Harmless interpolation of noisy data in regression (1903.09139v2)

Abstract: A continuing mystery in understanding the empirical success of deep neural networks is their ability to achieve zero training error and generalize well, even when the training data is noisy and there are more parameters than data points. We investigate this overparameterized regime in linear regression, where all solutions that minimize training error interpolate the data, including noise. We characterize the fundamental generalization (mean-squared) error of any interpolating solution in the presence of noise, and show that this error decays to zero with the number of features. Thus, overparameterization can be explicitly beneficial in ensuring harmless interpolation of noise. We discuss two root causes for poor generalization that are complementary in nature -- signal "bleeding" into a large number of alias features, and overfitting of noise by parsimonious feature selectors. For the sparse linear model with noise, we provide a hybrid interpolating scheme that mitigates both these issues and achieves order-optimal MSE over all possible interpolating solutions.

• The paper presents a novel analysis of OMP’s noise-fitting behavior compared to l2-norm minimization in high-dimensional regression.
• It leverages tight equiangular frames and an orthonormal polynomial basis to establish theoretical bounds on generalization error.
• The study outlines practical implications for sparse recovery and encourages adaptive methods to enhance robust feature selection.

An Analysis of Orthogonal Matching Pursuit's Generalization and Noise Fitting Capability

The research paper authored by Vidya Muthukumar, Kailas Vodrahalli, and Anant Sahai explores the intriguing topic of the noise-fitting capabilities of Orthogonal Matching Pursuit (OMP) when compared to solutions that minimize the $\ell_2$-norm. The examination is firmly rooted in high-dimensional statistics and sparse approximation theory, focusing on the generalization error introduced by incorporating noise into data modeling.

Error Representation in Fitting Noise

The paper initially establishes a framework for assessing the error incurred when OMP fits noise. It presents a comparative analysis to the minimal $\ell_2$-norm solution, highlighting the scenario where orthonormal columns are considered within a given deterministic matrix. The error in this context is characterized mathematically by:

$(_{\mathsf{OMP}) = W^\top (A(S) A(S)^\top)^{-1} W$

The inequality constraint is given by the term:

$\leq \frac{#2{W}{2}^2}{\lambda_{min}(A(S)A(S)^\top)}$

This exposition provides a fundamental understanding of how OMP may inadvertently fit noise, thereby elucidating on its variations in performance based on the characteristic properties of the design matrices used.

Equiangular Frames and Noise Fitting

The document further explores the use of tight equiangular frames to understand the test error properties of OMP. Tight equiangular frames, when achievable, provide a structured set of vectors that optimize intra-vector coherence, thereby maintaining a consistency in the sparse solutions chosen by OMP. Notably, the minimum eigenvalue of an appropriate matrix $A(S)^\top A(S)$ is manipulated for advantageous bounds, deriving conditions under which the excess test error maintains a linear relationship proportional to noise variance.

Analysis with Orthonormal Polynomial Basis

The paper extends its analysis to another structured setup using an orthonormal polynomial basis. It is suggested that the incoherence properties intrinsic to such a basis can serve as a protective measure against poor feature selection by OMP, ensuring that the chosen sparse representation remains effective and less susceptible to noise.

Random Gaussian Design Implications

Finally, the implications of using a random Gaussian design are scrutinized. The inherent incoherence of such randomly generated matrices does not suffice for the naive application of the previously discussed bounds due to potential spikes in error. Here, the paper calls for more sophisticated methodologies and suggests that the intuitive understanding of incoherence among selected columns might drive future analytical approaches to mitigate noise fitting further.

Theoretical and Practical Implications

This investigation into the OMP algorithm provides key insights into its ability to generalize while simultaneously capturing noise. Though theoretically grounded in the paper of signal processing and statistics, the conclusions drawn have practical applications in environments where sparse signal representation is desired alongside robust generalization, such as in machine learning model selection, compressed sensing, and algorithmic feature selection.

Future Directions

Given the rigorous nature of this work, future research could benefit from exploring adaptive or hybrid methodologies that combine the interpretational clarity of OMP with other regularization techniques to better manage noise. Further empirical validation in high-dimensional settings could establish more stable operational bounds for various classes of randomly generated or structured design matrices. Such advancements could support improved sparse recovery performance, especially in increasingly complex datasets typical in modern applications.