How close is the sample covariance matrix to the actual covariance matrix?

Abstract: Given a probability distribution in R^n with general (non-white) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent points. What is the optimal sample size N = N(n) that guarantees estimation with a fixed accuracy in the operator norm? Suppose the distribution is supported in a centered Euclidean ball of radius \sqrt{n}. We conjecture that the optimal sample size is N = O(n) for all distributions with finite fourth moment, and we prove this up to an iterated logarithmic factor. This problem is motivated by the optimal theorem of Rudelson which states that N = O(n \log n) for distributions with finite second moment, and a recent result of Adamczak, Litvak, Pajor and Tomczak-Jaegermann which guarantees that N = O(n) for sub-exponential distributions.

• The paper proves that for distributions with finite fourth moments, a sample size of O(n log log n) suffices to approximate the true covariance matrix with high probability.
• It extends classical sub-Gaussian results to sub-exponential and finite moment cases using novel decoupling and structure theorems for random matrices.
• The insights offer practical guidance for high-dimensional data analysis, impacting fields like machine learning, signal processing, and statistical learning theory.

Overview of "Approximating Covariance Matrices" by Roman Vershynin

The paper "Approximating Covariance Matrices" by Roman Vershynin tackles a fundamental problem in multivariate statistics, namely the estimation of covariance matrices of high-dimensional distributions. This problem is extensive in its applications, spanning fields like signal processing, genomics, financial mathematics, pattern recognition, and more. A classical approach to estimate the covariance matrix of a random vector is using the sample covariance matrix derived from a set of independent samples of the vector. However, the paper seeks to address a more specific question: What is the minimal sample size $N$ required for the sample covariance matrix to approximate, with high probability and given accuracy $\epsilon$, the actual covariance matrix in the operator norm?

Main Contributions

1. Conjecture and Theorem: The author conjectures that for all distributions with finite fourth moments, the optimal sample size required for approximation is $N = O(n)$. He provides a rigorous proof supporting this conjecture, albeit up to an iterated logarithmic factor, assuming $N = O(n \log \log n)$.
2. General Framework: Vershynin extends the understanding of sample covariance matrix approximations from sub-Gaussian distributions, where the optimal sample size is known to be linear in dimension, to sub-exponential and more generally, distributions with finite moments.
3. Methodology: Utilizing known results for sub-Gaussian and sub-exponential distributions, the paper orchestrates a framework combining structure theorems for divergent series and decoupling principles to identify necessary conditions for controls over norms of random matrices. This involves handling almost pairwise orthogonality of vectors and introducing novel decoupling mechanisms for vectors that satisfy specific weak orthogonality conditions.
4. Robustness Across Distribution Classes: The paper explores distributions with finite moments, elucidating that logarithmic oversampling is unavoidable for certain types. The study suggests this oversampling is unnecessary for distributions with appropriate q-th moments, proposing q = 4 as potentially sufficient.
5. Experimental Validation: Crucial results show the approximation can hold uniformly across independent, non-identically distributed vectors, providing broader applicability and theoretical grounding for practical estimation scenarios.

Implications

The findings in this paper offer significant theoretical advancements in understanding dimensionality's role in covariance matrix estimation. These insights are influential for theoretical developments in probability theory and statistical learning theory. Practically, the implications reach machine learning domains, where efficient high-dimensional data handling is paramount, particularly as models scale with larger data indices.

Future Directions

Several avenues for future exploration stem from this work. Firstly, refining bounds for different types of distributions may enhance understanding and potential applications. Secondly, exploring the practical algorithms that incorporate these theoretical bounds can bridge the gap between theory and application in fields like big data analytics and AI, where efficient estimation of covariance is critical.

In summary, Vershynin’s paper offers a comprehensive theoretical foundation for approximating covariance matrices, expanding the boundaries of current statistical methodologies for high-dimensional data. While bridging theory and practice remains a challenge, the paper provides clear steps and direction for continued research in this crucial domain.