IAN: Iterated Adaptive Neighborhoods for manifold learning and dimensionality estimation
This addresses the challenge of adapting neighborhoods to local structure in manifold learning for applications with limited or non-uniform data, though it appears incremental as it builds on existing graph-based methods.
The paper tackles the problem of inferring adaptive neighborhoods for manifold learning when data are limited and manifold properties are unknown, introducing an algorithm that sparsifies a Gabriel graph iteratively to compute minimal neighborhoods and detect outliers. It applies this to dimensionality reduction, geodesic computation, and dimension estimation, showing usefulness compared to standard methods like k-nearest neighbors.
Invoking the manifold assumption in machine learning requires knowledge of the manifold's geometry and dimension, and theory dictates how many samples are required. However, in applications data are limited, sampling may not be uniform, and manifold properties are unknown and (possibly) non-pure; this implies that neighborhoods must adapt to the local structure. We introduce an algorithm for inferring adaptive neighborhoods for data given by a similarity kernel. Starting with a locally-conservative neighborhood (Gabriel) graph, we sparsify it iteratively according to a weighted counterpart. In each step, a linear program yields minimal neighborhoods globally and a volumetric statistic reveals neighbor outliers likely to violate manifold geometry. We apply our adaptive neighborhoods to non-linear dimensionality reduction, geodesic computation and dimension estimation. A comparison against standard algorithms using, e.g., k-nearest neighbors, demonstrates their usefulness. Code for our algorithm will be available at https://github.com/dyballa/IAN