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.

Approximating invariant subspaces of generalized eigenvalue problems (GEPs) is a fundamental computational problem at the core of machine learning and scientific computing. It is, for example, the root of Principal Component Analysis (PCA) for dimensionality reduction, data visualization, and noise filtering, and of Density Functional Theory (DFT), arguably the most popular method to calculate the electronic structure of materials. Given Hermitian $H,S\in\mathbb{C}^{n\times n}$, where $S$ is positive-definite, let $\Pi_k$ be the true spectral projector on the invariant subspace that is associated with the $k$ smallest (or largest) eigenvalues of the GEP $HC=SC\Lambda$, for some $k\in[n]$.

Solving large-scale sparse linear systems is essential in fields like mathematics, science, and engineering.

Fundamental limits to predictability are central to our understanding of many physical and computational systems. Here we show that, despite its remarkable capabilities, deep learning exhibits such fundamental limits rooted in the fractal, riddled geometry of its basins of attraction: any initialization that leads to one solution lies arbitrarily close to another that leads to a different one. We derive sufficient conditions for the emergence of riddled basins by analytically linking features widely observed in deep learning, including chaotic learning dynamics and symmetry-induced invariant subspaces, to reveal a general route to riddling in realistic deep networks. The resulting basins of attraction possess an infinitely fine-scale fractal structure characterized by an uncertainty exponent near zero, so that even large increases in the precision of initial conditions yield only marginal gains in outcome predictability. Riddling thus imposes a fundamental limit on the predictability and hence reproducibility of neural network training, providing a unified account of many empirical observations. These results reveal a general organizing principle of deep learning with important implications for optimization and the safe deployment of artificial intelligence.

In this section we briefly introduce the representation theory of the three groups we used in this work. The complex irreducible representations are often used and correspond to the circular harmonics. The parallel transport operator transports vector fields defined over a space. These invariant subspaces can be identified as follows. SO(2) ambiguity we introduced in Section 2 lies.

In this section, we provide the proofs of the propositions stated in the main text. However, if an'inconsistent' decoder-encoder pair would be used, an encoder with a perturbed mean In the PCA case, the invariant subspace is explicitly known thanks to the linearity. "autoencoding" requires that realizations generated by the decoder are approximately invariant when The algorithm is shown in Algorithm 1. While SE introduced an'external selection mechanism' to generate adversarial examples, the analysis in this appendix shows that the approach could be viewed as a robust Bayesian We can employ a robust Bayesian approach to define a'pessimistic' bound in the sense of selecting With the given tighter bound the algorithm for SE is shown in Algorithm 2. From Equation 18 we This algorithm can be used for post training an already trained V AE. Figure 6 shows the graphical The algorithm is shown in Algorithm 4. We approximate the required expectations by their Monte C.5 SE-A V AE Figure 7 shows the graphical model describing A V AE-SS model. The algorithm is shown in Algorithm 3. We approximate the required expectations by their Monte In this example Section 3.1, we will assume that both the observations Convolutional architectures are stabilized using BatchNorm between each convolutional layer.