Search Dimension in Unlabeled Projection Pursuit: A Scaling Law for Subspace Restriction

Principal component analysis (PCA) is a fundamental tool to reduce the dimensionality of the data in many applications. PCA finds a few signal directions that contain most of the variability of the data by computing the eigenvectors of the sample covariance matrix. In this work, we focus on the spiked covariance model, in which the data vectors are defined by a few orthogonal signals plus an isotropic Gaussian noise, and our goal is to estimate one or more of the leading signals. Our main theoretical finding is that the subspace spanned by several leading eigenvectors of the sample covariance matrix contains significant information about the desired signals long before the individual eigenvectors converge to the population principal components. To prove this, we derive a posteriori bounds for the angle between the subspace spanned by the desired population signals and the subspace obtained from the sample using perturbation theory for singular vectors. This leads to a new algorithm, SuperPCA (SUbsPace subsamplER PCA), which capitalizes on an approximate eigenspace of the sample covariance matrix to find the leading signals far more efficiently and accurately than classical PCA in the high-dimensional, multi-signal setting. SuperPCA exploits only a small number of subsampled coordinates of the data, which can lead to tremendous savings in data acquisition cost, especially when the signals are approximately sparse. For the same number of measurements, SuperPCA can offer a factor $10$ improvement in accuracy compared to the classical PCA method.

This paper analyzes bidirectional random projections for ordinary least squares (OLS) regression under the fixed design setting. Let $(X,Y) \in \mathbb{R}^{n \times p} \times \mathbb{R}^n$ be a sample and $R \in \mathbb{R}^{n_1 \times n}, W \in \mathbb{R}^{p \times p_1}$ be two properly distributed random projections. We develop an expected excess loss bound for the OLS estimator built on $(WXR, WY)$. Compared to an established bound for OLS estimator built on $(XR, Y)$, the gap is approximately $O\left( p_1 + C \frac{1}{p_1} \right)$, where $C$ scales with $n_1/n$ and can be negative for small $n_1/n$. Its implications are confirmed by numerical results on real-world data.

Restricted eigenvalue (RE) bounds govern stable recovery by norm-regularized estimators. For isotropic sub-Gaussian measurements, the benchmark sample size is $1+w(A)^2$, where $w(A)$ is the Gaussian width of the normalized descent cone. The COLT 2015 open-problem note (Banerjee et al., 2015) asked whether the same law follows for heavy-tailed designs from a uniform small-ball condition alone. We give an explicit and systematic negative answer to the general question as formulated there: the proposed law fails in its full dimension-free, arbitrary-set form, and the missing obstruction is simultaneous threshold occupancy. A constant-width polyhedral descent cone with fixed small-ball constants has zero empirical RE on every sample path up to half the ambient dimension. More generally, every finite range space admits exact threshold encoding in an arbitrarily narrow spherical cap and a lift to a full polyhedral descent-cone section. For every fixed threshold VC dimension $d$, as $β\downarrow0$, the sharp worst-case sample complexity is $Θ(β^{-1}[d\log(1/β)+\log(1/δ)])$. The separation persists under exact isotropy and all finite moments: on the same constant-width cone, Gaussian measurements succeed with $O(1+\log(1/δ))$ samples, whereas an isotropic heavy-tailed design fails pathwise for $n\lesssim\sqrt{p/\log p}$. Gaussian smoothing yields an everywhere-positive $C^\infty$ density while retaining arbitrarily poor RE. Under isotropy, a distribution-free fallback governed by affine dimension times squared enclosing radius is sharp on this family.