On the Spectral Structure and Objective Equivalence of Orthogonal Multilabel Fisher Discriminants

• URL: http://arxiv.org/abs/2605.03283v1
• Date: Tue, 05 May 2026 02:14:33 GMT
• Title: On the Spectral Structure and Objective Equivalence of Orthogonal Multilabel Fisher Discriminants
• Abstract summary: We provide a unified theoretical analysis of Linear Discriminant Analysis with simultaneous multilabel scatter matrix formulations.<n>Our contributions span both algebraic structure and statistical guarantees.
• Score: 0.22917707112773592
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We provide a unified theoretical analysis of Linear Discriminant Analysis with simultaneous multilabel scatter matrix formulations and Stiefel orthogonality constraints. Our contributions span both algebraic structure and statistical guarantees. On the algebraic side, we characterize the rank of the multilabel between-class scatter matrix, showing that the effective discriminant dimensionality can strictly exceed the classical single-label bound of $C-1$; we establish a multilabel partition of variance and prove that all four Fisher objectives are equivalent under the $W^\top S_t^{ML} W = I_r$ constraint while characterizing their divergence under the Stiefel constraint; and we prove a two-sided label-distance preservation bound relating projected distances to Hamming distances in label space. On the statistical side, we establish a finite-sample $O(k_{\max}\sqrt{d\log d/n}/gap_r)$ bound on the subspace estimation error under sub-Gaussian noise with a matching $Ω(σ^2 d/(n\,gap_r))$ minimax lower bound, establishing a near-minimax-optimal rate (matching up to logarithmic and $k_{\max}$ factors) for multilabel discriminant subspace estimation. We further provide high-probability distance concentration, robustness guarantees under label interactions, and a regularization analysis preserving the spectral structure when $d \gg n$. All results are verified numerically on synthetic data generated from the linear label-effect model, covering both the algebraic identities and the multilabel-specific quantities ($k_{\max}$, $κ(S_t^{ML})$, $\|Γ/n\|_2$, $Δ_r$) that govern the statistical bounds. The numerical experiments are designed as a sanity check for the theorems rather than as an empirical benchmark; evaluation on real multilabel datasets is left to future work targeting application-oriented venues.

Related papers

• Is Spurious Correlation Removal Always Learnable? [56.28155520961125]
  Invariant learning can fail even when the structure is statistically identifiable.<n>We show a conditional computational barrier: under a black-box samplable supervised sparse recovery primitive, there exist emphamplable multi-dimensional environments.<n>Synthetic and real datasets illustrate the predicted gaps and transitions and motivate simple diversity diagnostics.
  arXiv  Detail & Related papers  (2026-06-11T05:49:43Z)
• Reflected diffusion models adapt to low-dimensional data [1.4273866043218157]
  This paper provides a statistical analysis of reflected diffusion models on the hypercube $[0,1]D$ supported on $d$-dimensional linear subspaces.<n>By employing an easily implementable infinite series expansion of the transition densities, we develop analytic tools to bound the score function and its approximation by sparse ReLU networks.
  arXiv  Detail & Related papers  (2026-03-25T16:37:33Z)
• Minimax Rates for the Estimation of Eigenpairs of Weighted Laplace-Beltrami Operators on Manifolds [7.639886528552829]
  We study the problem of estimating eigenpairs of elliptic differential operators from samples of a distribution $rho$ supported on a manifold $M$.<n>We find that eigenpairs of graph Laplacians induce regularity manifold estimators with an error of approximation that, up to logarithmic corrections, matches our lower bounds.
  arXiv  Detail & Related papers  (2025-05-30T19:19:25Z)
• Convergence Analysis of Probability Flow ODE for Score-based Generative Models [5.939858158928473]
  We study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives.<n>We prove the total variation between the target and the generated data distributions can be bounded above by $mathcalO(d3/4delta1/2)$ in the continuous time level.
  arXiv  Detail & Related papers  (2024-04-15T12:29:28Z)
• Semidefinite programming relaxations and debiasing for MAXCUT-based clustering [1.9761774213809036]
  We consider the problem of partitioning a small data sample of size $n$ drawn from a mixture of 2 sub-gaussian distributions in $mathbbRp$.<n>We use semidefinite programming relaxations of an integer quadratic program that is formulated as finding the maximum cut on a graph.
  arXiv  Detail & Related papers  (2024-01-16T03:14:24Z)
• Tackling Combinatorial Distribution Shift: A Matrix Completion Perspective [42.85196869759168]
  We study a setting we call distribution shift, where (a) under the test- and training-random data, the labels $z$ are determined by pairs of features $(x,y)$, (b) the training distribution has coverage of certain marginal distributions over $x$ and $y$ separately, but (c) the test distribution involves examples from a product distribution over $(x,y)$ that is not covered by the training distribution.
  arXiv  Detail & Related papers  (2023-07-12T21:17:47Z)
• How Does Pseudo-Labeling Affect the Generalization Error of the Semi-Supervised Gibbs Algorithm? [73.80001705134147]
  We provide an exact characterization of the expected generalization error (gen-error) for semi-supervised learning (SSL) with pseudo-labeling via the Gibbs algorithm. The gen-error is expressed in terms of the symmetrized KL information between the output hypothesis, the pseudo-labeled dataset, and the labeled dataset.
  arXiv  Detail & Related papers  (2022-10-15T04:11:56Z)
• Subspace clustering in high-dimensions: Phase transitions \& Statistical-to-Computational gap [24.073221004661427]
  A simple model to study subspace clustering is the high-dimensional $k$-Gaussian mixture model. We provide an exact characterization of the statistically optimal reconstruction error in this model in the high-dimensional regime with extensive sparsity.
  arXiv  Detail & Related papers  (2022-05-26T17:47:35Z)
• Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
  We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
  arXiv  Detail & Related papers  (2021-09-24T14:48:20Z)
• A Unified Joint Maximum Mean Discrepancy for Domain Adaptation [73.44809425486767]
  This paper theoretically derives a unified form of JMMD that is easy to optimize. From the revealed unified JMMD, we illustrate that JMMD degrades the feature-label dependence that benefits to classification. We propose a novel MMD matrix to promote the dependence, and devise a novel label kernel that is robust to label distribution shift.
  arXiv  Detail & Related papers  (2021-01-25T09:46:14Z)
• Consistent Structured Prediction with Max-Min Margin Markov Networks [84.60515484036239]
  Max-margin methods for binary classification have been extended to the structured prediction setting under the name of max-margin Markov networks ($M3N$) We overcome such limitations by defining the learning problem in terms of a "max-min" margin formulation, naming the resulting method max-min margin Markov networks ($M4N$) Experiments on multi-class classification, ordinal regression, sequence prediction and ranking demonstrate the effectiveness of the proposed method.
  arXiv  Detail & Related papers  (2020-07-02T10:48:42Z)
• Self-training Avoids Using Spurious Features Under Domain Shift [54.794607791641745]
  In unsupervised domain adaptation, conditional entropy minimization and pseudo-labeling work even when the domain shifts are much larger than those analyzed by existing theory. We identify and analyze one particular setting where the domain shift can be large, but certain spurious features correlate with label in the source domain but are independent label in the target.
  arXiv  Detail & Related papers  (2020-06-17T17:51:42Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.