Estimating the Spectral Moments of the Kernel Integral Operator from Finite Sample Matrices (2410.17998v3)

Abstract: Analyzing the structure of sampled features from an input data distribution is challenging when constrained by limited measurements in both the number of inputs and features. Traditional approaches often rely on the eigenvalue spectrum of the sample covariance matrix derived from finite measurement matrices; however, these spectra are sensitive to the size of the measurement matrix, leading to biased insights. In this paper, we introduce a novel algorithm that provides unbiased estimates of the spectral moments of the kernel integral operator in the limit of infinite inputs and features from finitely sampled measurement matrices. Our method, based on dynamic programming, is efficient and capable of estimating the moments of the operator spectrum. We demonstrate the accuracy of our estimator on radial basis function (RBF) kernels, highlighting its consistency with the theoretical spectra. Furthermore, we showcase the practical utility and robustness of our method in understanding the geometry of learned representations in neural networks.

• The paper introduces a dynamic programming algorithm that provides unbiased spectral moment estimates from finite sample matrices.
• It mitigates biases inherent in traditional eigenvalue methods, ensuring more reliable statistical inference in high-dimensional neural network analysis.
• The method is computationally efficient, aligns with theoretical spectra, and maintains robustness even in noisy, correlated measurement scenarios.

Estimating the Spectral Moments of the Kernel Integral Operator from Finite Sample Matrices

This paper introduces a novel approach for estimating the spectral moments of the kernel integral operator from finite sample matrices, addressing challenges in statistical inference when constrained by limited data. Traditional methods relying on eigenvalue spectra of sample covariance matrices often yield biased insights, especially when both inputs and features are finitely sampled. This research presents a computationally efficient algorithm leveraging dynamic programming to provide unbiased estimates of the spectral moments, offering a substantial enhancement in understanding the geometry of neural network representations.

Core Contributions

At the heart of this work is the estimation of spectral properties of kernel integral operators, which describe the expected covariance from infinite samples of inputs and features. The proposed estimator accounts for the dynamic nature of high-dimensional data distributions, ensuring accurate inference from finite samples. The approach centers on:

• Spectral Moments Unbiased Estimation: The paper proposes a dynamic programming algorithm that computes unbiased estimates of spectral moments. This is crucial in counteracting the biases introduced by conventional methods when the sample sizes of inputs and features are finite.
• Algorithmic Efficiency: By averaging over non-repeating cycles in the measurement matrix, the algorithm achieves polynomial computational efficiency, significantly reducing the complexity intrinsic to higher-order moment calculations.
• Theoretical Consistency: Demonstrated through the radial basis function (RBF) kernels, the proposed estimator aligns closely with theoretical spectra, proving its validity and robustness.

Technical Insights

The authors formulate an unbiased estimator by constructing cyclic paths over matrix entries, ensuring these paths do not revisit any input or feature index more than twice. This recursive procedure systematically aggregates these paths, enabling the estimation of spectral moments. Additionally, an intriguing aspect of the proposed work is its handling of noise, where the estimator remains unbiased even with noisy, correlated measurements.

In comparison to related work, particularly the approach by Kong and Valiant for fully observed features, this paper's method excels by resolving the biases present when both rows and columns are sampled. The application of random matrix theory elucidates how the naive estimators converge to the Marchenko-Pastur distribution under i.i.d. Gaussian kernels, reaffirming the systemic biases in conventional techniques.

Practical and Theoretical Implications

This advancement holds significant potential in AI development, particularly in the field of neural networks. By utilizing accurate spectral moment estimates, researchers can better understand neural network dynamics, feature learning processes, and potentially enhance generalization capabilities. The paper demonstrates this application by analyzing ReLU neural networks, illustrating the consistency of kernel operators across varying neural network widths—a critical criterion for model scalability and robustness.

Future Directions

The implications of this research extend to optimizing kernel-based learning algorithms and exploring advanced model performance metrics. Future studies might leverage this method to refine feature extraction processes, enhance computational efficiency in large-scale models, or further explore the learning dynamics observable in complex neural architectures. Moreover, extending this methodology to broader kernel types and input distributions could yield insights into universal patterns in statistical learning frameworks.

In summary, this paper advances the computational methodology for estimating spectral moments, addressing a gap in accurate inference from finite data matrices. Through its innovative approach, it lays foundational work for future research in machine learning, particularly in understanding high-dimensional data dynamics and enhancing neural network architectures.