מושגי ליבה
This paper proposes two novel semi-supervised Fréchet regression methods, semi-supervised Nadaraya-Watson (NW) Fréchet regression and semi-supervised k-nearest neighbor (kNN) Fréchet regression, which leverage graph distances to effectively model the regression relationship between Euclidean predictors and non-Euclidean responses when limited labeled data is available.
תקציר
The paper explores the field of semi-supervised Fréchet regression, which is motivated by the significant costs associated with obtaining non-Euclidean labels. The authors propose two novel semi-supervised methods:
Semi-supervised NW Fréchet regression: This method extends the classical NW regression by replacing Euclidean distances with graph distances estimated from all feature instances, including both labeled and unlabeled data.
Semi-supervised kNN Fréchet regression: This method extends the classical kNN regression in a similar manner, using graph distances instead of Euclidean distances.
The authors establish the convergence rates of these two semi-supervised methods, showing that they can adapt to the intrinsic dimension of the low-dimensional manifold underlying the feature space, even with limited labeled data. Through comprehensive simulations and real data applications, the authors demonstrate the superior performance of their semi-supervised methods over their supervised counterparts.
The key insights are:
Leveraging unlabeled data to accurately estimate the geodesic distances on the low-dimensional manifold enables effective semi-supervised Fréchet regression.
The semi-supervised methods can achieve faster convergence rates compared to supervised methods by exploiting the manifold structure, even with a small number of labeled samples.
The semi-supervised NW Fréchet regression slightly outperforms the semi-supervised kNN Fréchet regression when the size of unlabeled data is large enough.
סטטיסטיקה
The paper does not provide any specific numerical data, but rather focuses on the theoretical analysis and simulation results.