[Paper Review] Estimation in high dimensions: a geometric perspective

Roman Vershynin|arXiv (Cornell University)|May 20, 2014

Sparse and Compressive Sensing Techniques28 references14 citations

TL;DR

This paper presents a geometric framework for high-dimensional estimation problems with constraints, using tools from asymptotic convex geometry—particularly the $M^*$ bound and mean width—to unify and generalize approaches in sparse recovery, matrix completion, and regression. The key contribution is a robust, geometry-based theory that links the complexity of feasible sets to estimation error, enabling tight bounds under sub-Gaussian and noisy observations with minimal sample requirements.

ABSTRACT

This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of asymptotic convex geometry, and demonstrates connections between geometric results and estimation problems. The theory is illustrated with applications to sparse recovery, matrix completion, quantization, linear and logistic regression and generalized linear models.

Motivation & Objective

• To develop a geometric intuition for high-dimensional sets $K$ that represent low-complexity structures in estimation problems.
• To formalize connections between asymptotic convex geometry and high-dimensional estimation via the $M^*$ bound and mean width.
• To unify diverse estimation problems—such as sparse recovery, matrix completion, and logistic regression—under a common geometric framework.
• To extend results from Gaussian to sub-Gaussian and heavy-tailed distributions, broadening applicability.
• To demonstrate how local mean width improves estimation error bounds by capturing scale-dependent complexity.

Proposed method

• The framework uses the Gaussian mean width $w(K)$ as a measure of geometric complexity of the feasible set $K$.
• It applies the $M^*$ bound to control the diameter of high-dimensional sections $K \cap E$, where $E$ is a random subspace of dimension $m$.
• The $M^*$ bound is derived via symmetrization, contraction, and rotation invariance of Gaussian processes.
• Estimation is formulated as a convex feasibility or optimization problem, minimizing over $K$ subject to linear or non-linear observations.
• For non-linear observations, metric projection and hyperplane tessellations are used to model single-bit and generalized linear models.
• Local mean width $w_r(K)$ is introduced to refine global bounds, enabling tighter error control in localized regions of $K$.

Experimental results

Research questions

• RQ1 How can geometric properties of high-dimensional sets $K$ be used to derive sample complexity bounds for estimation?
• RQ2 What role does the mean width $w(K)$ play in quantifying the complexity of feasible sets in high-dimensional estimation?
• RQ3 Can the $M^*$ bound be generalized to non-Gaussian, sub-Gaussian, and heavy-tailed observation models?
• RQ4 How does the local mean width $w_r(K)$ improve upon the global mean width in estimation error bounds?
• RQ5 To what extent can the geometric framework unify seemingly disparate problems like sparse recovery, matrix completion, and logistic regression?

Key findings

• The $M^*$ bound ensures that the diameter of $K \cap E$ is small with high probability when $m \gtrsim w(K)^2$, enabling accurate recovery from few linear observations.
• For sub-Gaussian observations, the $M^*$ bound is extended with a similar dependence on $w(K)$, maintaining sample complexity guarantees.
• In sparse recovery, $m \sim s \log(n/s)$ measurements suffice for exact recovery when $K$ is the $\ell^1$-ball of $s$-sparse vectors.
• For low-rank matrix recovery, $m \sim r(d_1 + d_2)$ non-linear observations suffice, even with unknown non-linearities, via singular value thresholding.
• The local mean width $w_r(K)$ allows the $M^*$ bound to be localized, improving error bounds by capturing scale-dependent structure in $K$.
• The framework achieves optimal error rates in metric projection and generalized linear models by leveraging the local mean width as a proxy for complexity.