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.

In this section we briefly introduce the representation theory of the three groups we used in this work. Planar rotations group SO(2) The standard representation of r 2 SO(2) is as a 2 2 rotation matrix (r)= cos sin sin cos The complex irreducible representations are often used and correspond to the circular harmonics. Planar rotations and reflections group O(2) The standard representation of O(2) is as a 2 2 orthogonal matrix (r)= cos sin sin cos and (r f)= cos sin sin cos 10 01 Apart from the trivial representation 0,0(h)=1 8h 2 O(2) and the sign-flip representation 1,0(r)=1 and 1,0(f)= 1, all other irreps are 2 dimensional. These representations are isomorphic to the Wigner D matrices. In particular, 0 is the trivial representation and i is isomorphic to the standard representation of SO(3) as 3 3 rotation matrices. An element g =( m,r) 2 O(3) is a pair of a mirroring m 2{ e,mz} and a rotation r 2 SO(3). In general, if G is a group, we denote with bG the set of its irreducible representations. Recall the generative process for cryo-EM images: oi = (g 1i) with gi 2 SO(3) (12) 14 Let Rz = SO(2) < SO(3) the subgroup of SO(3) containing rotations around the Z axis and H = O(2) < SO(3) the subgroup containing also the rotation ry by around the Y axis.

In many applications, we desire neural networks to exhibit invariance or equivari-ance to certain groups due to symmetries inherent in the data. Recently, frame-averaging methods emerged to be a unified framework for attaining symmetries efficiently by averaging over input-dependent subsets of the group, i.e., frames. What we currently lack is a principled understanding of the design of frames.

However,thisoperation can greatly impair the performance of stochastic methods, e.g.

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