essential spectral structure. FSPA amplifies any nonzero warm-start overlap with the leading principal subspace and remains robust in small-gap and near-degenerate regimes without inducing artificial symmetry breaking in the absence of bias. To connect this approach to classical datasets, we show that for amplitude-encoded centered data, the ensemble density matrix $ρ=\sum_i p_i|ψ_i\rangle\langleψ_i|$ coincides with the covariance matrix. For uncentered data, $ρ$ corresponds to PCA without centering, and we derive eigenvalue interlacing bounds quantifying the deviation from standard PCA. We further show that ensembles of quantum states admit an equivalent centered covariance interpretation. Numerical demonstrations on benchmark datasets, including Breast Cancer Wisconsin and handwritten Digits, show that downstream performance remains stable whenever projection quality is preserved. These results suggest that, in a broad class of qPCA settings, spectral projection is the essential primitive, and explicit eigenvalue estimation is often unnecessary.

We study the problem of learning a low-degree spherical polynomial of degree $k_0 = Θ(1) \ge 1$ defined on the unit sphere in $\RR^d$ by training an over-parameterized two-layer neural network with augmented feature in this paper. Our main result is the significantly improved sample complexity for learning such low-degree polynomials. We show that, for any regression risk $\eps \in (0, Θ(d^{-k_0})]$, an over-parameterized two-layer neural network trained by a novel Gradient Descent with Projection (GDP) requires a sample complexity of $n \asymp Θ( \log(4/δ) \cdot d^{k_0}/\eps)$ with probability $1-δ$ for $δ\in (0,1)$, in contrast with the representative sample complexity $Θ(d^{k_0} \max\set{\eps^{-2},\log d})$. Moreover, such sample complexity is nearly unimprovable since the trained network renders a nearly optimal rate of the nonparametric regression risk of the order $\log({4}/δ) \cdot Θ(d^{k_0}/{n})$ with probability at least $1-δ$. On the other hand, the minimax optimal rate for the regression risk with a kernel of rank $Θ(d^{k_0})$ is $Θ(d^{k_0}/{n})$, so that the rate of the nonparametric regression risk of the network trained by GDP is nearly minimax optimal. In the case that the ground truth degree $k_0$ is unknown, we present a novel and provable adaptive degree selection algorithm which identifies the true degree and achieves the same nearly optimal regression rate. To the best of our knowledge, this is the first time that a nearly optimal risk bound is obtained by training an over-parameterized neural network with a popular activation function (ReLU) and algorithmic guarantee for learning low-degree spherical polynomials. Due to the feature learning capability of GDP, our results are beyond the regular Neural Tangent Kernel (NTK) limit.

In this paper, we study the problem of learning multi-dimensional Gaussian Mixture Models (GMMs), with a specific focus on model order selection and efficient mixing distribution estimation. We first establish an information-theoretic lower bound on the critical sample complexity required for reliable model selection. More specifically, we show that distinguishing a $k$-component mixture from a simpler model necessitates a sample size scaling of $Ω(Δ^{-(4k-4)})$. We then propose a thresholding-based estimation algorithm that evaluates the spectral gap of an empirical covariance matrix constructed from random Fourier measurement vectors. This parameter-free estimator operates with an efficient time complexity of $\mathcal{O}(k^2 n)$, scaling linearly with the sample size. We demonstrate that the sample complexity of our method matches the established lower bound, confirming its minimax optimality with respect to the component separation distance $Δ$. Conditioned on the estimated model order, we subsequently introduce a gradient-based minimization method for parameter estimation. To effectively navigate the non-convex objective landscape, we employ a data-driven, score-based initialization strategy that guarantees rapid convergence. We prove that this method achieves the optimal parametric convergence rate of $\mathcal{O}_p(n^{-1/2})$ for estimating the component means. To enhance the algorithm's efficiency in high-dimensional regimes where the ambient dimension exceeds the number of mixture components (i.e., \(d > k\)), we integrate principal component analysis (PCA) for dimension reduction. Numerical experiments demonstrate that our Fourier-based algorithmic framework outperforms conventional Expectation-Maximization (EM) methods in both estimation accuracy and computational time.

Several graph data mining, signal processing, and machine learning downstream tasks rely on information related to the eigenvectors of the associated adjacency or Laplacian matrix. Classical eigendecomposition methods are powerful when the matrix remains static but cannot be applied to problems where the matrix entries are updated or the number of rows and columns increases frequently. Such scenarios occur routinely in graph analytics when the graph is changing dynamically and either edges and/or nodes are being added and removed. This paper puts forth a new algorithmic framework to update the eigenvectors associated with the leading eigenvalues of an initial adjacency or Laplacian matrix as the graph evolves dynamically. The proposed algorithm is based on Rayleigh-Ritz projections, in which the original eigenvalue problem is projected onto a restricted subspace which ideally encapsulates the invariant subspace associated with the sought eigenvectors. Following ideas from eigenvector perturbation analysis, we present a new methodology to build the projection subspace. The proposed framework features lower computational and memory complexity with respect to competitive alternatives while empirical results show strong qualitative performance, both in terms of eigenvector approximation and accuracy of downstream learning tasks of central node identification and node clustering.

We present a novel deep neural network architecture for unsupervised subspace clustering. This architecture is built upon deep auto-encoders, which non-linearly map the input data into a latent space. Our key idea is to introduce a novel self-expressive layer between the encoder and the decoder to mimic the self-expressiveness property that has proven effective in traditional subspace clustering. Being differentiable, our new self-expressive layer provides a simple but effective way to learn pairwise affinities between all data points through a standard back-propagation procedure. Being nonlinear, our neural-network based method is able to cluster data points having complex (often nonlinear) structures. We further propose pre-training and fine-tuning strategies that let us effectively learn the parameters of our subspace clustering networks. Our experiments show that the proposed method significantly outperforms the state-of-the-art unsupervised subspace clustering methods.

Given samples lying on any of a number of subspaces, subspace clustering is the task of grouping the samples based on the their corresponding subspaces. Many subspace clustering methods operate by assigning a measure of affinity to each pair of points and feeding these affinities into a graph clustering algorithm. This paper proposes a new paradigm for subspace clustering that computes affinities based on the corresponding conic geometry. The proposed conic subspace clustering (CSC) approach considers the convex hull of a collection of normalized data points and the corresponding tangent cones. The union of subspaces underlying the data imposes a strong association between the tangent cone at a sample $x$ and the original subspace containing $x$. In addition to describing this novel geometric perspective, this paper provides a practical algorithm for subspace clustering that leverages this perspective, where a tangent cone membership test is used to estimate the affinities. This algorithm is accompanied with deterministic and stochastic guarantees on the properties of the learned affinity matrix, on the true and false positive rates and spread, which directly translate into the overall clustering accuracy.

In this paper, we study the mixed linear regression (MLR) problem, where the goal is to recover multiple underlying linear models from their unlabeled linear measurements. We propose a non-convex objective function which we show is {\em locally strongly convex} in the neighborhood of the ground truth. We use a tensor method for initialization so that the initial models are in the local strong convexity region. We then employ general convex optimization algorithms to minimize the objective function. To the best of our knowledge, our approach provides first exact recovery guarantees for the MLR problem with $K \geq 2$ components. Moreover, our method has near-optimal computational complexity $\tilde O (Nd)$ as well as near-optimal sample complexity $\tilde O (d)$ for {\em constant} $K$. Furthermore, we show that our non-convex formulation can be extended to solving the {\em subspace clustering} problem as well. In particular, when initialized within a small constant distance to the true subspaces, our method converges to the global optima (and recovers true subspaces) in time {\em linear} in the number of points. Furthermore, our empirical results indicate that even with random initialization, our approach converges to the global optima in linear time, providing speed-up of up to two orders of magnitude.