Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications (0807.3590v1)

Abstract: Let $A$ be an $n$ by $N$ real valued random matrix, and $\h$ denote the $N$-dimensional hypercube. For numerous random matrix ensembles, the expected number of $k$-dimensional faces of the random $n$-dimensional zonotope $A\h$ obeys the formula $E f_k(A\h) /f_k(\h) = 1-P_{N-n,N-k}$, where $P_{N-n,N-k}$ is a fair-coin-tossing probability. The formula applies, for example, where the columns of $A$ are drawn i.i.d. from an absolutely continuous symmetric distribution. The formula exploits Wendel's Theorem\cite{We62}. Let $\po$ denote the positive orthant; the expected number of $k$-faces of the random cone$A \po$ obeys $ {\cal E} f_k(A\po) /f_k(\po) = 1 - P_{N-n,N-k}$. The formula applies to numerous matrix ensembles, including those with iid random columns from an absolutely continuous, centrally symmetric distribution. There is an asymptotically sharp threshold in the behavior of face counts of the projected hypercube; thresholds known for projecting the simplex and the cross-polytope, occur at very different locations. We briefly consider face counts of the projected orthant when $A$ does not have mean zero; these do behave similarly to those for the projected simplex. We consider non-random projectors of the orthant; the 'best possible' $A$ is the one associated with the first $n$ rows of the Fourier matrix. These geometric face-counting results have implications for signal processing, information theory, inverse problems, and optimization. Most of these flow in some way from the fact that face counting is related to conditions for uniqueness of solutions of underdetermined systems of linear equations.

• The paper establishes exact thresholds for preserving the face count of projected hypercubes using probabilistic methods.
• It applies Wendel’s Theorem to connect geometric probability with the analysis of zonotopes formed by random projections.
• The findings clarify conditions for unique solutions in underdetermined systems, significantly aiding compressed sensing applications.

Counting the Faces of Randomly-Projected Hypercubes and Orthants: An Analytical Examination

This paper, authored by David L. Donoho and Jared Tanner, explores the intricate behavior of randomly-projected polytopes, specifically focusing on hypercubes and orthants, and their applications across several mathematical and engineering fields. The primary goal is to quantify and understand the dimensionality and face structure of these projections through the applications of probabilistic methods and geometric probability, with implications spanning signal processing and information theory.

The authors offer a robust framework based on probabilistic analysis to derive exact expressions for the expected number of faces of the projected zonotope, utilizing a fascinating connection to Wendel's Theorem in geometric probability. This theorem determines the probability that a collection of points in $\mathbb{R}^m$ all lie in a half-space of the dimension, which further facilitates the counting of faces in the zonotope formed from a hypercube projection.

Numerical Results and Theoretical Implications

Donoho and Tanner establish a set of clear thresholds that determine when the number of faces of the randomly-projected hypercube $AH^N$ remains close to the original hypercube $H^N$. The introduction of proportionally dimensional scaling, denoted as $k_n/n$ and $n/N_n$, introduces a sophisticated lens through which to analyze such projections, setting thresholds $\rho_W(\delta,H^N)$ and $\rho_S(\delta,)$ indicating distinct regimes of face-count preservation amidst random projections.

The strong numerical results in the paper demonstrate that, under specific conditions, such as the proportionally small dimension $n$ compared to the full dimension $N$, the randomly-projected zonotope has statistically fewer faces than the originating hypercube. This is primarily due to a sharp transition behavior in face counts that the authors document comprehensively for varying levels of dimensional perturbations. Notably, it is articulated that whenever the subdimension is less than half of the full dimension, the number of faces drastically reduces, underscoring a fundamental structural shift in the zonotope's geometry under projection.

Impact and Future Directions

These strong analytical results bear extensive implications for signal processing and optimization, especially in scenarios dealing with underdetermined systems of linear equations—a typical situation in compressed sensing. The projections' face counting relates directly to uniqueness conditions that govern such systems’ solutions, which can, in applications, simplify reconstruction problems massively when dealing with sparsity or bounded constraints.

Potential future developments emerging from this research include further exploration of random matrix ensembles beyond the commonly used Gaussian matrices, to evaluate the broader applicability of the derived thresholds. Additionally, examining the implications of random projections in higher-dimensional data analysis fields, such as machine learning and data science, where understanding the intrinsic geometry of data manifolds plays a critical role, stands as an intriguing avenue for these mathematical frameworks.

In summary, Donoho and Tanner's paper opens pathways to better comprehend the geometric intricacies involved in high-dimensional data projections, offering insights that elegantly bridge fundamental mathematical theory with pressing practical applications. The methodologies and results encapsulated here serve as a foundation upon which further exploration into the structural dynamics of random projections can be robustly anchored.