mode detection

Subspace clustering in high-dimensional data often yields multiple semantically distinct subspaces, yet existing methods require manual specification of both the number of subspaces and the number of clusters within each—rendering them parameter-sensitive and poorly interpretable. This paper proposes an automatic, non-redundant multi-subspace clustering framework. First, it introduces the Minimum Description Length (MDL) principle to non-redundant clustering, enabling joint, adaptive inference of both the optimal number of subspaces and the cluster count per subspace. Second, it designs a split-merge-based greedy search strategy coupled with a subspace-level outlier encoding mechanism, allowing simultaneous outlier detection. Evaluated on multiple benchmark datasets, the method achieves competitive accuracy against state-of-the-art approaches while significantly improving parameter robustness, model interpretability, and practical applicability.

This paper investigates the fundamental relationship between clustering risk and classification risk under nonparametric hidden Markov models (HMMs) and i.i.d. settings. Using Bayesian decision theory and nonparametric statistical analysis, we establish—for the first time in a nonparametric HMM framework—an upper bound on the clustering excess risk of a plug-in Bayesian classifier. We further introduce a unified information-theoretic quantity that characterizes the intrinsic difficulty of both tasks. Theoretically, we prove that, under broad regularity conditions, the Bayesian classifier is nearly optimal for clustering: its clustering performance loss is both controlled and universally bounded. Extensive simulations confirm the tightness of the derived bounds and the practical efficacy of the approach. Our core contribution lies in revealing an intrinsic equivalence between classification and clustering risks, thereby providing a theoretically grounded, supervised-style solution to unsupervised clustering.

This paper investigates the fundamental limits and algorithmic design of joint inference in high-dimensional multimodal learning: given two noisy, correlated spiked data matrices, how can shared latent variables be optimally recovered? First, it rigorously characterizes the Bayesian optimal recovery threshold under generalized priors and heterogeneous noise. Second, it proves that classical methods—Partial Least Squares (PLS) and Canonical Correlation Analysis (CCA)—exhibit suboptimal phase transitions and fail to achieve this theoretical limit. Third, it proposes a joint estimation algorithm based on Approximate Message Passing (AMP), with rigorous performance characterization via state evolution analysis and numerical validation. The AMP algorithm achieves Bayesian-optimal recovery with linear time complexity, significantly outperforming PLS and CCA. Collectively, this work establishes precise statistical boundaries for multimodal fusion and provides a principled algorithmic pathway to attain them.

To address the intractability of surrogate loss optimization in large-scale multiclass classification caused by high-dimensional embeddings, this paper proposes a low-dimensional convex polyhedral embedding framework, establishing theoretical trade-offs among embedding dimension, consistency regions, and data distribution assumptions. First, it rigorously proves that “hallucination”—i.e., spurious class predictions—necessarily occurs when the embedding dimension is less than $n-1$. Second, under a low-noise assumption, it derives a verifiable consistency criterion. Third, it designs structured embeddings—including hypercubes and permutahedra—that achieve dimensionality reductions from $2^d$ to $d$ and from $d!$ to $d$, respectively. Finally, in the multiple-instance learning setting, it shows that full simplex consistency is guaranteed with only $n/2$ embedding dimensions and proves the existence of consistent subsets around any point-mass distribution.

This work addresses the high computational complexity and lack of theoretical guarantees in multivariate (M ≥ 2) joint independence testing via the Hilbert–Schmidt Independence Criterion (HSIC). We propose the first Nyström-based HSIC estimator scalable to arbitrary M ≥ 2, built upon low-rank kernel matrix approximation and multivariate tensor kernel embeddings. We establish its statistical consistency under mild regularity conditions and reduce its time complexity from O(n²) to near-linear O(nm), where m ≪ n. Unlike existing methods—limited to bivariate (M = 2) settings and lacking theoretical foundations—our approach is the first to deliver a scalable, statistically consistent, and computationally efficient estimator for multivariate HSIC. Extensive experiments on synthetic data, media annotation dependency analysis, and causal discovery tasks demonstrate its effectiveness and practical utility.

This work investigates the breakdown of Gaussian universality for high-dimensional empirical risk minimization (ERM) under non-Gaussian designs. By extending the convex Gaussian min-max theorem to non-Gaussian settings and leveraging concentration inequalities, random matrix theory, and asymptotic analysis, the authors establish an asymptotic min-max characterization of the mean and covariance of ERM estimators and uncover the structure of their predictive distributions. A central contribution is delineating the precise boundary under which Gaussian universality remains valid, and proving that, asymptotically, any twice-differentiable regularizer is equivalent to a quadratic form determined by its gradient and Hessian. Numerical experiments confirm the theoretical predictions, demonstrating accurate characterization of ERM’s statistical behavior across diverse loss functions and data models.

This work addresses the high variance in local intrinsic dimensionality (LID) estimation caused by data sparsity in small neighborhoods, which undermines estimation accuracy. To mitigate this issue, the study introduces subbagging—a variant of bootstrap aggregation—into LID estimation for the first time, constructing an ensemble method that substantially reduces estimation variance while preserving the local distribution of nearest-neighbor distances. Through a systematic analysis of the interplay among sampling rate, neighborhood size (k), and ensemble scale, combined with a neighborhood smoothing strategy, the proposed approach effectively enhances estimation precision. Experimental results demonstrate that the method consistently achieves significantly lower mean squared error across a broad range of hyperparameters, while enabling controlled bias adjustment, thereby outperforming existing state-of-the-art techniques in overall performance.

This work addresses the challenge of characterizing the distribution of critical points in the empirical risk landscape and its impact on optimization dynamics under high-dimensional Gaussian single-index models. By leveraging the Kac–Rice formula within a proportional asymptotic regime, the authors reduce the original problem to a tractable finite-dimensional scalar variational problem. This approach yields the first complete topological phase diagram of the loss landscape for phase retrieval, along with a precise characterization of a BBP-type spectral instability transition in the Hessian along the signal direction. Integrating high-dimensional asymptotic analysis, Hessian spectral theory, and gradient flow simulations, the framework accurately predicts the distribution of critical points, their joint label statistics, and the dynamical behavior of local optimization algorithms, with theoretical predictions showing excellent agreement with finite-dimensional numerical experiments.

This work addresses a fundamental limitation of the traditional Smart PAC learning framework, which fails to provide verifiable guarantees for semi-supervised learning when the marginal distributions are indistinguishable, rendering it impractical in real-world settings. To circumvent this impossibility result, the paper introduces the “Optimally-Verifiable Intelligent Learning” (OIG) framework, which relaxes the learning objective to compete only against the best verifiably optimal semi-supervised learner. Theoretical analysis demonstrates that, in a distribution-free setting, an OIG learner can achieve relative intelligence with sample complexity quadratic in the relevant parameters. Moreover, the study reveals that under certain distribution families, the task may remain infeasible or require specialized treatment, and uncovers a non-monotonic relationship between learning difficulty and the structure of these distribution families.

This work proposes the GERVE framework to directly approximate Gibbs measures from samples in scenarios where the number of modes is unknown and full density estimation is infeasible. By combining an entropy-regularized variational objective with natural gradient updates, GERVE unifies kernel density estimation, mean shift, Gaussian mixture modeling, and annealing strategies to adaptively fit high-density regions and assign cluster memberships. Theoretically, the method provides convergence guarantees for mode recovery; practically, it incorporates bootstrap-based confidence ellipses and stability scores. Experiments on both synthetic and real-world data demonstrate that GERVE accurately recovers modal structures with high precision and exhibits robustness to over-parameterization of mixture components, substantially reducing reliance on pre-specified numbers of clusters.